// Numbas version: exam_results_page_options
{"name": "cormac's copy of Apply the sine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "advice": "<p>a) We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t0}}{\\var{s0}}=\\var{a0*t0/s0}$. The closest integer is then $\\var{b0}$.</p>\n    <p>Since $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC1}$. To 3dp, this gives $\\var{CC2}$.</p>\n    <p>We use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u2}}{\\var{s0}}=\\var{a0*u2/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC1}$ would give a slightly different long decimal value of $c$, but the integer value would be the same.</p>\n    <p>b) We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t3}}{\\var{u3}}=\\var{c3*t3/u3}$. The closest integer is then $\\var{b3}$.</p>\n    <p>Since $A+B+C=\\pi$, we calculate $A=\\pi-B-C=\\var{AA4}$. To 3dp, this gives $\\var{AA5}$.</p>\n    <p>We use the Sine Rule to find $a$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $a=\\dfrac{c \\sin A}{\\sin C}=\\dfrac{\\var{c3}* \\var{s5}}{\\var{u3}}=\\var{c3*s5/u3}$. The closest integer is then $\\var{a3}$. Note that this solution uses the 3dp value of $A$; the answer using $\\var{AA4}$ would give a slightly different long decimal value of $a$, but the integer value would be the same.</p>", "name": "cormac's copy of Apply the sine rule", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Two questions testing the application of the Sine Rule when given two angles and a side. In this question, the triangle is always acute.</p>"}, "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["c4", "s3", "cc0", "temp2", "temp1", "b0", "cc3", "b3", "u2", "q0", "q3", "c0", "cc5", "s2", "s0", "cc1", "u0", "u3", "cc2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "x2", "c31", "c32", "a0", "a3", "bb0", "s5", "c3", "c2", "c1", "x1", "c02", "x4", "x5", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "c5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "check1"], "statement": "<p>Suppose that $\\Delta ABC$ is a triangle with all interior angles $&lt; \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used (i.e. the sides opposite angles A,B and C are called a,b and c respectively).&nbsp;</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>", "variables": {"x4": {"templateType": "anything", "group": "Ungrouped variables", "name": "x4", "definition": "abs(a3^2-b3^2)", "description": ""}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "name": "s3", "definition": "sin(AA3)", "description": ""}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "name": "cc0", "definition": "precround(arccos(r0),4)", "description": ""}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "name": "aa5", "definition": "precround(AA4,3)", "description": ""}, "temp2": {"templateType": "anything", "group": "Ungrouped variables", "name": "temp2", "definition": "b0-temp1", "description": ""}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "name": "bb0", "definition": "precround(arccos(q0),4)", "description": ""}, "c5": 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"definition": "pi-aa0-bb0", "description": ""}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "name": "t0", "definition": "sin(bb0)", "description": ""}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "name": "bb5", "definition": "precround(BB4,3)", "description": ""}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "name": "bb2", "definition": "precround(bb1,3)", "description": ""}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "name": "u5", "definition": "sin(CC5)", "description": ""}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "name": "c32", "definition": "ceil(min(a3,b3)*0.05)", "description": ""}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "name": "r3", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": ""}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "name": "q3", "definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "description": ""}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "name": "u3", "definition": "sin(CC3)", "description": ""}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "name": "x2", "definition": "a0^2+b0^2", "description": ""}, "a0": {"templateType": "anything", "group": "Ungrouped variables", "name": "a0", "definition": "random(10..25)", "description": ""}, "cc4": {"templateType": "anything", "group": "Ungrouped variables", "name": "cc4", "definition": "pi-AA3-BB3", "description": ""}, "cc2": {"templateType": "anything", "group": "Ungrouped variables", "name": "cc2", "definition": "precround(cc1,3)", "description": ""}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "name": "bb1", "definition": "pi-aa0-cc0", "description": ""}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "name": "aa3", "definition": "precround(arccos(p3),4)", "description": ""}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "name": "a3", "definition": "random(7..20)", "description": ""}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "name": "p3", "definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "description": ""}, "c02": {"templateType": "anything", "group": "Ungrouped