// Numbas version: exam_results_page_options
{"showstudentname": true, "duration": 0, "name": "Trigonometric equations", "showQuestionGroupNames": false, "navigation": {"showresultspage": "oncompletion", "browse": true, "onleave": {"message": "", "action": "none"}, "reverse": true, "allowregen": true, "showfrontpage": true, "preventleave": true}, "feedback": {"allowrevealanswer": true, "showtotalmark": true, "intro": "", "feedbackmessages": [], "showactualmark": true, "showanswerstate": true, "advicethreshold": 0}, "metadata": {"description": "<p>A variety of trigonometric equations which can be solved using inverse operations.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "Simon's copy of Trigonometric Equations 1 - Simple (Degrees)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "variable_groups": [{"variables": ["a", "f1", "s11", "s12", "s13", "s14"], "name": "Q1"}, {"variables": ["b", "c", "f2", "s22", "s21", "s23", "s24"], "name": "Q2"}, {"variables": ["f3", "s31", "s32", "s33", "s34"], "name": "Q3"}, {"variables": ["f4", "s4", "s41", "s42", "s43", "s44"], "name": "Q4"}], "variables": {"s4": {"group": "Q4", "description": "", "definition": "round(arcsin(f4)*180/pi)", "templateType": "anything", "name": "s4"}, "s11": {"group": "Q1", "description": "", "definition": "round(arcsin(f1)*180/pi)", "templateType": "anything", "name": "s11"}, "f1": {"group": "Q1", "description": "", "definition": "(a-1)/a", "templateType": "anything", "name": "f1"}, "s21": {"group": "Q2", "description": "", "definition": "round(arccos(f2)*180/pi)", "templateType": "anything", "name": "s21"}, "s24": {"group": "Q2", "description": "", "definition": "720-s21", "templateType": "anything", "name": "s24"}, "s33": {"group": "Q3", "description": "", "definition": "360+s31", "templateType": "anything", "name": "s33"}, "b": {"group": "Q2", "description": "", "definition": "random(5..12)", "templateType": "anything", "name": "b"}, "s44": {"group": "Q4", "description": "", "definition": "720+s4", "templateType": "anything", "name": "s44"}, "s42": {"group": "Q4", "description": "", "definition": "360+s4", "templateType": "anything", "name": "s42"}, "s13": {"group": "Q1", "description": "", "definition": "360+s11", "templateType": "anything", "name": "s13"}, "c": {"group": "Q2", "description": "", "definition": "random(2..b-1)", "templateType": "anything", "name": "c"}, "f3": {"group": "Q3", "description": "", "definition": "(1-b)/(b+1)", "templateType": "anything", "name": "f3"}, "s31": {"group": "Q3", "description": "", "definition": "round(arccos(f3)*180/pi)", "templateType": "anything", "name": "s31"}, "s23": {"group": "Q2", "description": "", "definition": "360+s21", "templateType": "anything", "name": "s23"}, "s14": {"group": "Q1", "description": "", "definition": "540-s11", "templateType": "anything", "name": "s14"}, "a": {"group": "Q1", "description": "", "definition": "random(3..9)", "templateType": "anything", "name": "a"}, "f2": {"group": "Q2", "description": "", "definition": "c/b", "templateType": "anything", "name": "f2"}, "s34": {"group": "Q3", "description": "", "definition": "720-s31", "templateType": "anything", "name": "s34"}, "s32": {"group": "Q3", "description": "", "definition": "360-s31", "templateType": "anything", "name": "s32"}, "s22": {"group": "Q2", "description": "", "definition": "360-s21", "templateType": "anything", "name": "s22"}, "s41": {"group": "Q4", "description": "", "definition": "180-s4", "templateType": "anything", "name": "s41"}, "s12": {"group": "Q1", "description": "", "definition": "180-s11", "templateType": "anything", "name": "s12"}, "f4": {"group": "Q4", "description": "", "definition": "-c/(c+3)", "templateType": "anything", "name": "f4"}, "s43": {"group": "Q4", "description": "", "definition": "540-s4", "templateType": "anything", "name": "s43"}}, "preamble": {"js": "", "css": ""}, "tags": [], "rulesets": {}, "statement": "<p><strong>Solve the following equations in <em>degrees</em>, for $x$ in the range $0^\\circ\\leq x\\leq720^\\circ$.</strong></p>\n<p>Give your answers correct to the <strong>nearest degree</strong> in <strong>ascending order</strong>.</p>", "ungrouped_variables": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"type": "gapfill", "gaps": [{"allowFractions": true, "type": "numberentry", "maxValue": "{f1}+0.01", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{f1}-0.01", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": true}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s11}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s11}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s12}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s12}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s13}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s13}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s14}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s14}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}], "sortAnswers": false, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "<p>\\[\\var{a}\\sin(x)=\\var{a-1}\\]</p>\n<p>$\\sin(x)=$&nbsp;[[0]]</p>\n<p>$x=$&nbsp;[[1]]$^\\circ$,&nbsp;[[2]]$^\\circ$,&nbsp;[[3]]$^\\circ$ or&nbsp;[[4]]$^\\circ$</p>", "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": []}, {"type": "gapfill", "gaps": [{"allowFractions": true, "type": "numberentry", "maxValue": "{f2}+0.01", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{f2}-0.01", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": true}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s21}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s21}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s22}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s22}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s23}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s23}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s24}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s24}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}], "sortAnswers": false, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "<p>\\[\\var{b}\\cos(x)=\\var{c}\\]</p>\n<p>$\\cos(x)=$ [[0]]</p>\n<p>$x=$&nbsp;[[1]]$^\\circ$,&nbsp;[[2]]$^\\circ$,&nbsp;[[3]]$^\\circ$ or&nbsp;[[4]]$^\\circ$</p>", "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": []}, {"type": "gapfill", "gaps": [{"allowFractions": true, "type": "numberentry", "maxValue": "{f3}+0.01", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{f3}-0.01", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": true}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s31}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s31}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s32}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s32}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s33}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s33}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s34}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s34}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}], "sortAnswers": false, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "<p>\\[\\var{b+1}\\cos(x)=-\\var{b-1}\\]</p>\n<p>$\\cos(x)=$&nbsp;[[0]]</p>\n<p>$x=$&nbsp;[[1]]$^\\circ$,&nbsp;[[2]]$^\\circ$,&nbsp;[[3]]$^\\circ$ or&nbsp;[[4]]$^\\circ$</p>", "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": []}, {"type": "gapfill", "gaps": [{"allowFractions": true, "type": "numberentry", "maxValue": "{f4}+0.01", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{f4}-0.01", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": true}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s41}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s41}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s42}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s42}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s43}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s43}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}, {"allowFractions": false, "type": "numberentry", "maxValue": "{s44}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 1, "minValue": "{s44}", "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "scripts": {}, "mustBeReduced": false, "correctAnswerFraction": false}], "sortAnswers": false, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "<p>\\[\\var{c+3}\\sin(x)=-\\var{c}\\]</p>\n<p>$\\sin(x)=$&nbsp;[[0]]</p>\n<p>$x=$&nbsp;[[1]]$^\\circ$,&nbsp;[[2]]$^\\circ$,&nbsp;[[3]]$^\\circ$ or&nbsp;[[4]]$^\\circ$</p>\n<p>Discard any solutions out of range.