// Numbas version: finer_feedback_settings
{"name": "Exact values for sin, cos, tan (acute, radians)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/exact_values_radians.svg", "exact_values_radians_KF51lfI.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Exact values for sin, cos, tan (acute, radians)", "tags": [], "metadata": {"description": "<p>multiple choice testing sin, cos, tan&nbsp;of &nbsp;random(pi/6, pi/4, pi/3)&nbsp;radians</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>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 $\\dfrac{\\pi}{6}$, $\\dfrac{\\pi}{4}$ and $\\dfrac{\\pi}{3}$.</p>\n<p><img src=\"resources/question-resources/exact_values_radians.svg\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Since we are asked about $\\var{distheta}$ radians, we use the triangle on the <span data-jme-visible=\"theta&lt;&gt;45\">left</span><span data-jme-visible=\"theta=45\">right</span> and the mnemonic SOH CAH TOA to determine:</p>\n<p><span data-jme-visible=\"theta=30\">$\\sin\\left(\\var{distheta}\\right)=\\dfrac{\\text{Opposite}}{\\text{Hypotenuse}}=\\;\\;\\dfrac{1}{2}$</span><span data-jme-visible=\"theta=45\">$\\sin\\left(\\var{distheta}\\right)=\\dfrac{\\text{Opposite}}{\\text{Hypotenuse}}=\\dfrac{1}{\\sqrt{2}}$</span><span data-jme-visible=\"theta=60\">$\\sin\\left(\\var{distheta}\\right)=\\dfrac{\\text{Opposite}}{\\text{Hypotenuse}}=\\dfrac{\\sqrt{3}}{2}$</span></p>\n<p><span data-jme-visible=\"theta=30\">$\\cos\\left(\\var{distheta}\\right)=\\dfrac{\\text{Adjacent}}{\\text{Hypotenuse}}=\\dfrac{\\sqrt{3}}{2}$</span><span data-jme-visible=\"theta=45\">$\\cos\\left(\\var{distheta}\\right)=\\dfrac{\\text{Adjacent}}{\\text{Hypotenuse}}=\\dfrac{1}{\\sqrt{2}}$</span><span data-jme-visible=\"theta=60\">$\\cos\\left(\\var{distheta}\\right)=\\dfrac{\\text{Adjacent}}{\\text{Hypotenuse}}=\\;\\;\\dfrac{1}{2}$</span></p>\n<p><span data-jme-visible=\"theta=30\">$\\tan\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{\\text{Opposite}}{\\text{Adjacent}}\\;\\;=\\dfrac{1}{\\sqrt{3}}$</span><span data-jme-visible=\"theta=45\">$\\tan\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{\\text{Opposite}}{\\text{Adjacent}}\\;\\;=\\;\\;1$</span><span data-jme-visible=\"theta=60\">$\\tan\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{\\text{Opposite}}{\\text{Adjacent}}\\;\\;=\\sqrt{3}$</span></p>\n<hr/>\n<p>Alternatively, one can memorise the following table:&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: center;\">$\\dfrac{\\pi}{6}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\pi}{4}$</td>\n<td style=\"text-align: center;\">$\\dfrac{\\pi}{3}$</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>Since we are asked about $\\var{distheta}$ radians, we use the $\\var{distheta}$ column of the table to determine that:</p>\n<p><span data-jme-visible=\"theta=30\">$\\sin\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{1}{2}$</span><span data-jme-visible=\"theta=45\">$\\sin\\left(\\var{distheta}\\right)=\\dfrac{1}{\\sqrt{2}}$</span><span data-jme-visible=\"theta=60\">$\\sin\\left(\\var{distheta}\\right)=\\dfrac{\\sqrt{3}}{2}$</span></p>\n<p><span data-jme-visible=\"theta=30\">$\\cos\\left(\\var{distheta}\\right)=\\dfrac{\\sqrt{3}}{2}$</span><span data-jme-visible=\"theta=45\">$\\cos\\left(\\var{distheta}\\right)=\\dfrac{1}{\\sqrt{2}}$</span><span data-jme-visible=\"theta=60\">$\\cos\\left(\\var{distheta}\\right)=\\;\\;\\dfrac{1}{2}$</span></p>\n<p><span data-jme-visible=\"theta=30\">$\\tan\\left(\\var{distheta}\\right)=\\dfrac{1}{\\sqrt{3}}$</span><span data-jme-visible=\"theta=45\">$\\tan\\left(\\var{distheta}\\right)=\\;\\;1$</span><span data-jme-visible=\"theta=60\">$\\tan\\left(\\var{distheta}\\right)=\\sqrt{3}$</span></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"distheta": {"name": "distheta", "group": "Ungrouped variables", "definition": "if(theta=30,'\\$\\\\dfrac{\\\\pi}{6}\\$',if(theta=45,'\\$\\\\dfrac{\\\\pi}{4}\\$','\\$\\\\dfrac{\\\\pi}{3}\\$'))", "description": "<p>'\\$\\\\frac{\\pi}{6}\\$'</p>", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "Ungrouped variables", "definition": "random(30,45,60)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["theta", "distheta"], "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 $\\sin\\Large($$\\var{distheta}\\Large)$ is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showBlankOption": true, "showCellAnswerState": true, "choices": ["<p>$\\dfrac{1}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{2}}$</p>", "<p>$\\dfrac{\\sqrt{3}}{2}$</p>", "<p>$\\dfrac{1}{\\sqrt{3}}$</p>", "<p>$\\sqrt{3}$</p>", "<p>$1$</p>"], "matrix": ["if(theta=30,1,0)", "if(theta=45,1,0)", "if(theta=60,1,0)", 0, 0, 0], "distractors": ["", "", "", "", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The exact value of $\\cos\\Large($$\\var{distheta}\\Large)$&nbsp;is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showBlankOption": true, "showCellAnswerState": true, "choices": ["<p>$\\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>"], "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 $\\tan\\Large($$\\var{distheta}\\Large)$&nbsp;is</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showBlankOption": true, "showCellAnswerState": true, "choices": ["<p>$\\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>"], "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", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "resources": ["question-resources/exact_values_radians.svg"]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}