// Numbas version: exam_results_page_options
{"name": "Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //Angle C calculated from A,B\n    pi-aa0-bb0\n    \n    ", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(13..29)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a3^2+b3^2))+1", "description": "", "name": "c31"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB3)", "description": "", "name": "t3"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "check": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA0-BB0-CC0", "description": "", "name": "check"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c31..c32)", "description": "", "name": "c3"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "description": "", "name": "q3"}, "aa0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //The angle A\n    precround(arccos(p0),4)\n    \n    ", "description": "", "name": "aa0"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //Angle A calculated from B,C\n    pi-bb0-cc0\n    \n    ", "description": "", "name": "aa1"}, "area": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(b0*c0*s0/2,3)", "description": "", "name": "area"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //The angle C\n    precround(arccos(r0),4)\n    \n    ", "description": "", "name": "cc0"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "bb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-CC3", "description": "", "name": "bb4"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //Angle B calculated from A,C\n    pi-aa0-cc0\n    \n    ", "description": "", "name": "bb1"}, "c01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a0^2+b0^2))+1", "description": "", "name": "c01"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc2)", "description": "", "name": "u2"}, "cc3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r3),4)", "description": "", "name": "cc3"}, "bb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q3),4)", "description": "", "name": "bb3"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "description": "", "name": "p3"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa2)", "description": "", "name": "s2"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA3)", "description": "", "name": "s3"}, "c02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a0+0.9*b0,b0+0.9*a0))", "description": "", "name": "c02"}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": "", "name": "q0"}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC3)", "description": "", "name": "u3"}, "cc5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(CC4,3)", "description": "", "name": "cc5"}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //The angle B\n    precround(arccos(q0),4)\n    \n    ", "description": "", "name": "bb0"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c01..c02)", "description": "", "name": "c0"}, "t5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB5)", "description": "", "name": "t5"}, "a0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a0"}, "s0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa0)", "description": "", "name": "s0"}, "cc2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(cc1,3)", "description": "", "name": "cc2"}, "b0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(14..30)", "description": "", "name": "b0"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb2)", "description": "", "name": "t2"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb0)", "description": "", "name": "t0"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": "", "name": "r3"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc0)", "description": "", "name": "u0"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a3"}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC5)", "description": "", "name": "u5"}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(AA4,3)", "description": "", "name": "aa5"}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a3+0.9*b3,b3+0.9*a3))", "description": "", "name": "c32"}}, "ungrouped_variables": ["s3", "cc0", "b0", "cc3", "b3", "cc2", "check", "q0", "q3", "area", "cc5", "s2", "s0", "cc1", "u0", "u3", "u2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "c31", "c32", "a0", "a3", "s5", "c3", "c0", "c02", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "bb0"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Apply the cosine rule", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{aa0}-0.0001", "maxValue": "{aa0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb0}-0.0001", "maxValue": "{bb0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{cc0}-0.0001", "maxValue": "{cc0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>$a=\\var{a0}$, $b=\\var{b0}$, $c=\\var{c0}$</p>\n        <p>Angle $A=$ [[0]]</p>\n        <p>Angle $B=$ [[1]]</p>\n        <p>Angle $C=$ [[2]]</p>", "steps": [{"type": "information", "prompt": "<p>Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Then use $\\cos^{-1}$ to find $A$. Apply similar rules to find $B$ and $C$. Take care over $C$: $\\cos C &lt;0$, which means that $C$ lies between $\\dfrac{\\pi}{2}$ and $\\pi$.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{Area}-0.001", "maxValue": "{Area}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "<p>The area is [[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Suppose that $\\Delta ABC$ is a triangle with $C&gt; \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).</p>\n    <p style=\"margin: 0px;\">Given the following three side lengths, determine the three angles using the Cosine Rule. Write down the angles (in radians) as decimals to 4dp. [Before submitting answers, you can check that the sum of the three angles is $\\pi$.]&nbsp;</p>\n    <p style=\"margin: 0px;\"></p>\n    <p style=\"margin: 0px;\">Also calculate the area of the triangle, giving your answer as a decimal to 3dp.</p>\n    <p style=\"margin: 0px;\"></p>\n    <p style=\"margin: 0px;\"><img src=\"resources/images/Triangle_2.PNG\"/></p>", "tags": ["Area of a triangle", "checked2015", "cosine rule", "Cosine Rule", "SFY0001", "Solving triangles", "Three side lengths", "Triangle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n        \t\t    \t\t<p>I want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2&lt;c^2&lt;(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$.&nbsp;</p>\n        \t\t    \t\t                    <p></p>\n        \t\t    \n        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>A question testing the application of the Cosine Rule when given three side lengths. In this question, the triangle is always obtuse. A secondary application is finding the area of a triangle.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>(a) Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Therefore</p>\n    <p>\\[\\cos A =\\dfrac{\\var{b0}^2+\\var{c0}^2-\\var{a0}^2}{2 \\times \\var{b0} \\times \\var{c0}}=\\dfrac{\\var{b0^2+c0^2-a0^2}}{\\var{2 *b0*c0}}\\]</p>\n    <p>\\[=\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)}\\]</p>\n    <p>and so $A=\\cos^{-1}(\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)})=\\var{aa0}$.</p>\n    <p></p>\n    <p>Similarly $\\cos B =\\dfrac{a^2+c^2-b^2}{2ac}$ and $\\cos C =\\dfrac{a^2+b^2-c^2}{2ab}$. So</p>\n    <p>\\[\\cos B =\\dfrac{\\var{a0}^2+\\var{c0}^2-\\var{b0}^2}{2 \\times \\var{a0} \\times \\var{c0}}=\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)}\\]</p>\n    <p>and so $B=\\cos^{-1}(\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)})=\\var{bb0}$.</p>\n    <p>\\[\\cos C =\\dfrac{\\var{a0}^2+\\var{b0}^2-\\var{c0}^2}{2 \\times \\var{a0} \\times \\var{b0}}=\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)}\\]</p>\n    <p>and so $C=\\cos^{-1}(\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)})=\\var{cc0}$.</p>\n    <p>(b) We can use any of the formulae &nbsp;$\\dfrac{1}{2}ac \\sin B$, $\\dfrac{1}{2}bc \\sin A$ or $\\dfrac{1}{2}ab \\sin C$&nbsp;for the area. For example&nbsp;</p>\n    <p>\\[\\dfrac{1}{2}bc \\sin A = \\dfrac{1}{2} \\times \\var{b0} \\times \\var{c0} \\times \\sin \\var{aa0}\\]</p>\n    <p>\\[=\\dfrac{1}{2} \\times \\var{b0 * c0} \\times \\var{sin(aa0)}=\\var{Area}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}