// Numbas version: exam_results_page_options
{"metadata": {"description": "<p>Using the unit circle definition of sin, cos and tan, to calculate the exact value of trig functions evaluated at angles that depend on 0, 30, 45, 60 or 90 degrees.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "percentPass": 0, "name": "Trigonometry: Exact values and unit circle definitions", "duration": 0, "feedback": {"showactualmark": true, "allowrevealanswer": true, "showtotalmark": true, "intro": "", "showanswerstate": true, "feedbackmessages": [], "advicethreshold": 0}, "navigation": {"onleave": {"action": "none", "message": ""}, "preventleave": true, "showfrontpage": true, "showresultspage": "oncompletion", "allowregen": true, "browse": true, "reverse": true}, "timing": {"timedwarning": {"action": "none", "message": ""}, "allowPause": true, "timeout": {"action": "none", "message": ""}}, "showstudentname": true, "showQuestionGroupNames": false, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Exact values for sin, cos, tan (acute, degrees)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values.svg", "/srv/numbas/media/question-resources/exact_values.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "showQuestionGroupNames": false, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>multiple choice testing sin, cos, tan of &nbsp;random(30, 45, 60) degrees</p>"}, "parts": [{"scripts": {}, "displayColumns": 0, "matrix": ["if(theta=30,1,0)", "if(theta=45,1,0)", "if(theta=60,1,0)", 0, 0, 0], "showCorrectAnswer": true, "prompt": "<p>The exact value of $\\sin(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "variableReplacements": [], "marks": 0, "distractors": ["", "", "", "", "", ""], "type": "1_n_2", "displayType": "radiogroup", "maxMarks": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "choices": ["<p>$\\dfrac{1}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{2}}$</p>", "<p>$\\dfrac{\\sqrt{3}}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$1$</p>"]}, {"scripts": {}, "displayColumns": 0, "matrix": ["if(theta=60,1,0)", "if(theta=45,1,0)", "if(theta=30,1,0)", 0, 0, 0], "showCorrectAnswer": true, "prompt": "<p>The exact value of $\\cos(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "variableReplacements": [], "marks": 0, "distractors": ["", "", "", "", "", ""], "type": "1_n_2", "displayType": "radiogroup", "maxMarks": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "choices": ["<p>$\\dfrac{1}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{2}}$</p>", "<p>$\\dfrac{\\sqrt{3}}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$1$</p>"]}, {"scripts": {}, "displayColumns": 0, "matrix": ["0", "0", "0", "if(theta=30,1,0)", "if(theta=60,1,0)", "if(theta=45,1,0)"], "showCorrectAnswer": true, "prompt": "<p>The exact value of $\\tan(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "variableReplacements": [], "marks": 0, "distractors": ["", "", "", "", "", ""], "type": "1_n_2", "displayType": "radiogroup", "maxMarks": 0, "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "choices": ["<p>$\\dfrac{1}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{2}}$</p>", "<p>$\\dfrac{\\sqrt{3}}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$1$</p>"]}], "advice": "<p>By drawing the following triangles we can determine the exact values of $\\sin$, $\\cos$ and $\\tan$ (and their reciprocals $\\csc$, $\\sec$, $\\cot$) for the angles $30^\\circ$, $45^\\circ$ and $60^\\circ$.</p>\n<p><br/><img src=\"resources/question-resources/exact_values.svg\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Alternatively, one can memorise the following table:&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">$30^\\circ$</td>\n<td style=\"text-align: center;\">$45^\\circ$</td>\n<td style=\"text-align: center;\">$60^\\circ$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\sin$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\cos$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\tan$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{3}}$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">$\\sqrt{3}$</td>\n</tr>\n</tbody>\n</table>", "preamble": {"js": "", "css": ""}, "functions": {}, "tags": ["exact values", "trigonometry"], "variable_groups": [], "ungrouped_variables": ["theta"], "statement": "<p>Often we prefer to work with&nbsp;exact values rather than approximations from a calculator.</p>", "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "variables": {"theta": {"name": "theta", "templateType": "anything", "definition": "random(30,45,60)", "group": "Ungrouped variables", "description": ""}}, "rulesets": {}}, {"name": "Exact values for csc, sec, cot (acute, degrees)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values.svg", "/srv/numbas/media/question-resources/exact_values.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": [], "metadata": {"description": "<p>multiple choice testing csc, sec, cot&nbsp;of random(30, 45, 60) degrees</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Often we prefer to work with&nbsp;exact values rather than approximations from a calculator.