variables", "name": "c02", "definition": "ceil(min(a0,b0)*0.05)", "description": ""}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "name": "check2", "definition": "pi-AA3-BB3-CC3", "description": ""}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "name": "c2", "definition": "floor(sqrt(x2))", "description": ""}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "name": "r0", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": ""}, "s0": {"templateType": "anything", "group": "Ungrouped variables", "name": "s0", "definition": "sin(aa0)", "description": ""}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "name": 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variables", "name": "b3", "definition": "random(7..20)", "description": ""}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "name": "p0", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": ""}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "name": "c3", "definition": "random(c4..c5 except 0)", "description": ""}, "x1": {"templateType": "anything", "group": "Ungrouped variables", "name": "x1", "definition": "abs(a0^2-b0^2)", "description": ""}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "name": "c0", "definition": "random(c1..c2 except 0)", "description": ""}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "name": "q0", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": ""}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "name": "u0", "definition": "sin(cc0)", "description": ""}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "name": "t2", "definition": "sin(bb2)", "description": ""}}, "parts": [{"showCorrectAnswer": true, "stepsPenalty": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "prompt": "<p>$A=\\var{AA0}$, $B=\\var{BB0}$, $a=\\var{a0}$</p>\n        <p>Side length $b=$ [[0]]</p>\n        <p>Angle $C=$ [[1]]</p>\n        <p>Side length $c=$ [[2]]</p>", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "gaps": [{"correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "maxValue": "{b0}", "mustBeReduced": false, "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "minValue": "{b0}", "type": "numberentry", "marks": 1, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}, {"showCorrectAnswer": true, "marks": 1, "unitTests": [], "allowFractions": false, "scripts": {}, "precisionType": "dp", "correctAnswerStyle": "plain", "strictPrecision": true, "precision": 3, "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "extendBaseMarkingAlgorithm": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{CC2}+0.001", "mustBeReduced": false, "correctAnswerFraction": false, "variableReplacements": [], "showFeedbackIcon": true, "minValue": "{CC2}-0.001", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "customMarkingAlgorithm": ""}, {"correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "maxValue": "{c0}", "mustBeReduced": false, "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "minValue": "{c0}", "type": "numberentry", "marks": 1, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}], "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "steps": [{"showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.</p>", "type": "information", "marks": 0, "variableReplacements": [], "customMarkingAlgorithm": "", "unitTests": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}]}, {"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "prompt": "\n      <p>$B=\\var{BB3}$, $C=\\var{CC3}$, $c=\\var{c3}$</p>\n        <p>Side length $b=$ [[0]]</p>\n        <p>Angle $A=$ [[1]]</p>\n        <p>Side length $a=$ [[2]]</p>\n      \n        \n      ", "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "gaps": [{"correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "maxValue": "{b3}", "mustBeReduced": false, "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "minValue": "{b3}", "type": "numberentry", "marks": 1, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}, {"showCorrectAnswer": true, "marks": 1, "unitTests": [], "allowFractions": false, "scripts": {}, "precisionType": "dp", "correctAnswerStyle": "plain", "strictPrecision": true, "precision": 3, "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "extendBaseMarkingAlgorithm": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{AA5}+0.001", "mustBeReduced": false, "correctAnswerFraction": false, "variableReplacements": [], "showFeedbackIcon": true, "minValue": "{AA5}-0.001", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "customMarkingAlgorithm": ""}, {"correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "unitTests": [], "maxValue": "{a3}", "mustBeReduced": false, "allowFractions": false, "correctAnswerFraction": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "minValue": "{a3}", "type": "numberentry", "marks": 1, "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}], "extendBaseMarkingAlgorithm": true, "sortAnswers": false}], "functions": {}, "variable_groups": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}