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{"name": "Basic trigonometry", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Basic trigonometry", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Fill in the gap.</p>", "advice": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>if we write $\\boxed{\\var{a}=r\\sin\\alpha}$&nbsp; and&nbsp; $\\boxed{\\var{b}=r\\cos\\alpha}$ for some $r$ and $\\alpha$ then we have $(\\var{a})\\cos\\theta+(\\var{b})\\sin\\theta=(r\\sin\\alpha)\\cos\\theta+(r\\cos\\alpha)\\sin\\theta=r\\sin(\\alpha+\\theta)$ . Then the maximum and minimum value will be $r$&nbsp; and $-r$ since $-1\\leq\\sin(\\alpha+\\theta)\\leq 1$</p>\n<p>To find $r$ we do the following:&nbsp; $r^2\\sin^2\\alpha=(\\var{a})^2$ and&nbsp;$r^2\\cos^2\\alpha=(\\var{b})^2 \\implies r^2\\sin^2\\alpha+r^2\\cos^2\\alpha=(\\var{a})^2+(\\var{b})^2\\implies r^2(\\sin^2\\alpha+\\cos^2\\alpha)=(\\var{a})^2+(\\var{b})^2\\implies r^2=(\\var{a})^2+(\\var{b})^2\\\\ \\implies r=\\sqrt{({\\var{a})^2+(\\var{b})^2}}=\\var{c}$&nbsp;</p>", "rulesets": {}, "extensions": [], "variables": {"triplet": {"name": "triplet", "group": "Ungrouped variables", "definition": "random([[3,4,5],[5,12,13],[7,24,25],[8,15,17],[9,40,41],[11,60,61],[12,35,37],[16,63,65],[20,21,29],[20,99,101]])\n", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p></p>", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random([-1,1])*triplet[0]", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p></p>", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random([-1,1])*triplet[1]", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p></p>", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "triplet[2]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["triplet", "a", "b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>The maximum of $(\\var{a})\\cos\\theta +(\\var{b})\\sin\\theta$=[[0]]&nbsp;</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "triplet[2]", "maxValue": "triplet[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>The minimum of $(\\var{a})\\cos\\theta +(\\var{b})\\sin\\theta$=[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "-triplet[2]", "maxValue": "-triplet[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "AJAY OTTA", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3976/"}]}]}], "contributors": [{"name": "AJAY OTTA", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3976/"}]}