</p>", "advice": "<p>Recall that $\\csc\\theta=\\dfrac{1}{\\sin\\theta}$, $\\sec\\theta=\\dfrac{1}{\\cos\\theta}$, and&nbsp;$\\cot\\theta=\\dfrac{1}{\\tan\\theta}$.</p>\n<p></p>\n<p>By drawing the following triangles we can determine the exact values of $\\sin$, $\\cos$ and $\\tan$ (and their reciprocals $\\csc$, $\\sec$, $\\cot$) for the angles $30^\\circ$, $45^\\circ$ and $60^\\circ$.</p>\n<p><br/><img src=\"resources/question-resources/exact_values.svg\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Alternatively, one can memorise the following table:&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">$30^\\circ$</td>\n<td style=\"text-align: center;\">$45^\\circ$</td>\n<td style=\"text-align: center;\">$60^\\circ$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\sin$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\cos$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\tan$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{3}}$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">$\\sqrt{3}$</td>\n</tr>\n</tbody>\n</table>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"theta": {"name": "theta", "group": "Ungrouped variables", "definition": "random(30,45,60)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["theta"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "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>The exact value of $\\csc(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$2$</p>", "<p>$\\sqrt{2}$</p>", "<p>$\\dfrac{2}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$1$</p>"], "matrix": ["if(theta=30,1,0)", "if(theta=45,1,0)", "if(theta=60,1,0)", 0, 0, 0], "distractors": ["", "", "", "", "", ""]}, {"type": "1_n_2", "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>The exact value of $\\sec(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$2$</p>", "<p>$\\sqrt{2}$</p>", "<p>$\\dfrac{2}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$1$</p>"], "matrix": ["if(theta=60,1,0)", "if(theta=45,1,0)", "if(theta=30,1,0)", 0, 0, 0], "distractors": ["", "", "", "", "", ""]}, {"type": "1_n_2", "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>The exact value of $\\cot(\\var{theta}^\\circ)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>$2$</p>", "<p>$\\sqrt{2}$</p>", "<p>$\\dfrac{2}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$1$</p>"], "matrix": ["0", "0", "0", "if(theta=30,1,0)", "if(theta=60,1,0)", "if(theta=45,1,0)"], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Unit circle definition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "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>", "metadata": {"description": "<p>Unit circle definition of sin, cos, tan using degrees</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": [], "parts": [{"gaps": [{"vsetrangepoints": 5, "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answer": "cos(theta)", "checkingtype": "absdiff", "variableReplacements": [], "showFeedbackIcon": true, "showpreview": true, "expectedvariablenames": [], "type": "jme", "scripts": {}, "marks": 1, "checkingaccuracy": 0.001}, {"vsetrangepoints": 5, "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answer": "sin(theta)", "checkingtype": "absdiff", "variableReplacements": [], "showFeedbackIcon": true, "showpreview": true, "expectedvariablenames": [], "type": "jme", "scripts": {}, "marks": 1, "checkingaccuracy": 0.001}], "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "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>", "type": "gapfill", "scripts": {}, "marks": 0}, {"gaps": [{"vsetrangepoints": 5, "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "answer": "tan(theta)", "checkingtype": "absdiff", "variableReplacements": [], "showFeedbackIcon": true, "showpreview": true, "expectedvariablenames": [], "type": "jme", "scripts": {}, "marks": 1, "checkingaccuracy": 0.001}], "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>What is the slope of the line segment $OB$?</p>\n<p>$m_{OB}=$&nbsp;[[0]]</p>", "type": "gapfill", "scripts": {}, "marks": 0}], "variable_groups": [], "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>", "rulesets": {}, "functions": {}, "variables": {}, "type": "question"}, {"name": "Exact values for sin, cos, tan (0 to 330, degrees)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values.svg", "/srv/numbas/media/question-resources/exact_values.svg"], ["question-resources/unit_circle_working.svg", "/srv/numbas/media/question-resources/unit_circle_working.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["trig", "trigonometry", "Trigonometry"], "metadata": {"description": "<p>multiple choice testing sin, cos, tan of &nbsp;random(0,90,120,135,150,180,210,225,240,270,300,315,330) degrees</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Often we prefer to work with&nbsp;exact values rather than approximations from a calculator. In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example to input the exact value of $\\sin(60^\\circ)$, which is $\\dfrac{\\sqrt{3}}{2}$, you would input&nbsp;<strong>sqrt(3)/2</strong></p>", "advice": "<p>By drawing the following triangles we can determine the <strong>exact values</strong> of $\\sin$, $\\cos$ and $\\tan$ (and their reciprocals $\\csc$, $\\sec$, $\\cot$) for the angles $30^\\circ$, $45^\\circ$ and $60^\\circ$.