// Numbas version: exam_results_page_options
{"name": "Solving trigonometric equations using identities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["a", "s11", "s12"], "name": "Q1"}, {"variables": ["b", "d", "c", "positivex", "negativex", "theta1", "theta2", "theta3", "theta4"], "name": "Q2"}, {"variables": ["x", "tanx", "Rangenegative", "Rangepositive", "Answerforx", "Multipliedrangenegative", "Multipliedrangepositive", "Answer"], "name": "Question range of x"}], "name": "Solving trigonometric equations using identities", "parts": [{"sortAnswers": false, "prompt": "<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>", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"answer": "sqrt({a})", "answerSimplification": "all", "showPreview": true, "vsetRangePoints": 5, "checkingType": "absdiff", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "expectedVariableNames": [], "type": "jme", "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "vsetRange": [0, 1], "variableReplacements": [], "checkVariableNames": false}, {"correctAnswerFraction": false, "maxValue": "{s11}*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "{s11}*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "{s12}*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "{s12}*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}, {"sortAnswers": false, "prompt": "<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>", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"answer": "{b}*cos^2(theta)+{d}cos(theta)-{c}", "answerSimplification": "all", "showPreview": true, "vsetRangePoints": 5, "checkingType": "absdiff", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "expectedVariableNames": [], "type": "jme", "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "vsetRange": [0, 1], "variableReplacements": [], "checkVariableNames": false}, {"correctAnswerFraction": false, "maxValue": "{positivex}*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "{positivex}*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "{negativex}*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "{negativex}*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "theta1*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "theta1*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "theta2*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "theta2*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "theta3*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "theta3*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}, {"correctAnswerFraction": false, "maxValue": "theta4*1.01", "allowFractions": false, "correctAnswerStyle": "plain", "unitTests": [], "showFeedbackIcon": false, "showCorrectAnswer": true, "scripts": {}, "minValue": "theta4*0.99", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReducedPC": 0, "mustBeReduced": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": []}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}, {"sortAnswers": false, "prompt": "<p>$\\cos^2\\theta-\\var{a}\\sin^2\\theta=1$</p>\n<p>$\\theta=$&nbsp;[[0]],&nbsp;[[1]] or&nbsp;[[2]]</p>", "unitTests": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"unitTests": [], "showFeedbackIcon": true, "precision": "2", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "mustBeReduced": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "precisionPartialCredit": "50", "showPrecisionHint": false, "correctAnswerStyle": "plain", "maxValue": "0", "scripts": {}, "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "0", "type": "numberentry", "mustBeReducedPC": 0, "precisionType": "dp", "marks": 1, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"]}, {"unitTests": [], "showFeedbackIcon": true, "precision": "2", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "mustBeReduced": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "precisionPartialCredit": "50", "showPrecisionHint": false, "correctAnswerStyle": "plain", "maxValue": "3.1416", "scripts": {}, "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "3.1416", "type": "numberentry", "mustBeReducedPC": 0, "precisionType": "dp", "marks": 1, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"]}, {"unitTests": [], "showFeedbackIcon": true, "precision": "2", "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "mustBeReduced": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": [], "precisionPartialCredit": "50", "showPrecisionHint": false, "correctAnswerStyle": "plain", "maxValue": "6.2832", "scripts": {}, "showCorrectAnswer": true, "correctAnswerFraction": false, "minValue": "6.2832", "type": "numberentry", "mustBeReducedPC": 0, "precisionType": "dp", "marks": 1, "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"]}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacements": []}], "variablesTest": {"maxRuns": 100, "condition": "and(abs(positivex)<1, abs(negativex)<1)"}, "extensions": [], "functions": {}, "advice": "<p>a) 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>b)&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>c) 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>", "rulesets": {}, "metadata": {"description": "<p>Using trig identities to find solutions to equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "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": {"s12": {"templateType": "anything", "description": "", "definition": "s11+pi", "name": "s12", "group": "Q1"}, "Rangenegative": {"templateType": "anything", "description": "", "definition": "-90", "name": "Rangenegative", "group": "Question range of x"}, "theta3": {"templateType": "anything", "description": "", "definition": "2*pi-theta1", "name": "theta3", "group": "Q2"}, "tanx": {"templateType": "anything", "description": "", "definition": "-1", "name": "tanx", "group": "Question range of x"}, "c": {"templateType": "randrange", "description": "", "definition": "random(1..5#1)", "name": "c", "group": "Q2"}, "theta1": {"templateType": "anything", "description": "", "definition": "(arccos(positivex))", "name": "theta1", "group": "Q2"}, "Answerforx": {"templateType": "anything", "description": "", "definition": "arctan(tanx)*180/PI", "name": "Answerforx", "group": "Question range of x"}, "x": {"templateType": "anything", "description": "", "definition": "random(2..7)", "name": "x", "group": "Question range of x"}, "theta2": {"templateType": "anything", "description": "", "definition": "(arccos(negativex))", "name": "theta2", "group": "Q2"}, "a": {"templateType": "anything", "description": "", "definition": "random(2..9)", "name": "a", "group": "Q1"}, "Multipliedrangenegative": {"templateType": "anything", "description": "", "definition": "Rangenegative*x", "name": "Multipliedrangenegative", "group": "Question range of x"}, "Rangepositive": {"templateType": "anything", "description": "", "definition": "+90", "name": "Rangepositive", "group": "Question range of x"}, "positivex": {"templateType": "anything", "description": "", "definition": "(-d+sqrt(d^2+4*b*c))/(2*b)", "name": "positivex", "group": "Q2"}, "Answer": {"templateType": "anything", "description": "", "definition": "Answerforx/x", "name": "Answer", "group": "Question range of x"}, "d": {"templateType": "randrange", "description": "", "definition": "random(1..5#1)", "name": "d", "group": "Q2"}, "Multipliedrangepositive": {"templateType": "anything", "description": "", "definition": "Rangepositive*x", "name": "Multipliedrangepositive", "group": "Question range of x"}, "negativex": {"templateType": "anything", "description": "", "definition": "(-d-sqrt(d^2+4*b*c))/(2*b)", "name": "negativex", "group": "Q2"}, "theta4": {"templateType": "anything", "description": "", "definition": "2*pi-theta2", "name": "theta4", "group": "Q2"}, "s11": {"templateType": "anything", "description": "", "definition": "arctan(sqrt(a))", "name": "s11", "group": "Q1"}, "b": {"templateType": "randrange", "description": "", "definition": "random(1..5#1)", "name": "b", "group": "Q2"}}, "preamble": {"js": "", "css": ""}, "tags": [], "ungrouped_variables": [], "type": "question", "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/"}]}]}], "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/"}]}