// Numbas version: exam_results_page_options
{"name": "Trig equation with tan conversion", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trig equation with tan conversion", "tags": [], "metadata": {"description": "<p>Solve a trigonometric equation involving a conversion to tangent by division by cosine.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Solve the following equation for $x$ in radians in the range $0\\leq x\\leq\\var[fractionNumbers]{2/n}\\pi$:</p>\n<p>\\[\\simplify{{A}*sin({n}*x)}=\\simplify{{B}*cos({n}*x)}\\]</p>", "advice": "<p>Divide both sides by $\\cos(\\var{n}x)$:</p>\n<p>\\[\\simplify{{A}*tan({n}*x)}=\\var{B}\\]</p>\n<p>Divide by $\\var{A}$:</p>\n<p>\\[\\simplify{tan({n}*x)}=\\var[fractionNumbers]{B/A}\\]</p>\n<p>Take inverse tangent:</p>\n<p>\\[\\var{n}x=\\tan^{-1}\\left(\\var[fractionNumbers]{B/A}\\right)=\\var{precround(p,3)}\\text{ or }\\var{precround(p+pi,3)}\\text{ or }\\var{precround(p+2pi,3)}\\]</p>\n<p>Divide by {n}:</p>\n<p>\\[\\var{n}x=\\var{precround(p/n,3)}\\text{ or }\\var{precround((p+pi)/n,3)}\\text{ or }\\var{precround((p+2pi)/n,3)}\\]</p>\n<p>Note that $\\var[fractionNumbers]{2/n}\\pi=\\var{precround(2*pi/n,3)}$ so we discard any solutions which lie out of the range $0\\leq x\\leq\\var{precround(2*pi/n,3)}$ and round to 2 decimal places:</p>\n<p>\\[x=\\var{precround(x[0],2)}\\text{ or }\\var{precround(x[1],2)}\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,a])", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "arctan(B/A)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "p/n", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "(p+pi)/n", "description": "", "templateType": "anything", "can_override": false}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "(p+2*pi)/n", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "filter(x>=0,x,[x1,x2,x3])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "B", "A", "p", "x1", "x2", "x3", "x"], "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>Divide both sides by $\\cos(\\var{n}x)$ and use the identity $\\tan(\\var{n}x)=\\frac{\\cos(\\var{n}x)}{\\sin(\\var{n}x)}$:</p>\n<p><br/>[[0]]$\\tan(\\var{n}x)=\\;$[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "A", "maxValue": "A", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "B", "maxValue": "B", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>Divide by {A}:</p>\n<p>$\\tan(x)=\\;$[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "B/A", "maxValue": "B/A", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": "100", "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>Take inverse tangent. 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