</p>\n<p><br/><img src=\"resources/question-resources/exact_values.svg\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Alternatively, one can memorise the following table:&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">$30^\\circ$</td>\n<td style=\"text-align: center;\">$45^\\circ$</td>\n<td style=\"text-align: center;\">$60^\\circ$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\sin$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\cos$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\tan$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{3}}$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">$\\sqrt{3}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>That combined with the <strong>unit circle definitions</strong>:</p>\n<ul>\n<li>$\\cos\\theta$ is the $x$ coordinate of the&nbsp;point on the unit circle $\\theta$ degrees anti-clockwise from the positive $x$-axis&nbsp;</li>\n<li>$\\sin\\theta$ is the $y$ coordinate of the&nbsp;point on the unit circle $\\theta$ degrees anti-clockwise from the positive $x$-axis&nbsp;</li>\n<li>$\\tan\\theta$ is the slope of the line through the origin and the point&nbsp;on the unit circle $\\theta$ degrees anti-clockwise from the positive $x$-axis, or equivalently, $\\tan\\theta=\\dfrac{\\sin\\theta}{\\cos\\theta}$</li>\n</ul>\n<p>and some understanding of <strong>congruent triangles</strong>:</p>\n<div style=\"text-align: center;\"><iframe width=\"808px\" height=\"462px\" style=\"border: 0px;\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/e4kaj8M3/width/808/height/462/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/false/at/auto\"> </iframe></div>\n<p></p>\n<p>allows us to work out $\\sin$, $\\cos$ and $\\tan$ for certain angles regardless of what&nbsp;quadrant the point is in. Because&nbsp;whatever angle we are asked about, we can always use the triangle in the first quadrant to determine the side lengths and then consider the signs of the coordinates separately.</p>\n<p></p>\n<p><strong>For example</strong>, to determine $\\sin(210^\\circ)$, $\\cos(210^\\circ)$ and $\\tan(210^\\circ)$ we first draw the following:</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/unit_circle_working.svg\" width=\"812\" height=\"536\"/></p>\n<p>From this diagram, we can see&nbsp;that $\\cos(210^\\circ)=-\\cos(30^\\circ)$, and&nbsp;$\\sin(210^\\circ)=-\\sin(30^\\circ)$ since the triangles are congruent and we are in the 3rd quadrant where both the $x$ and $y$ values (and hence the $\\cos$ and $\\sin$ values) are negative.&nbsp;</p>\n<p>But given we know these&nbsp;exact values, we can conclude \\[\\cos(210^\\circ)=-\\cos(30^\\circ)=-\\dfrac{\\sqrt{3}}{2},\\]&nbsp;\\[\\sin(210^\\circ)=-\\sin(30^\\circ)=-\\dfrac{1}{2},\\] and finally \\[\\tan(210^\\circ)=\\dfrac{\\sin(210^\\circ)}{\\cos(210^\\circ)}=\\dfrac{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}=\\dfrac{1}{\\sqrt{3}}.\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"theta": {"name": "theta", "group": "Ungrouped variables", "definition": "random(0,90,120,135,150,180,210,225,240,270,300,315,330)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["theta"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The exact value of $\\sin(\\var{theta}^\\circ)$ is&nbsp;[[0]].</p>\n<p>The exact value of $\\cos(\\var{theta}^\\circ)$ is&nbsp;[[1]].</p>\n<p>If $\\tan(\\var{theta}^\\circ)$ is defined, what is its exact value? If it isn't enter <em>infinity</em> (even though it doesn't equal that).&nbsp; [[2]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sin({theta}*pi/180)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos({theta}*pi/180)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{if(theta=90 or theta=270,infinity,tan({theta}*pi/180))}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Exact values for sin, cos, tan (-1080 to 1080, degrees) ", "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values.svg", "/srv/numbas/media/question-resources/exact_values.svg"], ["question-resources/unit_circle_working.svg", "/srv/numbas/media/question-resources/unit_circle_working.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": [], "metadata": {"description": "<p>multiple choice testing sin, cos, tan of angles that are negative or greater than 360 degrees that result in nice exact values.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Often we prefer to work with&nbsp;exact values rather than approximations from a calculator. In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example to input the exact value of $\\sin(60^\\circ)$, which is $\\dfrac{\\sqrt{3}}{2}$, you would input&nbsp;<strong>sqrt(3)/2</strong></p>", "advice": "<p>By drawing the following triangles we can determine the <strong>exact values</strong> of $\\sin$, $\\cos$ and $\\tan$ (and their reciprocals $\\csc$, $\\sec$, $\\cot$) for the angles $30^\\circ$, $45^\\circ$ and $60^\\circ$.