</p>", "extendBaseMarkingAlgorithm": true, "marks": 0, "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "unitTests": []}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Simple trig equations with degrees</p>"}, "advice": "<p><strong>(a)</strong></p>\n<p>\\[\\var{a}\\sin(x)=\\var{a-1}\\]</p>\n<p>Dividing both sides by $\\var{a}$ gives&nbsp;$\\sin(x)=\\frac{\\var{a-1}}{\\var{a}}$</p>\n<p>Hence</p>\n<p>$x=\\sin^{-1}(\\frac{\\var{a-1}}{\\var{a}})=\\var{precround((180/pi)*arcsin((a-1)/a),3)}$ is one solution.</p>\n<p></p>\n<p>We use periodicity of sin to obtain the other solutions in the range $0^\\circ\\leq x\\leq720^\\circ$.</p>\n<p>$x=180-\\var{precround((180/pi)*arcsin((a-1)/a),3)}=\\var{precround(180-(180/pi)*arcsin((a-1)/a),3)}$</p>\n<p>$x=360+\\var{precround((180/pi)*arcsin((a-1)/a),3)}=\\var{precround(360+(180/pi)*arcsin((a-1)/a),3)}$</p>\n<p>$x=540-\\var{precround((180/pi)*arcsin((a-1)/a),3)}=\\var{precround(540-(180/pi)*arcsin((a-1)/a),3)}$</p>\n<p></p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>\\[\\var{b}\\cos(x)=\\var{c}\\]</p>\n<p>Dividing both sides by $\\var{b}$ gives&nbsp;$\\cos(x)=\\frac{\\var{c}}{\\var{b}}$</p>\n<p>Hence</p>\n<p>$x=\\cos^{-1}(\\frac{\\var{c}}{\\var{b}})=\\var{precround((180/pi)*arccos((c)/b),3)}$ is one solution.</p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $0^\\circ\\leq x\\leq720^\\circ$.</p>\n<p>$x=360-\\var{precround((180/pi)*arccos((c)/b),3)}=\\var{precround(360-(180/pi)*arccos((c)/b),3)}$</p>\n<p>$x=360+\\var{precround((180/pi)*arccos((c)/b),3)}=\\var{precround(360+(180/pi)*arccos((c)/b),3)}$</p>\n<p>$x=720-\\var{precround((180/pi)*arccos((c)/b),3)}=\\var{precround(720-(180/pi)*arccos((c)/b),3)}$</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>\\[\\var{b+1}\\cos(x)=-\\var{b-1}\\]</p>\n<p>Dividing both sides by $\\var{b+1}$ gives&nbsp;$\\cos(x)=\\frac{-\\var{b-1}}{\\var{b+1}}$</p>\n<p>Hence</p>\n<p>$x=\\cos^{-1}(\\frac{-\\var{b-1}}{\\var{b+1}})=\\var{precround((180/pi)*arccos(-(b-1)/(b+1)),3)}$ is one solution.</p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $0^\\circ\\leq x\\leq720^\\circ$.</p>\n<p>$x=360-\\var{precround((180/pi)*arccos(-(b-1)/(b+1)),3)}=\\var{precround(360-(180/pi)*arccos(-(b-1)/(b+1)),3)}$</p>\n<p>$x=360+\\var{precround((180/pi)*arccos(-(b-1)/(b+1)),3)}=\\var{precround(360+(180/pi)*arccos(-(b-1)/(b+1)),3)}$</p>\n<p>$x=720-\\var{precround((180/pi)*arccos(-(b-1)/(b+1)),3)}=\\var{precround(720-(180/pi)*arccos(-(b-1)/(b+1)),3)}$</p>\n<p></p>\n<p></p>\n<p><strong>(d)</strong></p>\n<p>\\[\\var{c+3}\\sin(x)=-\\var{c}\\]</p>\n<p>Dividing both sides by $\\var{c+3}$ gives&nbsp;$\\sin(x)=\\frac{-\\var{c}}{\\var{c+3}}$</p>\n<p>Hence</p>\n<p>$x=\\sin^{-1}(\\frac{-\\var{c}}{\\var{c+3}})=\\var{precround((180/pi)*arcsin(-(c)/(c+3)),3)}$.</p>\n<p></p>\n<p>Since we requre&nbsp;$0^\\circ\\leq x\\leq720^\\circ$, this solution is not in the given range. So we need to find the 4 required solutions as follows:</p>\n<p></p>\n<p>We use periodicity of sin to obtain the other solutions in the range $0^\\circ\\leq x\\leq720^\\circ$.</p>\n<p>$x=180-(\\var{precround((180/pi)*arcsin(-(c)/(c+3)),3)})=\\var{precround(180-(180/pi)*arcsin(-(c)/(c+3)),3)}$</p>\n<p>$x=360+(\\var{precround((180/pi)*arcsin(-(c)/(c+3)),3)})=\\var{precround(360+(180/pi)*arcsin(-(c)/(c+3)),3)}$</p>\n<p>$x=540-(\\var{precround((180/pi)*arcsin(-(c)/(c+3)),3)})=\\var{precround(540-(180/pi)*arcsin(-(c)/(c+3)),3)}$</p>\n<p>$x=720+(\\var{precround((180/pi)*arcsin(-(c)/(c+3)),3)})=\\var{precround(720+(180/pi)*arcsin(-(c)/(c+3)),3)}$</p>", "type": "question"}, {"name": "Simon's copy of Trigonometric Equations 2 - Simple (Radians)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "advice": "<p><strong>(a)</strong></p>\n<p>\\[\\var{a}\\sin(x)=\\var{a-1}\\]</p>\n<p>Dividing both sides by $\\var{a}$ gives&nbsp;$\\sin(x)=\\frac{\\var{a-1}}{\\var{a}}$</p>\n<p>Hence</p>\n<p>$x=\\sin^{-1}(\\frac{\\var{a-1}}{\\var{a}})=\\var{precround(arcsin((a-1)/a),3)}$ is one solution.</p>\n<p></p>\n<p>We use periodicity of sin to obtain the other solutions in the range $0\\leq\\theta\\leq2\\pi$.</p>\n<p>$x=\\pi-\\var{precround(arcsin((a-1)/a),3)}=\\var{precround(pi-arcsin((a-1)/a),3)}$</p>\n<p></p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>\\[\\var{b}\\cos(x)=\\var{c}\\]</p>\n<p>Dividing both sides by $\\var{b}$ gives&nbsp;$\\cos(x)=\\frac{\\var{c}}{\\var{b}}$</p>\n<p>Hence</p>\n<p>$x=\\cos^{-1}(\\frac{\\var{c}}{\\var{b}})=\\var{precround(arccos((c)/b),3)}$ is one solution.</p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $0\\leq\\theta\\leq2\\pi$.</p>\n<p>$x=2\\pi-\\var{precround(arccos((c)/b),3)}=\\var{precround(2pi-arccos((c)/b),3)}$</p>\n<p></p>\n<p><strong></strong></p>\n<p><strong>(c)</strong></p>\n<p></p>\n<p>\\[\\var{c+3}\\sin(x)=-\\var{c}\\]</p>\n<p>Dividing both sides by $\\var{c+3}$ gives&nbsp;$\\sin(x)=\\frac{-\\var{c}}{\\var{c+3}}$</p>\n<p>Hence</p>\n<p>$x=\\sin^{-1}(\\frac{-\\var{c}}{\\var{c+3}})=\\var{precround(arcsin(-(c)/(c+3)),3)}$.</p>\n<p></p>\n<p>Since we requre $0\\leq\\theta\\leq2\\pi$, this solution is not in the given range. So we need to find the 2 required solutions as follows:</p>\n<p></p>\n<p>We use periodicity of sin to obtain the other solutions in the range $0\\leq\\theta\\leq2\\pi$.</p>\n<p>$x=\\pi-(\\var{precround(arcsin(-(c)/(c+3)),3)})=\\var{precround(pi-arcsin(-(c)/(c+3)),3)}$</p>\n<p>$x=2\\pi+(\\var{precround(arcsin(-(c)/(c+3)),3)})=\\var{precround(2pi+arcsin(-(c)/(c+3)),3)}$</p>\n<p></p>\n<p><strong>(d)</strong></p>\n<p>\\[\\var{b+1}\\cos(x)=-\\var{b-1}\\]</p>\n<p>Dividing both sides by $\\var{b+1}$ gives&nbsp;$\\cos(x)=\\frac{-\\var{b-1}}{\\var{b+1}}$</p>\n<p>Hence</p>\n<p>$x=\\cos^{-1}(\\frac{-\\var{b-1}}{\\var{b+1}})=\\var{precround(arccos(-(b-1)/(b+1)),3)}$ is one solution.</p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $0\\leq\\theta\\leq2\\pi$.</p>\n<p>$x=2\\pi-\\var{precround(arccos(-(b-1)/(b+1)),3)}=\\var{precround(2pi-arccos(-(b-1)/(b+1)),3)}$</p>", "statement": "<p><strong>Solve the following equations in <em>radians</em>, for $\\theta$ in the range $0\\leq\\theta\\leq2\\pi$.</strong></p>\n<p>Give your answers correct to 2 decimal places in ascending order.</p>", "variable_groups": [{"name": "Q1", "variables": ["a", "f1", "s1", "s11", "s12"]}, {"name": "Q2", "variables": ["b", "c", "f2", "s2", "s21", "s22"]}, {"name": "Q4", "variables": ["f3", "s31", "s32", "s3"]}, {"name": "Q3", "variables": ["f4", "s4", "s41", "s42"]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Simple trig equations with