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{"name": "nuExam05 - Ttrigonometric Equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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"]}], "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "useCustomName": false, "type": "gapfill", "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>", "scripts": {}, 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"2*pi-theta1", "group": "Q2", "templateType": "anything", "name": "theta3", "description": ""}, "Answerforx": {"definition": "arctan(tanx)*180/PI", "group": "Question range of x", "templateType": "anything", "name": "Answerforx", "description": ""}, "x": {"definition": "random(2..7)", "group": "Question range of x", "templateType": "anything", "name": "x", "description": ""}, "Multipliedrangepositive": {"definition": "Rangepositive*x", "group": "Question range of x", "templateType": "anything", "name": "Multipliedrangepositive", "description": ""}, "positivex": {"definition": "(-d+sqrt(d^2+4*b*c))/(2*b)", "group": "Q2", "templateType": "anything", "name": "positivex", "description": ""}, "a": {"definition": "random(2..9)", "group": "Q1", "templateType": "anything", "name": "a", "description": ""}, "Rangenegative": {"definition": "-90", "group": "Question range of x", "templateType": "anything", "name": "Rangenegative", "description": ""}, "c": {"definition": "random(1 .. 5#1)", "group": "Q2", "templateType": "randrange", "name": "c", "description": ""}, "Multipliedrangenegative": {"definition": "Rangenegative*x", "group": "Question range of x", "templateType": "anything", "name": "Multipliedrangenegative", "description": ""}, "negativex": {"definition": "(-d-sqrt(d^2+4*b*c))/(2*b)", "group": "Q2", "templateType": "anything", "name": "negativex", "description": ""}, "b": {"definition": "random(1 .. 5#1)", "group": "Q2", "templateType": "randrange", "name": "b", "description": ""}, "d": {"definition": "random(1 .. 5#1)", "group": "Q2", "templateType": "randrange", "name": "d", "description": ""}, "s12": {"definition": "s11+pi", "group": "Q1", "templateType": "anything", "name": "s12", "description": ""}, "theta1": {"definition": "(arccos(positivex))", "group": "Q2", "templateType": "anything", "name": "theta1", "description": ""}, "theta4": {"definition": "2*pi-theta2", "group": "Q2", "templateType": "anything", "name": "theta4", "description": ""}, "s11": {"definition": "arctan(sqrt(a))", "group": "Q1", "templateType": "anything", "name": "s11", "description": ""}, "tanx": {"definition": "-1", "group": "Question range of x", "templateType": "anything", "name": "tanx", "description": ""}, "theta2": {"definition": "(arccos(negativex))", "group": "Q2", "templateType": "anything", "name": "theta2", "description": ""}, "Rangepositive": {"definition": "+90", "group": "Question range of x", "templateType": "anything", "name": "Rangepositive", "description": ""}}, "tags": [], "extensions": [], "name": "nuExam05 - Ttrigonometric Equations", "metadata": {"description": "<p>Using trig identities to find solutions to equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "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>", "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": "and(abs(positivex)<1, abs(negativex)<1)"}, "rulesets": {}, "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 entered as \"sqrt($a$)\". $\\theta$ is entered 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 up to 3 decimal places.&nbsp;</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "ungrouped_variables": [], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"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": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"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/"}]}