// Numbas version: finer_feedback_settings
{"name": "Trigonometry: Solving Trigonometric Equations 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry: Solving Trigonometric Equations 2", "tags": [], "metadata": {"description": "<p>Solving $\\sin(2x)-\\tan(x)=0$ for $x\\in \\left(0,\\frac{\\pi}{2}\\right)$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve $\\sin(2x)-\\tan(x)=0$, for $x \\in \\left(0,\\frac{\\pi}{2}\\right)$.&nbsp;</p>", "advice": "<p>To solve \\[ \\sin(2x)-\\tan(x)=0, \\]</p>\n<p>we want to rewrite the the&nbsp;equation using the following trigonometric identities:</p>\n<ul>\n<li>$\\sin(2x) = 2\\sin(x)\\cos(x)$;</li>\n<li>$\\tan(x)=\\frac{\\sin(x)}{\\cos(x)}$.</li>\n</ul>\n<p>\\[ \\begin{split} \\sin(2x) - \\tan(x) &amp;\\,=0&nbsp; \\\\ 2\\sin(x)\\cos(x) - \\dfrac{\\sin(x)}{\\cos(x)}&nbsp;&amp;\\,= 0 \\\\ 2\\sin(x)\\cos^2(x) - \\sin(x)&nbsp;&amp;\\,= 0 \\\\ 2\\cos^2(x) - 1 &amp;\\,=0. \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split} \\cos^2(x) &amp;\\,=\\frac{1}{2} \\\\ \\cos(x) &amp;\\,= \\frac{1}{\\sqrt{2}}\\\\ x &amp;\\, = \\cos^{-1}\\left(\\frac{1}{\\sqrt{2}}\\right). \\end{split} \\]</p>\n<p>So our solution is \\[ \\begin{split} x &amp;\\, =&nbsp;\\cos^{-1}\\left(\\frac{1}{\\sqrt{2}}\\right) \\\\ &amp;\\,= \\frac{\\pi}{4} \\\\ &amp;\\,= 0.785 \\, (3\\text{ d.p.}).&nbsp; \\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "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>$x=$[[0]] (<em>Give your answer to 3 decimal places, or in terms of $\\pi$.)</em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "pi/4", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}