// Numbas version: exam_results_page_options
{"name": "Trig equations 3 (sin2x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trig equations 3 (sin2x)", "tags": [], "metadata": {"description": "<p>solve trig equation involving sin2x=2sinxcosx in a given interval</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "<p>First replace $sin2x$ with $2sinxcosx$ and rearrange</p>\n<p>\\begin{align}<br/><br/>\\var{a}sin2x &amp; =\\simplify{0+{b} cosx}&nbsp; \\\\\\\\<br/>\\var{a}\\times 2sinxcosx &amp; =\\simplify{0+{b} cosx} \\\\\\\\<br/>\\simplify{2{a}}sinxcosx \\simplify{0-{b}cosx} &amp; =0 \\\\\\\\<br/>cosx(\\simplify{2{a}}sinx-\\var{b}) &amp;=0<br/>\\end{align}</p>\n<p>This can now be solved to give</p>\n<p>\\[ cosx=0 \\text{&nbsp; or&nbsp; } sinx=\\frac{\\var{b}}{\\simplify{2{a}}} \\]</p>\n<p>Taking $cosx=0$ and remembering that for $cosx$ a second angle can be found by talking the first away from $2 \\pi$</p>\n<p>\\begin{align}<br/><br/>cosx=0 \\implies x=cos^{-1}(0)=\\frac{\\pi}{2} \\text{&nbsp; or&nbsp; } \\frac{3 \\pi}{2}<br/><br/>\\end{align}</p>\n<p></p>\n<p>Next taking $cosx=\\frac{\\var{b}}{\\simplify{2{a}}}$ and remembering that for $sinx$ a second angle can be found by talking the first away from $ \\pi$</p>\n<p>\\begin{align}<br/><br/>sinx=\\frac{\\var{b}}{\\simplify{2{a}}} \\implies x=sin^{-1}\\left(\\frac{\\var{b}}{\\simplify{2{a}}}\\right)=\\var{x1} \\text{&nbsp; or&nbsp; } \\var{x3}<br/><br/>\\end{align}</p>\n<p>To 3 significant figures and in radians the answers are therefore</p>\n<p>$\\var{x1_3} \\text{&nbsp; and&nbsp; } \\var{x2_3} \\text{&nbsp; and&nbsp; } \\var{x3_3} \\text{&nbsp; and&nbsp; } \\var{x4_3}$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "arcsin(b/(2a))", "description": "", "templateType": "anything", "can_override": false}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "pi-x1", "description": "", "templateType": "anything", "can_override": false}, "x1_3": {"name": "x1_3", "group": "Ungrouped variables", "definition": "sigformat(x1,3)", "description": "", "templateType": "anything", "can_override": false}, "x3_3": {"name": "x3_3", "group": "Ungrouped variables", "definition": "sigformat(x3,3)", "description": "", "templateType": "anything", "can_override": false}, "x2_3": {"name": "x2_3", "group": "Ungrouped variables", "definition": "sigformat(pi/2,3)", "description": "", "templateType": "anything", "can_override": false}, "x4_3": {"name": "x4_3", "group": "Ungrouped variables", "definition": "sigformat(1.5pi,3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "x1", "x3", "x1_3", "x3_3", "x2_3", "x4_3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve the equation $\\var{a}sin2x=\\simplify{0+{b} cosx}$ in the range $0\\leq x \\leq 2\\pi$</p>\n<p></p>\n<p>Give your answers below in <strong>ascending order</strong> and to 3 significant figures.</p>\n<p></p>\n<p>$x_1=$[[0]]</p>\n<p>$x_2=$[[1]]</p>\n<p>$x_3=$[[2]]</p>\n<p>$x_4=$[[3]]</p>\n<p></p>", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The first setp is to replace $sin2x$ with $2sinxcosx$</p>"}], "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "x1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, 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