radians</p>"}, "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s11": {"group": "Q1", "templateType": "anything", "definition": "precround(s1,2)", "name": "s11", "description": ""}, "f1": {"group": "Q1", "templateType": "anything", "definition": "(a-1)/a", "name": "f1", "description": ""}, "b": {"group": "Q2", "templateType": "anything", "definition": "random(5..12)", "name": "b", "description": ""}, "s22": {"group": "Q2", "templateType": "anything", "definition": "precround(2*pi-s2,2)", "name": "s22", "description": ""}, "s1": {"group": "Q1", "templateType": "anything", "definition": "arcsin(f1)", "name": "s1", "description": ""}, "s21": {"group": "Q2", "templateType": "anything", "definition": "precround(s2,2)", "name": "s21", "description": ""}, "a": {"group": "Q1", "templateType": "anything", "definition": "random(3..9)", "name": "a", "description": ""}, "s31": {"group": "Q4", "templateType": "anything", "definition": "precround(s3,2)", "name": "s31", "description": ""}, "s12": {"group": "Q1", "templateType": "anything", "definition": "precround(pi-s1,2)", "name": "s12", "description": ""}, "s41": {"group": "Q3", "templateType": "anything", "definition": "precround(pi-s4,2)", "name": "s41", "description": ""}, "s32": {"group": "Q4", "templateType": "anything", "definition": "precround(2*pi-s3,2)", "name": "s32", "description": ""}, "s4": {"group": "Q3", "templateType": "anything", "definition": "arcsin(f4)", "name": "s4", "description": ""}, "f3": {"group": "Q4", "templateType": "anything", "definition": "(1-b)/(b+1)", "name": "f3", "description": ""}, "f2": {"group": "Q2", "templateType": "anything", "definition": "c/b", "name": "f2", "description": ""}, "f4": {"group": "Q3", "templateType": "anything", "definition": "-c/(c+3)", "name": "f4", "description": ""}, "c": {"group": "Q2", "templateType": "anything", "definition": "random(2..b-1)", "name": "c", "description": ""}, "s42": {"group": "Q3", "templateType": "anything", "definition": "precround(2*pi+s4,2)", "name": "s42", "description": ""}, "s2": {"group": "Q2", "templateType": "anything", "definition": "arccos(f2)", "name": "s2", "description": ""}, "s3": {"group": "Q4", "templateType": "anything", "definition": "arccos(f3)", "name": "s3", "description": ""}}, "functions": {}, "parts": [{"showCorrectAnswer": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "variableReplacements": [], "mustBeReduced": false, "marks": 1, "unitTests": [], "maxValue": "{f1}+0.01", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "correctAnswerFraction": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "minValue": "{f1}-0.01"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s11}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s11}", "precisionPartialCredit": "100"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s12}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s12}", "precisionPartialCredit": "100"}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>\\[\\var{a}\\sin(\\theta)=\\var{a-1}\\]</p>\n<p>$\\sin(\\theta)=$ [[0]]</p>\n<p>$\\theta=$ [[1]]&nbsp;or&nbsp;[[2]]</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "variableReplacements": [], "mustBeReduced": false, "marks": 1, "unitTests": [], "maxValue": "{f2}+0.01", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "correctAnswerFraction": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "minValue": "{f2}-0.01"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s21}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s21}", "precisionPartialCredit": "100"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s22}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s22}", "precisionPartialCredit": "100"}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>\\[\\var{b}\\cos(\\theta)=\\var{c}\\]</p>\n<p>$\\cos(\\theta)=$ [[0]]</p>\n<p>$\\theta=$ [[1]]&nbsp;or [[2]]</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "variableReplacements": [], "mustBeReduced": false, "marks": 1, "unitTests": [], "maxValue": "{f4}+0.01", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "correctAnswerFraction": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "minValue": "{f4}-0.01"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s41}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s41}", "precisionPartialCredit": "100"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s42}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s42}", "precisionPartialCredit": "100"}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>\\[\\var{c+3}\\sin(\\theta)=-\\var{c}\\]</p>\n<p>$\\sin(\\theta)=$ [[0]]</p>\n<p>$\\theta=$ [[1]]&nbsp;or&nbsp;[[2]]</p>\n<p>Discard any solutions out of range.</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "variableReplacements": [], "mustBeReduced": false, "marks": 1, "unitTests": [], "maxValue": "{f3}+0.01", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "correctAnswerFraction": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "minValue": "{f3}-0.01"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s31}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s31}", "precisionPartialCredit": "100"}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": true, "minValue": "{s32}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{s32}", "precisionPartialCredit": "100"}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>\\[\\var{b+1}\\cos(\\theta)=-\\var{b-1}\\]</p>\n<p>$\\cos(\\theta)=$ [[0]]</p>\n<p>$\\theta=$ [[1]]&nbsp;or&nbsp;[[2]]</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}], "tags": [], "rulesets": {}, "ungrouped_variables": [], "type": "question"}, {"name": "Simon's copy of Trigonometric Equations 3 - Intermediate (Degrees)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "preamble": {"js": "", "css": ""}, "advice": "<p>\\[\\var{a+1}\\cos(x-\\var{b})=\\var{a-2}\\]</p>\n<p></p>\n<p><strong>(a)</strong></p>\n<p>Dividing both sides by $\\var{a+1}$ gives&nbsp;$\\cos(x-\\var{b})=\\frac{\\var{a-2}}{\\var{a+1}}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>Hence</p>\n<p>$x-\\var{b}=\\cos^{-1}(\\frac{\\var{a-2}}{\\var{a+1}})=\\var{precround((180/pi)*arccos((a-2)/(a+1)),3)}$ is a solution.</p>\n<p></p>\n<p>Since we want solutions&nbsp;$0^\\circ\\leq x\\leq720^\\circ$, this corresponds to&nbsp;$-\\var{b}^\\circ\\leq x-\\var{b} \\leq\\var{720-b}^\\circ$</p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $-\\var{b}^\\circ\\leq x-\\var{b} \\leq\\var{720-b}^\\circ$:</p>\n<p>$x-\\var{b}=360-\\var{precround((180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(360-(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p>$x-\\var{b}=360+\\var{precround((180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(360+(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p>$x-\\var{b}=720-\\var{precround((180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(720-(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>$x-\\var{b}=\\var{precround((180/pi)*arccos((a-2)/(a+1)),3)} \\implies x = \\var{b} + \\var{precround((180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(b+(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p>$x-\\var{b}=\\var{precround(360-(180/pi)*arccos((a-2)/(a+1)),3)}\\implies x = \\var{b} +\\var{precround(360-(180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(b+360-(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p>$x-\\var{b}=\\var{precround(360+(180/pi)*arccos((a-2)/(a+1)),3)}\\implies x = \\var{b} +\\var{precround(360+(180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(b+360+(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p>$x-\\var{b}=\\var{precround(720-(180/pi)*arccos((a-2)/(a+1)),3)}\\implies x = \\var{b} +\\var{precround(720-(180/pi)*arccos((a-2)/(a+1)),3)}=\\var{precround(b+720-(180/pi)*arccos((a-2)/(a+1)),3)}$</p>\n<p></p>", "variablesTest": {"condition": "{b4}<720", "maxRuns": 100}, "statement": "<p><strong>Solve the following equation in <em>degrees</em>, for $x$ in the range $0^\\circ\\leq x\\leq720^\\circ$.</strong></p>\n<p>\\[\\var{a+1}\\cos(x-\\var{b})=\\var{a-2}\\]</p>\n<p>Give your final answers correct to the nearest degree in ascending order.