// 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    ", "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    ", "description": "", "name": "aa0"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //Angle A calculated from B,C\n    pi-bb0-cc0\n    ", "description": "", "name": "aa1"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //The angle C\n    precround(arccos(r0),4)\n    ", "description": "", "name": "cc0"}, "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"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa2)", "description": "", "name": "s2"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n    //Angle B calculated from A,C\n    pi-aa0-cc0\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"}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "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    ", "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"}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc0)", "description": "", "name": "u0"}, "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"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "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", "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": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{c0}", "maxValue": "{c0}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$a=\\var{a0}$, $b=\\var{b0}$, $C=\\var{CC0}$</p>\n        <p>Side length $c=$ [[0]]</p>", "steps": [{"type": "information", "prompt": "<p>Use the Cosine Rule to find $c$: $c^2=a^2+b^2-2ab \\cos C$. Take care over the fact that $\\cos(\\var{cc0})$ is negative.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n    <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 two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.</p>\n  <p style=\"margin: 0px;\"></p>\n  <p style=\"margin: 0px;\"></p>\n  <p style=\"margin: 0px;\"><img src=\"resources/images/Triangle_2.PNG\"/></p>\n    ", "tags": ["checked2015", "cosine rule", "Cosine Rule", "SFY0001", "Solving triangles", "Triangle", "Two sides and an angle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n        \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                    <p></p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>A question testing the application of the Cosine Rule when given two sides and an angle. In this question, the triangle is always obtuse and both of the given side lengths are adjacent to the given angle (which is the obtuse angle).</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>Use the Cosine Rule to find $c$: $c^2=a^2+b^2-2ab \\cos C$. &nbsp;&nbsp;</p>\n    <p>\\[c^2=\\var{a0}^2+\\var{b0}^2-2 \\times \\var{a0}\\times\\var{b0} \\times \\cos (\\var{cc0})=\\var{a0^2}+\\var{b0^2}-\\var{2*a0*b0} \\times (\\var{cos (cc0)})\\]</p>\n    <p>\\[=\\var{a0^2+b0^2-2*a0*b0* cos (cc0)}.\\]</p>\n    <p>Hence $a=\\sqrt{\\var{a0^2+b0^2-2*a0*b0* cos (cc0)}}=\\var{sqrt(a0^2+b0^2-2*a0*b0* cos (cc0))}$. To the nearest integer, this is $\\var{c0}$.</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/"}]}