</p>\n<p><br/><img src=\"resources/question-resources/exact_values.svg\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Alternatively, one can memorise the following table:&nbsp;</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">$30^\\circ$</td>\n<td style=\"text-align: center;\">$45^\\circ$</td>\n<td style=\"text-align: center;\">$60^\\circ$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\sin$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\cos$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\sqrt{3}}{2}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{2}}$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{2}$</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">&nbsp;</td>\n</tr>\n<tr>\n<td style=\"text-align: center;\">$\\tan$</td>\n<td style=\"text-align: center;\">$\\dfrac{1}{\\sqrt{3}}$</td>\n<td style=\"text-align: center;\">$1$</td>\n<td style=\"text-align: center;\">$\\sqrt{3}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>That combined with the <strong>unit circle definitions</strong>:</p>\n<ul>\n<ul>\n<li>$\\cos\\theta$ is the $x$ coordinate of the&nbsp;point on the unit circle $\\theta$ degrees anti-clockwise from the positive $x$-axis&nbsp;</li>\n<li>$\\sin\\theta$ is the $y$ coordinate of the&nbsp;point on the unit circle $\\theta$ degrees anti-clockwise from the positive $x$-axis&nbsp;</li>\n<li>$\\tan\\theta$ is the slope of the line through the origin and the point&nbsp;on the unit circle $\\theta$ degrees anti-clockwise from the positive $x$-axis, or equivalently, $\\tan\\theta=\\dfrac{\\sin\\theta}{\\cos\\theta}$. Note that when the line is vertical, which corresponds to when $\\cos\\theta=0$,&nbsp;the slope of the line is undefined and therefore so is $\\tan\\theta$.</li>\n</ul>\n</ul>\n<p></p>\n<p>and some understanding of <strong>congruent triangles</strong>:</p>\n<div style=\"text-align: center;\"><iframe width=\"808px\" height=\"462px\" style=\"border: 0px;\" scrolling=\"no\" src=\"https://www.geogebra.org/material/iframe/id/e4kaj8M3/width/808/height/462/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/false/at/auto\"> </iframe></div>\n<p></p>\n<p>allows us to work out $\\sin$, $\\cos$ and $\\tan$ for certain angles regardless of what&nbsp;quadrant the point is in. Because&nbsp;whatever angle we are asked about, we can always use the triangle in the first quadrant to determine the side lengths and then consider the signs of the coordinates separately.</p>\n<p></p>\n<p><strong>For example</strong>, to determine $\\sin(210^\\circ)$, $\\cos(210^\\circ)$ and $\\tan(210^\\circ)$ we first draw the following:</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/unit_circle_working.svg\" width=\"812\" height=\"536\"/></p>\n<p>From this diagram, we can see&nbsp;that $\\cos(210^\\circ)=-\\cos(30^\\circ)$, and&nbsp;$\\sin(210^\\circ)=-\\sin(30^\\circ)$ since the triangles are congruent and we are in the 3rd quadrant where both the $x$ and $y$ values (and hence the $\\cos$ and $\\sin$ values) are negative.&nbsp;</p>\n<p>But given we know these&nbsp;exact values, we can conclude \\[\\cos(210^\\circ)=-\\cos(30^\\circ)=-\\dfrac{\\sqrt{3}}{2},\\]&nbsp;\\[\\sin(210^\\circ)=-\\sin(30^\\circ)=-\\dfrac{1}{2},\\] and finally \\[\\tan(210^\\circ)=\\dfrac{\\sin(210^\\circ)}{\\cos(210^\\circ)}=\\dfrac{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}=\\dfrac{1}{\\sqrt{3}}.\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"phi": {"name": "phi", "group": "Ungrouped variables", "definition": "random(0,30,45,60,90,120,135,150,180,210,225,240,270,300,315,330)", "description": "", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "Ungrouped variables", "definition": "phi+rev", "description": "", "templateType": "anything", "can_override": false}, "rev": {"name": "rev", "group": "Ungrouped variables", "definition": "random(360,720,-360,-720,-1080)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["theta", "phi", "rev"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The exact value of $\\sin(\\var{theta}^\\circ)$ is&nbsp;[[0]].</p>\n<p>The exact value of $\\cos(\\var{theta}^\\circ)$ is&nbsp;[[1]].</p>\n<p>If $\\tan(\\var{theta}^\\circ)$ is defined, what is its exact value? If it isn't, then enter <em>infinity</em> (even though it doesn't equal that).&nbsp; &nbsp;[[2]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sin({phi}/180*pi)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "cos({phi}/180*pi)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{if(phi=90 or phi=270,infinity,tan({phi}*pi/180))}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": [".", "sin", "cos", "tan", "cosec", "sec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "type": "exam", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Paul Emanuel", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2551/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values.svg", "/srv/numbas/media/question-resources/exact_values.svg"], ["question-resources/unit_circle_working.svg", "/srv/numbas/media/question-resources/unit_circle_working.svg"]]}