</p>", "metadata": {"description": "<p>Trigonometric equations with degrees</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variables": {"p": {"name": "p", "definition": "arccos(f)*180/pi", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b4": {"name": "b4", "definition": "a4+b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a3": {"name": "a3", "definition": "360+p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "f": {"name": "f", "definition": "(a-2)/(a+1)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b1": {"name": "b1", "definition": "a1+b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b3": {"name": "b3", "definition": "a3+b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b2": {"name": "b2", "definition": "a2+b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a": {"name": "a", "definition": "random(3..9)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a2": {"name": "a2", "definition": "360-p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a1": {"name": "a1", "definition": "p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"name": "b", "definition": "random(10..p-5)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a4": {"name": "a4", "definition": "720-p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "tags": [], "variable_groups": [], "rulesets": {}, "ungrouped_variables": ["a", "a1", "a2", "a3", "a4", "b", "b1", "b2", "b3", "b4", "f", "p"], "parts": [{"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": true, "mustBeReducedPC": 0, "marks": 1, "mustBeReduced": false, "scripts": {}, "type": "numberentry", "maxValue": "{f}+0.01", "minValue": "{f}-0.01", "correctAnswerStyle": "plain", "unitTests": [], "variableReplacementStrategy": "originalfirst", "allowFractions": true}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$\\cos(x-\\var{b})=$ [[0]]</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a1}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a1}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a2}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a2}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a3}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a3}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a4}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a4}", "allowFractions": false}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$x-\\var{b}=$ [[0]]$^\\circ$, [[1]]$^\\circ$, [[2]]$^\\circ$ or [[3]]$^\\circ$</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{b1}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{b1}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{b2}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{b2}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{b3}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{b3}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{b4}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{b4}", "allowFractions": false}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$x=$ [[0]]$^\\circ$, [[1]]$^\\circ$, [[2]]$^\\circ$ or [[3]]$^\\circ$</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "type": "question"}, {"name": "Simon's copy of Trigonometric Equations 4 - Intermediate (Radians)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "advice": "<p>\\[\\var{b}\\sin(\\theta-\\var{d})=\\var{c}\\]</p>\n<p></p>\n<p><strong>(a)</strong></p>\n<p>Dividing both sides by $\\var{b}$ gives&nbsp;$\\sin(\\theta-\\var{d})=\\frac{\\var{c}}{\\var{b}}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>Hence</p>\n<p>$\\theta-\\var{d}=\\sin^{-1}(\\frac{\\var{c}}{\\var{b}})=\\var{precround(arcsin(c/b),3)}$ is a solution.</p>\n<p></p>\n<p>Since we want solutions &nbsp;$0\\leq\\theta\\leq4\\pi$, this corresponds to&nbsp;$-\\var{d}\\leq \\theta-\\var{d} \\leq\\var{4pi}-\\var{d}$</p>\n<p></p>\n<p>We use periodicity of sin to obtain the other solutions in the range $-\\var{d}\\leq \\theta-\\var{d} \\leq\\var{4pi}-\\var{d}$:</p>\n<p>$\\theta-\\var{d}=\\pi-\\var{precround(arcsin(c/b),3)}=\\var{precround(pi-arcsin(c/b),3)}$</p>\n<p>$\\theta-\\var{d}=2\\pi+\\var{precround(arcsin(c/b),3)}=\\var{precround(2pi+arcsin(c/b),3)}$</p>\n<p>$\\theta-\\var{d}=3\\pi-\\var{precround(arcsin(c/b),3)}=\\var{precround(3pi-arcsin(c/b),3)}$</p>\n<p><strong></strong></p>\n<p><strong>(c)</strong></p>\n<p>$\\theta-\\var{d}=\\var{precround(arcsin(c/b),3)} \\implies \\theta = \\var{precround(arcsin(c/b),3)} + \\var{d} = \\var{precround(d+arcsin(c/b),3)}$</p>\n<p>$\\theta-\\var{d}=\\var{precround(pi-arcsin(c/b),3)} \\implies \\theta =\\var{precround(pi-arcsin(c/b),3)} + \\var{d} =\\var{precround(d+pi-arcsin(c/b),3)}$</p>\n<p>$\\theta-\\var{d}=\\var{precround(2pi+arcsin(c/b),3)} \\implies \\theta =\\var{precround(2pi+arcsin(c/b),3)} + \\var{d} =\\var{precround(d+2pi+arcsin(c/b),3)}$</p>\n<p>$\\theta-\\var{d}=\\var{precround(3pi-arcsin(c/b),3)} \\implies \\theta =\\var{precround(3pi-arcsin(c/b),3)} + \\var{d} =\\var{precround(d+3pi-arcsin(c/b),3)}$</p>", "statement": "<p><strong>Solve the following equations in <em>radians</em>, for $\\theta$ in the range $0\\leq\\theta\\leq4\\pi$:</strong></p>\n<p>\\[\\var{b}\\sin(\\theta-\\var{d})=\\var{c}\\]</p>\n<p>Give your final answers correct to 2 decimal places in ascending order.</p>", "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Trigonometric&nbsp;equations with radians</p>"}, "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "b4<4*pi", "maxRuns": 100}, "variables": {"b4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a4+d", "name": "b4", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "c/b", "name": "f", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p", "name": "a1", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "pi-p", "name": "a2", "description": ""}, "b3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a3+d", "name": "b3", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..b-1)", "name": "c", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(5..12)", "name": "b", "description": ""}, "a4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "3*pi-p", "name": "a4", "description": ""}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "2*pi+p", "name": "a3", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a2+d", "name": "b2", "description": ""}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "arcsin(f)", "name": "p", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a1+d", "name": "b1", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..p+2 #0.1)", "name": "d", "description": ""}}, "functions": {}, "parts": [{"showCorrectAnswer": true, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "variableReplacements": [], "mustBeReduced": false, "marks": 1, "unitTests": [], "maxValue": "{f}+0.01", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": "", "correctAnswerFraction": true, "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "minValue": "{f}-0.01"}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>$\\sin(\\theta-\\var{d})=$ [[0]]</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{a1}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{a1}", "precisionPartialCredit": 0}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{a2}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{a2}", "precisionPartialCredit": 0}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{a3}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{a3}", "precisionPartialCredit": 0}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{a4}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": true, "maxValue": "{a4}", "precisionPartialCredit": 0}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>$\\theta-\\var{d}=$ [[0]]&nbsp;,&nbsp;[[1]], [[2]] or [[3]]</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "gaps": [{"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{b1}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{b1}", "precisionPartialCredit": 0}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{b2}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{b2}", "precisionPartialCredit": 0}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{b3}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "maxValue": "{b3}", "precisionPartialCredit": 0}, {"allowFractions": false, "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "customMarkingAlgorithm": "", "correctAnswerFraction": false, "strictPrecision": false, "minValue": "{b4}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "type": "numberentry", "mustBeReducedPC": 0, "precision": "2", "variableReplacementStrategy": "originalfirst", "showPrecisionHint": true, "maxValue": "{b4}", "precisionPartialCredit": 0}], "sortAnswers": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "marks": 0, "unitTests": [], "prompt": "<p>$\\theta=$ [[0]]&nbsp;,&nbsp;[[1]], [[2]] or [[3]]</p>", "extendBaseMarkingAlgorithm": true, "scripts": {}, "customMarkingAlgorithm": ""}], "tags": [], "rulesets": {}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b", "b1", "b2", "b3", "b4", "c", "d", "f", "p"], "type": "question"}, {"name": "Simon's copy of Trigonometric Equations 5 - Complex (Degrees)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "preamble": {"js": "", "css": ""}, "advice": "<p>\\[\\var{a-1}\\sin(2x+\\var{b})=\\var{a-2}\\]</p>\n<p></p>\n<p><strong>(a)</strong></p>\n<p>Dividing both sides by $\\var{a-1}$ gives&nbsp;$\\sin(2x+\\var{b})=\\frac{\\var{a-2}}{\\var{a-1}}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>Hence</p>\n<p>$2x+\\var{b}=\\sin^{-1}(\\frac{\\var{a-2}}{\\var{a-1}})=\\var{precround((180/pi)*arcsin((a-2)/(a-1)),3)}$ is a solution.