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{"name": "Unit circle definition of trig functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Unit circle definition of trig functions", "statement": "<p>Let $A$ be the point $(1,0)$, $O$ be the origin $(0,0)$, and $B$ be a point on the&nbsp;unit circle (the circle centred at the origin, $O$, with radius 1).&nbsp;</p>\n<p>Suppose&nbsp;that the line segment $OA$ would have&nbsp;to travel $\\theta$ degrees anti-clockwise around the origin to get to the line segment $OB$. Or in other words, $A$ is&nbsp;$\\theta$ degrees anti-clockwise from the positive $x$-axis.</p>", "advice": "<p>The point on&nbsp;the unit circle, $\\theta$ degrees anti-clockwise from the positive $x$-axis, is&nbsp;$(\\cos(\\theta),\\sin(\\theta))$. This is the unit circle definition of sine and cosine. You can think of this as being a generalisation of the right-angled trigonometry that takes place in the first quadrant of the cartesian plane.</p>\n<p></p>\n<p>The definition of $\\tan(\\theta)$ can be thought of as $\\dfrac{\\sin(\\theta)}{\\cos(\\theta)}$ but this is just the gradient of the line segment connecting the origin to the point on the unit circle.</p>\n<p>The following applet is for you to investigate the relationship between the trigonometric functions and the unit circle&nbsp;by moving the point $B$ around the circle.</p>\n<p></p>\n<div style=\"text-align: center;\"><iframe width=\"924px\" height=\"479px\" style=\"border: 0px;\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/QmzGnBT9/width/924/height/479/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/false/at/auto\"> </iframe></div>", "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Unit circle definition of sin, cos, tan using degrees</p>"}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "variables": {}, "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>What are the coordinates of the point $B$?</p>\n<p>$B=\\Large($ [[0]],&nbsp;[[1]]&nbsp;$\\Large)$</p>\n<p></p>\n<p><strong>Note:&nbsp;</strong>Suppose you wanted to enter $\\tan(\\theta)$, then you would&nbsp;type <strong>tan(theta)</strong> including the brackets.</p>", "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"checkingtype": "absdiff", "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "answer": "cos(theta)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "showpreview": true, "vsetrangepoints": 5, "scripts": {}, "marks": 1}, {"checkingtype": "absdiff", "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "answer": "sin(theta)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "showpreview": true, "vsetrangepoints": 5, "scripts": {}, "marks": 1}], "scripts": {}, "marks": 0}, {"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>What is the slope of the line segment $OB$?</p>\n<p>$m_{OB}=$&nbsp;[[0]]</p>", "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"checkingtype": "absdiff", "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "answer": "tan(theta)", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "vsetrange": [0, 1], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "showpreview": true, "vsetrangepoints": 5, "scripts": {}, "marks": 1}], "scripts": {}, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "rulesets": {}, "functions": {}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}