// Numbas version: finer_feedback_settings
{"name": "Trig Equations 3 (sin^2x+cos^2x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trig Equations 3 (sin^2x+cos^2x)", "tags": [], "metadata": {"description": "<p>solve trig equation that requires use of s^s+c^2=1. with worked solutions.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "<p>\\begin{align}<br/><br/>\\simplify{{a}sin^2(x)+({a}{c}-{b})cos(x)+({b}{c}-{a})}&amp;=0 \\\\\\\\<br/><br/>\\text{first replace $sin^2(x) \\text{&nbsp; with&nbsp; } 1-cos^2(x)$} \\\\\\\\<br/><br/>\\simplify{{a}(1-cos^2 ( x) )+({a}{c}-{b})cos(x)+({b}{c}-{a})}&amp;=0 \\\\\\\\<br/><br/>\\text{then move everything to the other side and simplify} \\\\\\\\<br/><br/>0&amp;=\\simplify{{a}cos^2(x)+({b}-{a}{c})cos(x)+(-{b}{c})} \\\\\\\\<br/><br/>\\text{now you can factorise} \\\\\\\\<br/><br/>0&amp;=(\\simplify{{a}cos(x)+{b}})(cos(x)-\\var{c}) \\\\\\\\<br/><br/>cos(x)&amp;=-\\frac{\\var{b}}{\\var{a}} \\text{&nbsp; or&nbsp; } \\var{c} \\\\\\\\<br/><br/>\\text{there are no solutions to $cos(x)=\\var{c}$ so this leaves only}\\\\\\\\<br/><br/>x_1 &amp;=cos^{-1}-\\frac{\\var{b}}{\\var{a}}= \\var{x1_3} \\\\\\\\<br/><br/>x_2 &amp;=2\\pi-\\var{x1_3}=\\var{x2_3}<br/><br/>\\end{align}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,1,-2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2 .. 7#1)", "description": "", "templateType": "randrange", "can_override": false}, "cosx": {"name": "cosx", "group": "Ungrouped variables", "definition": "-{b}/{a}", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "(arccos({cosx}))*180/pi", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "360-{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}, "x2_3": {"name": "x2_3", "group": "Ungrouped variables", "definition": "sigformat(x2,3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "cosx", "x1", "x2", "x1_3", "x2_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</p>\n<p>$\\simplify{{a}sin^2(x)+({a}{c}-{b})cos(x)+({b}{c}-{a})}=0$ betwwen $0\\leq x\\leq 360^o$</p>\n<p></p>\n<p>Give your answers below in <strong>ascending</strong> order and to 3 significant figures</p>\n<p></p>\n<p>[[0]]$^o$ or&nbsp; [[1]]$^o$</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "x1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{x1}", "maxValue": "{x1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "x2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{x2}", "maxValue": "{x2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Mike Phipps", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14139/"}]}]}], "contributors": [{"name": "Mike Phipps", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14139/"}]}