</p>\n<p></p>\n<p>Since we want solutions&nbsp;$0^\\circ\\leq x\\leq360^\\circ$, this corresponds to&nbsp;$\\var{b}^\\circ\\leq 2x+\\var{b} \\leq\\var{720+b}^\\circ$</p>\n<p></p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $\\var{b}^\\circ\\leq 2x+\\var{b} \\leq\\var{720+b}^\\circ$:</p>\n<p>$2x+\\var{b}=180-\\var{precround((180/pi)*arcsin((a-2)/(a-1)),3)}=\\var{precround(180-(180/pi)*arcsin((a-2)/(a-1)),3)}$</p>\n<p>$2x+\\var{b}=360+\\var{precround((180/pi)*arcsin((a-2)/(a-1)),3)}=\\var{precround(360+(180/pi)*arcsin((a-2)/(a-1)),3)}$</p>\n<p>$2x+\\var{b}=540-\\var{precround((180/pi)*arcsin((a-2)/(a-1)),3)}=\\var{precround(540-(180/pi)*arcsin((a-2)/(a-1)),3)}$</p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>$2x+\\var{b}=\\var{precround((180/pi)*arcsin((a-2)/(a-1)),3)} \\implies x = \\frac{\\var{precround((180/pi)*arcsin((a-2)/(a-1)),3)}-\\var{b}}{2}=\\var{precround((-b+(180/pi)*arcsin((a-2)/(a-1)))/2,3)}$</p>\n<p>$2x+\\var{b}=\\var{precround(180-(180/pi)*arcsin((a-2)/(a-1)),3)} \\implies x =&nbsp;\\frac{\\var{precround(180-(180/pi)*arcsin((a-2)/(a-1)),3)}-\\var{b}}{2}=\\var{precround((-b+180-(180/pi)*arcsin((a-2)/(a-1)))/2,3)}$</p>\n<p>$2x+\\var{b}=\\var{precround(360+(180/pi)*arcsin((a-2)/(a-1)),3)} \\implies x =&nbsp;\\frac{\\var{precround(360+(180/pi)*arcsin((a-2)/(a-1)),3)}-\\var{b}}{2}=\\var{precround((-b+360+(180/pi)*arcsin((a-2)/(a-1)))/2,3)}$</p>\n<p>$2x+\\var{b}=\\var{precround(540-(180/pi)*arcsin((a-2)/(a-1)),3)} \\implies x =&nbsp;\\frac{\\var{precround(540-(180/pi)*arcsin((a-2)/(a-1)),3)}-\\var{b}}{2}=\\var{precround((-b+540-(180/pi)*arcsin((a-2)/(a-1)))/2,3)}$</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p><strong>Solve the following equation in <em>degrees</em>, for $x$ in the range $0^\\circ\\leq x\\leq360^\\circ$.</strong></p>\n<p>\\[\\var{a-1}\\sin(2x+\\var{b})=\\var{a-2}\\]</p>\n<p>Give your final answers correct to the nearest degree in ascending order.</p>", "metadata": {"description": "<p>More difficult trigonometric&nbsp;equations with&nbsp;degrees</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variables": {"c3": {"name": "c3", "definition": "b3/2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "p": {"name": "p", "definition": "arcsin(f1)*180/pi", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b4": {"name": "b4", "definition": "a4-b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c4": {"name": "c4", "definition": "b4/2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a3": {"name": "a3", "definition": "360+p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b1": {"name": "b1", "definition": "a1-b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b3": {"name": "b3", "definition": "a3-b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c1": {"name": "c1", "definition": "b1/2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b2": {"name": "b2", "definition": "a2-b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a": {"name": "a", "definition": "random(3..9)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a2": {"name": "a2", "definition": "180-p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a1": {"name": "a1", "definition": "p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"name": "b", "definition": "random(10..p-5)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a4": {"name": "a4", "definition": "540-p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "f1": {"name": "f1", "definition": "(a-2)/(a-1)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c2": {"name": "c2", "definition": "b2/2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "tags": [], "variable_groups": [], "rulesets": {}, "ungrouped_variables": ["a", "a1", "a2", "a3", "a4", "b", "b1", "b2", "b3", "b4", "f1", "p", "c1", "c2", "c3", "c4"], "parts": [{"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": true, "mustBeReducedPC": 0, "marks": 1, "mustBeReduced": false, "scripts": {}, "type": "numberentry", "maxValue": "{f1}+0.01", "minValue": "{f1}-0.01", "correctAnswerStyle": "plain", "unitTests": [], "variableReplacementStrategy": "originalfirst", "allowFractions": true}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$\\sin(2x+\\var{b})=$ [[0]]</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a1}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a1}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a2}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a2}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a3}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a3}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a4}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a4}", "allowFractions": false}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$2x+\\var{b}=$ [[0]]$^\\circ$, [[1]]$^\\circ$, [[2]]$^\\circ$ or [[3]]$^\\circ$</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c1}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c1}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c2}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c2}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c3}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c3}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": 0, "showPrecisionHint": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c4}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c4}", "allowFractions": false}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$x=$ [[0]]$^\\circ$, [[1]]$^\\circ$, [[2]]$^\\circ$ or [[3]]$^\\circ$</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "type": "question"}, {"name": "Simon's copy of Trigonometric Equations 6 - Complex (Radians)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "preamble": {"js": "", "css": ""}, "advice": "<p>\\[\\var{b}\\cos(3\\theta+\\var{d})=\\var{c}\\]</p>\n<p></p>\n<p><strong>(a)</strong></p>\n<p>Dividing both sides by $\\var{b}$ gives&nbsp;$\\cos(3\\theta+\\var{d})=\\frac{\\var{c}}{\\var{b}}$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>Hence</p>\n<p>$3\\theta+\\var{d}=\\cos^{-1}(\\frac{\\var{c}}{\\var{b}})=\\var{precround(arccos(c/b),3)}$ is a solution.</p>\n<p></p>\n<p>Since we want solutions&nbsp; &nbsp;$0\\leq\\theta\\leq\\pi$, this corresponds to&nbsp;$\\var{d}\\leq 3\\theta+\\var{d} \\leq\\var{3pi}+\\var{d}$</p>\n<p></p>\n<p>We use periodicity of cos to obtain the other solutions in the range $\\var{d}\\leq 3\\theta+\\var{d} \\leq\\var{3pi}+\\var{d}$:</p>\n<p>$3\\theta+\\var{d}=2\\pi-\\var{precround(arccos(c/b),3)}=\\var{precround(2pi-arccos(c/b),3)}$</p>\n<p>$3\\theta+\\var{d}=2\\pi+\\var{precround(arccos(c/b),3)}=\\var{precround(2pi+arccos(c/b),3)}$</p>\n<p></p>\n<p><strong></strong></p>\n<p><strong>(c)</strong></p>\n<p>$3\\theta+\\var{d}=\\var{precround(arccos(c/b),3)} \\implies \\theta = \\frac{\\var{precround(arccos(c/b),3)} - \\var{d}}{3} = \\var{precround((-d+arccos(c/b))/3,3)}$</p>\n<p>$3\\theta+\\var{d}=\\var{precround(2pi-arccos(c/b),3)} \\implies \\theta =\\frac{\\var{precround(2pi-arccos(c/b),3)} - \\var{d}}{3} =\\var{precround((-d+2pi-arccos(c/b))/3,3)}$</p>\n<p>$3\\theta+\\var{d}=\\var{precround(2pi+arccos(c/b),3)} \\implies \\theta =\\frac{\\var{precround(2pi+arccos(c/b),3)} - \\var{d}}{3} =\\var{precround((-d+2pi+arccos(c/b))/3,3)}$</p>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p><strong>Solve the following equations in <em>radians</em>, for $\\theta$ in the range $0\\leq\\theta\\leq\\pi$:</strong></p>\n<p>\\[\\var{b}\\cos(3\\theta+\\var{d})=\\var{c}\\]</p>\n<p>Give your final answers correct to 2 decimal places in ascending order.