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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": 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"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, 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"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", "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/"}]}