// Numbas version: exam_results_page_options
{"name": "Pythagorean Identity: find cos given sin", "extensions": [], "custom_part_types": [], "resources": [["question-resources/right_angled_triangle_4erLEm1.svg", "/srv/numbas/media/question-resources/right_angled_triangle_4erLEm1.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Pythagorean Identity: find cos given sin", "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Use $\\cos^2\\theta+\\sin^2\\theta=1$ and/or an understanding on the unit circle definitions to determine $\\cos\\theta$ given $\\sin\\theta$ and the quadrant theta is in.</p>"}, "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "<p>You are told&nbsp;that $\\sin\\theta=\\simplify{{a}/sqrt({cc})}$.</p>", "variable_groups": [], "functions": {}, "variables": {"cc": {"name": "cc", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(144..624 except [169,196,225,256,289,324,361,400,441,484,529,576])"}, "diff": {"name": "diff", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "1-aa/cc"}, "aa": {"name": "aa", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "a^2"}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "'\\$\\\\simplify{sqrt({cc})\\}$'"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(-11..11 except 0)"}}, "tags": [], "parts": [{"variableReplacements": [], "gaps": [{"variableReplacements": [], "notallowed": {"showStrings": true, "strings": [".", "sin", "cos", "tan", "sec", "cosec", "cot"], "partialCredit": 0, "message": ""}, "answerSimplification": "fractionNumbers", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "sqrt({diff})", "unitTests": [], "scripts": {}, "vsetRange": [0, 1], "marks": 1, "vsetRangePoints": 5, "valuegenerators": [], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}], "type": "gapfill", "useCustomName": false, "prompt": "<p>If&nbsp;$\\theta$ is in the <span data-jme-visible=\"a&gt;0\"><strong>first</strong></span>&nbsp;<span data-jme-visible=\"a&lt;0\"><strong>fourth</strong></span> quadrant, then the exact value of $\\cos\\theta$ is&nbsp;[[0]].</p>\n<p><strong>Note:</strong>&nbsp;In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example, if your answer is $\\frac{\\sqrt{5}}{\\sqrt{17}}$, then enter <strong>sqrt(5)/sqrt(17)</strong></p>", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}, {"variableReplacements": [], "gaps": [{"variableReplacements": [], "notallowed": {"showStrings": true, "strings": [".", "sin", "cos", "tan", "sec", "cosec", "cot"], "partialCredit": 0, "message": ""}, "answerSimplification": "fractionNumbers", "variableReplacementStrategy": "originalfirst", "checkVariableNames": false, "showCorrectAnswer": true, "showPreview": true, "answer": "-sqrt({diff})", "unitTests": [], "scripts": {}, "vsetRange": [0, 1], "marks": 1, "vsetRangePoints": 5, "valuegenerators": [], "type": "jme", "useCustomName": false, "customName": "", "failureRate": 1, "checkingAccuracy": 0.001, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "customMarkingAlgorithm": ""}], "type": "gapfill", "useCustomName": false, "prompt": "<p>If&nbsp;$\\theta$ is in the <span data-jme-visible=\"a&gt;0\"><strong>second</strong></span>&nbsp;<span data-jme-visible=\"a&lt;0\"><strong>third</strong></span>&nbsp;quadrant, then the exact value of $\\cos\\theta$ is&nbsp;[[0]].</p>\n<p><strong>Note:</strong>&nbsp;In this question we require you input your answer without decimals and without entering the words sin, cos or tan. For example, if your answer is $\\frac{\\sqrt{5}}{\\sqrt{17}}$, then enter <strong>sqrt(5)/sqrt(17)</strong></p>", "unitTests": [], "customName": "", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The equation of the unit circle is \\[x^2+y^2=1\\]</p>\n<p>This equation is a consequence of Pythagoras' Theorem, $a^2+b^2=c^2$,&nbsp;on a triangle in the unit circle with hypotenuse 1.</p>\n<p>Given that $\\cos\\theta$ is the $x$ coordinate of a point on the unit circle, and that $\\sin\\theta$ is the $y$ coordinate of the same point, by substitution we have the following Pythagorean identity \\[\\cos^2\\theta+\\sin^2\\theta=1\\]</p>\n<p>where $\\cos^2\\theta$ and $\\sin^2\\theta$ is the notation used to represent $(\\cos\\theta)^2$ and&nbsp;$(\\sin\\theta)^2$ respectively.</p>\n<p>Using the vaue of $\\sin\\theta$&nbsp;given in the question we have that $\\cos^2\\theta+\\left(\\simplify{{a}/sqrt({cc})}\\right)^2=1$. We solve for $\\sin\\theta$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\cos^2\\theta+\\simplify{{aa}/{cc}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$1$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\cos^2\\theta$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$1-\\simplify{{aa}/{cc}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\cos^2\\theta$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify[fractionNumbers]{{diff}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\cos\\theta$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\pm\\simplify[fractionNumbers]{sqrt({diff})}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Because we are told&nbsp;that $\\sin\\theta$ is <span data-jme-visible=\"a&gt;0\">positive, we know that $\\theta$ is in the first or second quadrant (since $\\sin\\theta$ is the $y$ coordinate). If $\\theta$ is in the first quadrant, then (since $\\cos\\theta$ is the $x$ coordinate) $\\cos\\theta$ is positive, that is, $\\cos\\theta=\\simplify[fractionNumbers]{sqrt({diff})}$. However, if $\\theta$ is in the second quadrant, then (since $\\cos\\theta$ is the $x$ coordinate) $\\cos\\theta$ is negative, that is, $\\cos\\theta=-\\simplify[fractionNumbers]{sqrt({diff})}$.&nbsp;</span> <span data-jme-visible=\"a&lt;0\">negative, we know that $\\theta$ is in the third or fourth quadrant (since $\\sin\\theta$ is the $y$ coordinate).&nbsp;If $\\theta$ is in the fourth&nbsp;quadrant, then (since $\\cos\\theta$ is the $x$ coordinate) $\\cos\\theta$ is positive, that is, $\\cos\\theta=\\simplify[fractionNumbers]{sqrt({diff})}$. However, if $\\theta$ is in the third&nbsp;quadrant, then (since $\\cos\\theta$ is the $x$ coordinate) $\\cos\\theta$ is&nbsp;negative, that is, $\\cos\\theta=-\\simplify[fractionNumbers]{sqrt({diff})}$.&nbsp;</span>&nbsp;</p>\n<p></p>", "rulesets": {}, "ungrouped_variables": ["a", "c", "aa", "cc", "diff"], "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/"}]}