</p>", "metadata": {"description": "<p>More difficult trigonometric&nbsp;equations with radians</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variables": {"c3": {"name": "c3", "definition": "b3/3", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "p": {"name": "p", "definition": "arccos(f)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b4": {"name": "b4", "definition": "a4-d", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c4": {"name": "c4", "definition": "b4/3", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a3": {"name": "a3", "definition": "2*pi+p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "d": {"name": "d", "definition": "random(0.1..p-0.1 #0.1)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a2": {"name": "a2", "definition": "2*pi-p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "f": {"name": "f", "definition": "c/b", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b1": {"name": "b1", "definition": "a1-d", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c": {"name": "c", "definition": "random(2..b-1)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c1": {"name": "c1", "definition": "b1/3", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b2": {"name": "b2", "definition": "a2-d", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c2": {"name": "c2", "definition": "b2/3", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b3": {"name": "b3", "definition": "a3-d", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a1": {"name": "a1", "definition": "p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"name": "b", "definition": "random(5..12)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a4": {"name": "a4", "definition": "4*pi-p", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "tags": [], "variable_groups": [], "rulesets": {}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b", "b1", "b2", "b3", "b4", "c", "c1", "c2", "c3", "c4", "d", "f", "p"], "parts": [{"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": true, "mustBeReducedPC": 0, "marks": 1, "mustBeReduced": false, "scripts": {}, "type": "numberentry", "maxValue": "{f}+0.01", "minValue": "{f}-0.01", "correctAnswerStyle": "plain", "unitTests": [], "variableReplacementStrategy": "originalfirst", "allowFractions": true}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$\\cos(3\\theta+\\var{d})=$ [[0]]</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a1}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a1}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a2}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a2}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": 0, "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{a3}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{a3}", "allowFractions": false}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$3\\theta+\\var{d}=$ [[0]]&nbsp;,&nbsp;[[1]]&nbsp;or&nbsp;[[2]]</p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}, {"showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "gaps": [{"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": "100", "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c1}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c1}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": "100", "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": false, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c2}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c2}", "allowFractions": false}, {"strictPrecision": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "mustBeReduced": false, "marks": 1, "scripts": {}, "correctAnswerFraction": false, "unitTests": [], "precisionPartialCredit": "100", "showFeedbackIcon": true, "variableReplacements": [], "customMarkingAlgorithm": "", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": true, "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "minValue": "{c3}", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "maxValue": "{c3}", "allowFractions": false}], "extendBaseMarkingAlgorithm": true, "prompt": "<p>$\\theta=$ [[0]]&nbsp;,&nbsp;[[1]] or&nbsp;[[2]]</p>\n<p></p>", "marks": 0, "scripts": {}, "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst"}], "type": "question"}, {"name": "Simon's copy of Solving trigonometric equations using identities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "tags": [], "variable_groups": [{"name": "Q1", "variables": ["a", "s11", "s12"]}, {"name": "Q2", "variables": ["b", "d", "c", "positivex", "negativex", "theta1", "theta2", "theta3", "theta4"]}, {"name": "Question range of x", "variables": ["x", "tanx", "Rangenegative", "Rangepositive", "Answerforx", "Multipliedrangenegative", "Multipliedrangepositive", "Answer"]}], "rulesets": {}, "variablesTest": {"condition": "and(abs(positivex)<1, abs(negativex)<1)", "maxRuns": 100}, "parts": [{"extendBaseMarkingAlgorithm": true, "sortAnswers": false, "variableReplacements": [], "prompt": "<p>Solve in&nbsp;<b><i>radians</i></b>, for $\\theta$ in the range $0\\leq \\theta\\leq2\\pi$.</p>\n<p></p>\n<p>$\\sin\\theta-\\sqrt{\\var{a}}\\cos\\theta=0$</p>\n<p>$\\tan\\theta=$&nbsp;[[0]]</p>\n<p>$\\theta=$&nbsp;[[1]]&nbsp;or&nbsp;[[2]]</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "marks": 0, "scripts": {}, "useCustomName": false, "showCorrectAnswer": true, "customName": "", "gaps": [{"failureRate": 1, "vsetRange": [0, 1], "valuegenerators": [], "checkVariableNames": false, "checkingType": "absdiff", "variableReplacements": [], "vsetRangePoints": 5, "scripts": {}, "showPreview": true, "marks": 1, "useCustomName": false, "type": "jme", "unitTests": [], "extendBaseMarkingAlgorithm": true, "answerSimplification": "all", "variableReplacementStrategy": "originalfirst", "customName": "", "customMarkingAlgorithm": "", "showFeedbackIcon": false, "answer": "sqrt({a})", "checkingAccuracy": 0.001, "showCorrectAnswer": true}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "{s11}", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "{s11}", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "{s12}", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": true, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "{s12}", "notationStyles": ["plain", "en", "si-en"]}]}, {"extendBaseMarkingAlgorithm": true, "sortAnswers": false, "variableReplacements": [], "prompt": "<p>Solve in&nbsp;<b><i>radians</i></b>, for $\\theta$ in the range $0\\leq \\theta\\leq2\\pi$.</p>\n<p></p>\n<p>$\\var{b}\\cos^2\\theta=\\var{c}-\\var{d}\\cos\\theta$</p>\n<p>Rearrange the equation so that all terms are on the left hand side.</p>\n<p>[[0]] $=0$</p>\n<p>Use the quadratic formula or factorise the equation to get two values of $\\cos\\theta$</p>\n<p>$\\cos \\theta$=[[1]]and[[2]] (put the most positive value first)</p>\n<p>Using the above result, find four values of the $\\theta$</p>\n<p>$\\theta=$&nbsp;[[3]] ,&nbsp;[[4]],&nbsp;&nbsp;[[5]],&nbsp;[[6]] (start with the smallest angle first)</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "marks": 0, "scripts": {}, "useCustomName": false, "showCorrectAnswer": true, "customName": "", "gaps": [{"failureRate": 1, "vsetRange": [0, 1], "valuegenerators": [{"name": "theta", "value": ""}], "checkVariableNames": false, "checkingType": "absdiff", "variableReplacements": [], "vsetRangePoints": 5, "scripts": {}, "showPreview": true, "marks": 1, "useCustomName": false, "type": "jme", "unitTests": [], "extendBaseMarkingAlgorithm": true, "answerSimplification": "all", "variableReplacementStrategy": "originalfirst", "customName": "", "customMarkingAlgorithm": "", "showFeedbackIcon": false, "answer": "{b}*cos(theta)^2+{d}cos(theta)-{c}", "checkingAccuracy": 0.001, "showCorrectAnswer": true}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "{positivex}", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "{positivex}", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "{negativex}", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": true, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "{negativex}", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "theta1", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "theta1", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "theta2", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "theta2", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "theta4", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "theta4", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacements": [], "maxValue": "theta3", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": true, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": false, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "theta3", "notationStyles": ["plain", "en", "si-en"]}]}, {"extendBaseMarkingAlgorithm": true, "sortAnswers": false, "variableReplacements": [], "prompt": "<p>Solve in&nbsp;<b><i>radians</i></b>, for $\\theta$ in the range $0\\leq \\theta\\leq2\\pi$.