// Numbas version: finer_feedback_settings
{"name": "Pythagorean Identity: find sin given cos", "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 sin given cos", "tags": ["pythagoras", "Trigonometry", "trigonometry"], "metadata": {"description": "<p>Use $\\cos^2\\theta+\\sin^2\\theta=1$ and/or an understanding on the unit circle definitions to determine $\\sin\\theta$ given $\\cos\\theta$ and the quadrant theta is in.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>You are told&nbsp;that $\\cos\\theta=\\simplify{{a}/sqrt({cc})}$.</p>", "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 value of $\\cos\\theta$&nbsp;given in the question we have that $\\left(\\simplify{{a}/sqrt({cc})}\\right)^2+\\sin^2\\theta=1$. We solve for $\\sin\\theta$:</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{aa}/{cc}}+\\sin^2\\theta$</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;\">$\\sin^2\\theta$</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;\">$\\sin^2\\theta$</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;\">$\\sin\\theta$</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 $\\cos\\theta$ is <span data-jme-visible=\"a&gt;0\">positive, we know that $\\theta$ is in the first or fourth&nbsp;quadrant (since $\\cos\\theta$ is the $x$ coordinate). If $\\theta$ is in the first quadrant, then (since $\\sin\\theta$ is the $y$ coordinate) $\\sin\\theta$ is positive, that is, $\\sin\\theta=\\simplify[fractionNumbers]{sqrt({diff})}$. However, if $\\theta$ is in the fourth&nbsp;quadrant, then (since $\\sin\\theta$ is the $y$ coordinate) $\\sin\\theta$ is negative, that is, $\\sin\\theta=-\\simplify[fractionNumbers]{sqrt({diff})}$.&nbsp;</span>&nbsp;<span data-jme-visible=\"a&lt;0\">negative, we know that $\\theta$ is in the second or third quadrant (since $\\cos\\theta$ is the $x$ coordinate).&nbsp;If $\\theta$ is in the second&nbsp;quadrant, then (since $\\sin\\theta$ is the $y$ coordinate) $\\sin\\theta$ is positive, that is, $\\sin\\theta=\\simplify[fractionNumbers]{sqrt({diff})}$. However, if $\\theta$ is in the third&nbsp;quadrant, then (since $\\sin\\theta$ is the $y$ coordinate) $\\sin\\theta$ is&nbsp;negative, that is, $\\sin\\theta=-\\simplify[fractionNumbers]{sqrt({diff})}$.&nbsp;</span>&nbsp;</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "'\\$\\\\simplify{sqrt({cc})\\}$'", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-11..11 except 0)", "description": "", "templateType": "anything", "can_override": false}, "aa": {"name": "aa", "group": "Ungrouped variables", "definition": "a^2", "description": "", "templateType": "anything", "can_override": false}, "cc": {"name": "cc", "group": "Ungrouped variables", "definition": "random(144..624 except [169,196,225,256,289,324,361,400,441,484,529,576])", "description": "", "templateType": "anything", "can_override": false}, "diff": {"name": "diff", "group": "Ungrouped variables", "definition": "1-aa/cc", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "aa", "cc", "diff"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>If&nbsp;$\\theta$ is in the <span data-jme-visible=\"a&gt;0\"><strong>first</strong></span><strong><span data-jme-visible=\"a&lt;0\">second</span></strong> quadrant, then the exact value of $\\sin\\theta$ 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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "sqrt({diff})", "answerSimplification": "fractionNumbers", "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", "sec", "cosec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "sortAnswers": false}, {"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": "<p>If&nbsp;$\\theta$ is in the <span data-jme-visible=\"a&lt;0\"><strong>third</strong></span><span data-jme-visible=\"a&gt;0\"><strong>fourth</strong></span>&nbsp;quadrant, then the exact value of $\\sin\\theta$ 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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-sqrt({diff})", "answerSimplification": "fractionNumbers", "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", "sec", "cosec", "cot"], "showStrings": true, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}