// Numbas version: finer_feedback_settings {"name": "Converting angles from degrees to radians", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Converting angles from degrees to radians", "tags": [], "metadata": {"description": "
Convert from degrees to radians
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Note: to enter the value $\\pi$ simply type pi. For example, $\\frac{3\\pi}{2}$ could be entered as 3*pi/2 or even as 3pi/2
", "advice": "$180^\\circ$ is equal to $\\pi$ radians. This means for each $180^\\circ$ we can replace it with $\\pi$ radians. To determine how many $180^\\circ$s are
For example, $\\displaystyle 72^\\circ=\\frac{72^\\circ}{180^\\circ}\\times \\pi=\\frac{2\\pi}{5}$.
\n\nIt is useful to memorise some of the very common angles, for example, $30^\\circ=\\displaystyle \\frac{\\pi}{6},\\, 45^\\circ=\\frac{\\pi}{4},\\, 60^\\circ=\\frac{\\pi}{3}, \\,90^\\circ=\\frac{\\pi}{2}, \\,180^\\circ=\\pi$ and $360^\\circ=2\\pi$.
", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"med": {"name": "med", "group": "Ungrouped variables", "definition": " random(30, 45, 60, 90)", "description": "", "templateType": "anything", "can_override": false}, "easy1": {"name": "easy1", "group": "Ungrouped variables", "definition": "random(180,360)", "description": "", "templateType": "anything", "can_override": false}, "easyans": {"name": "easyans", "group": "Ungrouped variables", "definition": "if(easy1=180, pi, 2*pi)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "medansden": {"name": "medansden", "group": "Ungrouped variables", "definition": "switch(med=30, 6, med=45, 4, med=60, 3, med=90, 2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["easy1", "easyans", "med", "medansden", "a", "b"], "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": "$\\var{easy1}^\\circ=$ [[0]] radians
", "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": "Recall that $180^\\circ$ is equal to $\\pi$ radians. Alternatively, recall that $360^\\circ$ is equal to $2\\pi$ radians and then halve those angles.
\nRecall that $360^\\circ$ is equal to $2\\pi$ radians. Alternatively, recall that $180^\\circ$ is equal to $\\pi$ radians radians and then double those angles.
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", "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": "Ultimately, it is beneficial to memorise that
\n$30^\\circ=\\dfrac{\\pi}{6}\\text{ radians}$
\n$45^\\circ=\\dfrac{\\pi}{4}\\text{ radians}$
\n$60^\\circ=\\dfrac{\\pi}{3}\\text{ radians}$
\n$90^\\circ=\\dfrac{\\pi}{2}\\text{ radians}$
\nOtherwise, recall each $180^\\circ$ is equal to $\\pi$ radians. So to convert from degrees to radians we find out how many $180^\\circ$s fit into our angle and then multiply by $\\pi$ radians, we can do this in one step by multiplying by $\\dfrac{\\pi}{180^\\circ}$:
\n\\begin{align}\\var{med}^\\circ&=\\var{med}^\\circ\\times\\frac{\\pi}{180^\\circ}\\text{ radians}\\\\&=\\frac{\\pi}{\\var{medansden}}\\text{ radians}.\\end{align}
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", "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": "Recall each $180^\\circ$ is equal to $\\pi$ radians. So to convert from degrees to radians we find out how many $180^\\circ$s fit into our angle and then multiply by $\\pi$ radians, we can do this in one step by multiplying by $\\dfrac{\\pi}{180^\\circ}$:
\n\\begin{align}\\var{a*b}^\\circ&=\\var{a*b}^\\circ\\times\\frac{\\pi}{180^\\circ}\\text{ radians}\\\\&=\\simplify[fractionnumbers]{{a*b/180}*pi}\\text{ radians}.\\end{align}
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