</p>\n<p></p>\n<p>$\\cos^2\\theta-\\var{a}\\sin^2\\theta=1$</p>\n<p>$\\theta=$&nbsp;[[0]],&nbsp;[[1]] or&nbsp;[[2]]</p>", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "marks": 0, "scripts": {}, "useCustomName": false, "showCorrectAnswer": true, "customName": "", "gaps": [{"precisionPartialCredit": "50", "strictPrecision": false, "variableReplacements": [], "maxValue": "0", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": true, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "0", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": "50", "strictPrecision": false, "variableReplacements": [], "maxValue": "3.1416", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": false, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": true, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "3.1416", "notationStyles": ["plain", "en", "si-en"]}, {"precisionPartialCredit": "50", "strictPrecision": false, "variableReplacements": [], "maxValue": "6.2832", "scripts": {}, "correctAnswerStyle": "plain", "marks": 1, "useCustomName": false, "precision": "3", "type": "numberentry", "showPrecisionHint": true, "mustBeReduced": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "customName": "", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "showFeedbackIcon": true, "precisionType": "dp", "correctAnswerFraction": false, "showCorrectAnswer": true, "minValue": "6.2832", "notationStyles": ["plain", "en", "si-en"]}]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Using trig identities to find solutions to equations</p>"}, "ungrouped_variables": [], "preamble": {"js": "", "css": ""}, "statement": "<p>Solve the following equations in <b><i>radians</i></b>, for $\\theta$ in the range $0\\leq \\theta\\leq2\\pi$.</p>\n<p>The square root of $a$ is written as sqrt($a$). $\\theta$ is written as theta.</p>\n<p></p>\n<p>You may want to use the following trigonometric identities.</p>\n<p>$\\tan(x)=\\frac{\\sin(x)}{\\cos(x)}$</p>\n<p>$\\sin^2(x)+\\cos^2(x)=1$</p>\n<p>$\\cos(2x)=\\cos^2(x)-\\sin^2(x)=2\\cos^2(x)-1=1-2\\sin^2(x)$</p>\n<p>$\\sin(2x)=2\\sin(x)\\cos(x)$</p>\n<p></p>\n<p>For angles, insert values in increasing order.&nbsp;DO NOT write your answer in an exact form with $\\pi$ but rather in decimal form to up 3 decimal places.&nbsp;</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "variables": {"theta1": {"name": "theta1", "group": "Q2", "description": "", "templateType": "anything", "definition": "(arccos(positivex))"}, "theta4": {"name": "theta4", "group": "Q2", "description": "", "templateType": "anything", "definition": "2*pi-theta2"}, "positivex": {"name": "positivex", "group": "Q2", "description": "", "templateType": "anything", "definition": "(-d+sqrt(d^2+4*b*c))/(2*b)"}, "theta3": {"name": "theta3", "group": "Q2", "description": "", "templateType": "anything", "definition": "2*pi-theta1"}, "d": {"name": "d", "group": "Q2", "description": "", "templateType": "randrange", "definition": "random(1 .. 5#1)"}, "x": {"name": "x", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "random(2..7)"}, "Rangenegative": {"name": "Rangenegative", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "-90"}, "Multipliedrangenegative": {"name": "Multipliedrangenegative", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "Rangenegative*x"}, "s11": {"name": "s11", "group": "Q1", "description": "", "templateType": "anything", "definition": "arctan(sqrt(a))"}, "theta2": {"name": "theta2", "group": "Q2", "description": "", "templateType": "anything", "definition": "(arccos(negativex))"}, "negativex": {"name": "negativex", "group": "Q2", "description": "", "templateType": "anything", "definition": "(-d-sqrt(d^2+4*b*c))/(2*b)"}, "Multipliedrangepositive": {"name": "Multipliedrangepositive", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "Rangepositive*x"}, "a": {"name": "a", "group": "Q1", "description": "", "templateType": "anything", "definition": "random(2..9)"}, "b": {"name": "b", "group": "Q2", "description": "", "templateType": "randrange", "definition": "random(1 .. 5#1)"}, "Answerforx": {"name": "Answerforx", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "arctan(tanx)*180/PI"}, "Answer": {"name": "Answer", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "Answerforx/x"}, "Rangepositive": {"name": "Rangepositive", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "+90"}, "tanx": {"name": "tanx", "group": "Question range of x", "description": "", "templateType": "anything", "definition": "-1"}, "s12": {"name": "s12", "group": "Q1", "description": "", "templateType": "anything", "definition": "s11+pi"}, "c": {"name": "c", "group": "Q2", "description": "", "templateType": "randrange", "definition": "random(1 .. 5#1)"}}, "functions": {}, "advice": "<p><strong>(a)</strong></p>\n<p>Rearrange the equation to give&nbsp;$\\sin\\theta=\\sqrt{\\var{a}}\\cos\\theta$</p>\n<p>Divide both sides by&nbsp;$\\cos\\theta$ to give&nbsp;$\\tan\\theta=\\sqrt{\\var{a}}$</p>\n<p>Take $\\tan^{-1}\\sqrt{\\var{a}}$ to give $\\theta=\\var{precround(s11,3)}$. The second solution in the range $0\\leq \\theta\\leq2\\pi$ is given by $\\theta=\\var{precround(s11,3)}+\\pi = \\var{precround(s12,3)}$</p>\n<p></p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>&nbsp;Rearrange the equation to give&nbsp;$\\var{b}\\cos^2\\theta+\\var{d}\\cos\\theta-\\var{c}=0$</p>\n<p>This is a quadratic in $\\cos\\theta$ which we can solve using the quadratic formula to give $\\cos\\theta =\\frac{-\\var{d}\\pm\\sqrt{\\var{d}^2+4\\times{\\var{b}\\times\\var{c}}}}{2\\times\\var{b}} =&nbsp;\\var{precround(positivex,3)}$ or $\\var{precround(negativex,3)}$</p>\n<p>Take&nbsp;$\\cos^{-1}\\var{precround(positivex,3)}=\\var{precround(theta1,3)}$ and&nbsp;$\\cos^{-1}\\var{precround(negativex,3)}=\\var{precround(theta2,3)}$.</p>\n<p>Now recall that the periodicity of cos means that if $\\theta$ is a solution then so is $\\ 2\\pi-\\theta$. Hence we obtain 4 solutions in the range $0\\leq \\theta\\leq2\\pi:$</p>\n<p>$\\theta =&nbsp;\\var{precround(theta1,3)}$&nbsp; or&nbsp; $2\\pi-\\var{precround(theta1,3)}=\\var{precround(theta3,3)}$&nbsp; or&nbsp; $\\var{precround(theta2,3)}$&nbsp; or&nbsp; $2\\pi-\\var{precround(theta2,3)}=\\var{precround(theta4,3)}$</p>\n<p></p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>Use the identity $\\cos^2\\theta=1-\\sin^2\\theta$ and substituting this in place of&nbsp;$\\cos^2\\theta$ gives $1-\\sin^2\\theta-\\var{a}\\sin^2\\theta=1$</p>\n<p>Hence&nbsp;$\\var{a+1}\\sin^2\\theta=0$</p>\n<p>So $\\sin^2\\theta=0$ and in turn&nbsp;$\\sin\\theta=0$, which has 3 solutions $\\theta = 0, \\pi, 2\\pi$ in the range&nbsp;$0\\leq \\theta\\leq2\\pi$</p>\n<p></p>\n<p></p>", "type": "question"}, {"name": "Trigonometry equations: Solve asinx + bcosx = c", "extensions": ["chemistry", "geogebra", "jsxgraph", "codewords", "permutations", "polynomials", "quantities", "random_person", "stats", "visjs"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "preamble": {"js": "", "css": ""}, "parts": [{"useCustomName": false, "prompt": "<p>Write $\\simplify{ {a} sin theta + {b}cos theta}$ in the form $\\ r \\sin(\\theta+ \\alpha)$</p>\n<p></p>\n<p>Write&nbsp;$r$ and $\\alpha$ correct to 3 d.p.</p>", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "scripts": {}, "answer": "{precround(r,3)}sin(theta+{precround(alpha,3)})", "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "checkingAccuracy": "0.002", "unitTests": [], "customName": "", "valuegenerators": [{"value": "", "name": "theta"}], "marks": "2", "customMarkingAlgorithm": "", "notallowed": {"strings": ["cos"], "message": "", "showStrings": false, "partialCredit": 0}, "vsetRangePoints": 5, "showFeedbackIcon": true, "musthave": {"strings": ["sin", "(", ")", "theta"], "message": "", "showStrings": false, "partialCredit": 0}, "vsetRange": [0, 1], "checkVariableNames": false, "failureRate": 1, "type": "jme", "showPreview": true}, {"useCustomName": false, "prompt": "<p>Hence solve $\\simplify{{a}sin theta +{b}cos theta={c}}$.&nbsp;</p>\n<p>First find the smallest value of $\\theta$ such that&nbsp;$\\ 0\\leq\\theta&lt;2\\pi$.</p>\n<p>[[0]]</p>\n<p>Now find the largest value of $\\theta$ such that&nbsp;$\\ 0\\leq\\theta&lt;2\\pi$.</p>\n<p>[[1]]</p>", "extendBaseMarkingAlgorithm": true, "gaps": [{"strictPrecision": false, "useCustomName": false, "precisionType": "dp", "extendBaseMarkingAlgorithm": true, "allowFractions": false, "unitTests": [], "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precision": "3", "customName": "", "showPrecisionHint": true, "precisionPartialCredit": 0, "marks": "2", "customMarkingAlgorithm": "", "minValue": "min(theta1,theta2)", "variableReplacements": [], "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReduced": false, "correctAnswerFraction": false, "maxValue": "min(theta1,theta2)"}, {"strictPrecision": false, "useCustomName": false, "precisionType": "dp", "extendBaseMarkingAlgorithm": true, "allowFractions": false, "unitTests": [], "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precision": "3", "customName": "", "showPrecisionHint": true, "precisionPartialCredit": 0, "marks": "2", "customMarkingAlgorithm": "", "minValue": "max(theta1,theta2)", "variableReplacements": [], "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "mustBeReduced": false, "correctAnswerFraction": false, "maxValue": "max(theta1,theta2)"}], "sortAnswers": false, "unitTests": [], "scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "customName": "", "marks": 0, "customMarkingAlgorithm": ""}], "ungrouped_variables": ["a", "b", "c", "rold", "alphaold", "theta1", "theta2", "asign", "bsign", "alphatemp", "alpha", "r", "alphaexplain", "thetatemp1", "thetatemp2"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "rulesets": {}, "statement": "", "variable_groups": [], "advice": "<p><strong>(a)</strong></p>\n<p>Compare&nbsp;$\\simplify{ {a}sin theta + {b}cos theta}$ with the identity $\\ r \\sin(\\theta+ \\alpha) \\equiv r \\sin \\theta \\cos \\alpha + r \\cos \\theta \\sin \\alpha$</p>\n<p>Hence $ \\var{a} = r \\cos \\alpha $</p>\n<p>and $ \\var{b} = r \\sin \\alpha $</p>\n<p>So $ r^2 = \\var{abs(a)}^2 + \\var{abs(b)}^2$</p>\n<p>$r = \\sqrt{\\var{abs(a)}^2 + \\var{abs(b)}^2} = \\var{precround(abs(r),3)}$</p>\n<p>Calculate $ \\tan^{-1}&nbsp;({\\var{abs(b)}/\\var{abs(a)}}) = \\var{precround({alphatemp},3)} $</p>\n<p>So our possible values of&nbsp;$\\alpha$ are:</p>\n<p>$\\var{precround(alphatemp,3)}$&nbsp; &nbsp; &nbsp; &nbsp;,&nbsp; &nbsp; &nbsp; &nbsp;$\\pi-\\var{precround(alphatemp,3)}=\\var{precround(pi-alphatemp,3)}$&nbsp; &nbsp; &nbsp; &nbsp;,&nbsp; &nbsp; &nbsp; &nbsp;$\\pi+\\var{precround(alphatemp,3)}=\\var{precround(pi+alphatemp,3)}$&nbsp; &nbsp; &nbsp; &nbsp;,&nbsp; &nbsp; &nbsp; &nbsp;$2\\pi-\\var{precround(alphatemp,3)}=\\var{precround(2pi-alphatemp,3)}$</p>\n<p></p>\n<p>Since $ r\\cos\\alpha =\\var{a} \\var{asign} 0$ and $ r \\sin \\alpha =&nbsp; \\var{b} \\var{bsign} 0$ we choose $\\alpha = \\var{precround(switch(a&lt;0 and b&lt;0,pi+{alphatemp},b&lt;0,2pi-{alphatemp},a&lt;0,pi-{alphatemp},{alphatemp}),3)}$ to be in the correct quadrant.</p>\n<p>Hence&nbsp;$\\simplify{ {a} sin theta + {b} cos theta} \\equiv \\var{precround(r,3)} \\sin (\\simplify{theta +{precround(alpha,3)}})$</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>We now have $\\var{precround(r,3)} \\sin (\\simplify{theta +{precround(alpha,3)}}) = \\var{c} $</p>\n<p>So&nbsp;&nbsp;$ \\sin (\\simplify{theta +{precround(alpha,3)}}) = \\var{c} /&nbsp;\\var{precround(r,3)} =&nbsp;\\var{precround(c/r,3)} $</p>\n<p>We obtain 2 solutions:&nbsp; $\\simplify{theta&nbsp; +{precround(alpha,3)}}=&nbsp; \\sin^{-1}(\\var{precround(c/r,3)}) = \\var{precround(arcsin(c/r),3)}$&nbsp; &nbsp; or&nbsp; &nbsp; $\\simplify{theta +{precround(alpha,3)}}=&nbsp; \\pi - \\var{precround(arcsin(c/r),3)} = \\var{precround(pi - arcsin(c/r),3)}$</p>\n<p>Hence $\\ \\theta =&nbsp; \\var{precround(arcsin(c/r),3)} - \\var{precround(alpha,3)} = \\var{precround(thetatemp1,3)}&nbsp;$&nbsp; &nbsp; &nbsp;or&nbsp; &nbsp; &nbsp;$\\ \\theta =&nbsp; \\var{precround(pi - arcsin(c/r),3)} - \\var{precround(alpha,3)} =&nbsp;\\var{precround(thetatemp2,3)} $</p>\n<p></p>\n<p>Remember we require solutions&nbsp;such that&nbsp;$\\ 0\\leq\\theta&lt;2\\pi$. Since sin is periodic with period $2 \\pi$ we can add or subtract&nbsp;$2 \\pi$ to our answers if necessary to obtain final answers of</p>\n<p>$\\ \\theta = \\var{precround(theta1,3)}$&nbsp; &nbsp;or&nbsp; &nbsp;$\\var{precround(theta2,3)}$</p>", "functions": {}, "variables": {"alphaold": {"group": "Ungrouped variables", "name": "alphaold", "definition": "arctan(b/a)", "templateType": "anything", "description": ""}, "theta2": {"group": "Ungrouped variables", "name": "theta2", "definition": "mod(thetatemp2,2pi)", "templateType": "anything", "description": ""}, "alphaexplain": {"group": "Ungrouped variables", "name": "alphaexplain", "definition": "switch(a<0 and b<0,-pi+alphatemp,b<0,-alphatemp,a<0,pi-alphatemp,alphatemp)", "templateType": "anything", "description": ""}, "rold": {"group": "Ungrouped variables", "name": "rold", "definition": "{a}/cos({alpha})", "templateType": "anything", "description": ""}, "a": {"group": "Ungrouped variables", "name": "a", "definition": "random(-9..9 except 0)", "templateType": "anything", "description": ""}, "thetatemp1": {"group": "Ungrouped variables", "name": "thetatemp1", "definition": "arcsin({c}/{r})-{alpha}", "templateType": "anything", "description": ""}, "r": {"group": "Ungrouped variables", "name": "r", "definition": "sqrt({a}^2+{b}^2)", "templateType": "anything", "description": ""}, "asign": {"group": "Ungrouped variables", "name": "asign", "definition": "if(a>0,\" > \",\" < \")", "templateType": "anything", "description": ""}, "bsign": {"group": "Ungrouped variables", "name": "bsign", "definition": "if(b>0,\" > \",\" < \")", "templateType": "anything", "description": ""}, "theta1": {"group": "Ungrouped variables", "name": "theta1", "definition": "mod(thetatemp1,2pi)", "templateType": "anything", "description": ""}, "b": {"group": "Ungrouped variables", "name": "b", "definition": "random(-9..9 except 0 except a)", "templateType": "anything", "description": ""}, "alpha": {"group": "Ungrouped variables", "name": "alpha", "definition": "switch(a<0 and b<0,pi+alphatemp,b<0,2pi-alphatemp,a<0,pi-alphatemp,alphatemp)", "templateType": "anything", "description": ""}, "thetatemp2": {"group": "Ungrouped variables", "name": "thetatemp2", "definition": "pi-arcsin({c}/{r})-{alpha}", "templateType": "anything", "description": ""}, "alphatemp": {"group": "Ungrouped variables", "name": "alphatemp", "definition": "arctan(abs({b}/{a}))", "templateType": "anything", "description": ""}, "c": {"group": "Ungrouped variables", "name": "c", "definition": "random(-9..9 except 0)", "templateType": "anything", "description": ""}}, "tags": [], "variablesTest": {"condition": "c^2<a^2+b^2", "maxRuns": "20"}, "type": "question"}]}], "timing": {"timeout": {"message": "", "action": "none"}, "allowPause": true, "timedwarning": {"message": "", "action": "none"}}, "percentPass": "0", "type": "exam", "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}], "extensions": ["chemistry", "geogebra", "jsxgraph", "codewords", "permutations", "polynomials", "quantities", "random_person", "stats"], "custom_part_types": [], "resources": []}