// Numbas version: finer_feedback_settings
{"name": "Robert'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": [{"functions": {}, "ungrouped_variables": ["bb11", "cc1", "cc0", "b0", "cc2", "check", "q0", "s2", "s0", "u1", "u0", "u2", "aa1", "aa0", "aa2", "cc01", "bb21", "a0", "c0", "p0", "r0", "t2", "t0", "t21", "c01", "c02", "bb2", "bb1", "bb0"], "name": "Robert's copy of Apply the sine rule", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "Solving triangles", "Triangle", "Two sides and an angle"], "advice": "<p>We use the Sine Rule to find $C$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $\\sin C=\\dfrac{c \\sin A}{a}=\\dfrac{\\var{c0}* \\var{s0}}{\\var{a0}}=\\var{c0*s0/a0}$. To find $C$ we need to calculate $\\sin^{-1} (\\var{c0*s0/a0})$, calculating the angle between $\\dfrac{\\pi}{2}$ and $\\pi$, so $C=\\var{cc01}$ (to 3 decimal places). [Using a calculator, $\\sin^{-1} (\\var{c0*s0/a0})$ returns $\\var{arcsin(u1)}$, which lies between $0$ and $\\dfrac{\\pi}{2}$, so the value we need is $\\pi-\\var{arcsin(u1)}=\\var{pi-arcsin(u1)}$.]</p>\n    <p>Since $A+B+C=\\pi$, we calculate $B=\\pi-A-C=\\var{bb11}$. To 3dp, this gives $\\var{bb21}$.</p>\n    <p>We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t21}}{\\var{s0}}=\\var{a0*t21/s0}$. The closest integer is then $\\var{b0}$. Note that this solution uses the 3dp value of $B$; the answer using $\\var{bb11}$ would give a slightly different long decimal value of $b$, but the integer value would be the same.</p>\n    <p></p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "<p>$A=\\var{AA0}$, $a=\\var{a0}$, $c=\\var{c0}$</p>\n        <p>Angle $C=$ [[0]]</p>\n        <p>Angle $B=$ [[1]]</p>\n        <p>Side length $b=$ [[2]]</p>", "marks": 0, "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": 3, "maxValue": "{cc0}+0.001", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "minValue": "{cc0}-0.001", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "scripts": {}, "precision": 3, "maxValue": "{bb2}+0.001", "strictPrecision": true, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "marks": 1, "minValue": "{bb2}-0.001", "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "scripts": {}, "maxValue": "{b0}", "minValue": "{b0}", "correctAnswerFraction": false, "showCorrectAnswer": true, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"showCorrectAnswer": true, "prompt": "<p>Use the Sine Rule to find $\\sin C$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$, and then find $C$. Note that $C$ lies between $\\dfrac{\\pi}{2}$ and $\\pi$, whereas $\\sin^{-1}$ returns a value between $0$ and $\\dfrac{\\pi}{2}$; this means that $C$ will be $\\pi -$ (value calculated using $\\sin^{-1}$). Remember that $A+B+C=\\pi$. Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$.</p>", "marks": 0, "type": "information", "scripts": {}}], "type": "gapfill"}], "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    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"bb11": {"definition": "pi-aa0-cc01", "templateType": "anything", "group": "Ungrouped variables", "name": "bb11", "description": ""}, "cc1": {"definition": "\n    //Angle C calculated from A,B\n    pi-aa0-bb0\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "cc1", "description": ""}, "cc0": {"definition": "\n    //The angle C\n    precround(arccos(r0),4)\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "cc0", "description": ""}, "b0": {"definition": "random(14..30)", "templateType": "anything", "group": "Ungrouped variables", "name": "b0", "description": ""}, "cc2": {"definition": "precround(cc1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc2", "description": ""}, "check": {"definition": "pi-AA0-BB0-CC0", "templateType": "anything", "group": "Ungrouped variables", "name": "check", "description": ""}, "q0": {"definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "templateType": "anything", "group": "Ungrouped variables", "name": "q0", "description": ""}, "s2": {"definition": "sin(aa2)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s0": {"definition": "sin(aa0)", "templateType": "anything", "group": "Ungrouped variables", "name": "s0", "description": ""}, "u1": {"definition": "c0*sin(aa0)/a0", "templateType": "anything", "group": "Ungrouped variables", "name": "u1", "description": ""}, "u0": {"definition": "sin(cc0)", "templateType": "anything", "group": "Ungrouped variables", "name": "u0", "description": ""}, "u2": {"definition": "sin(cc2)", "templateType": "anything", "group": "Ungrouped variables", "name": "u2", "description": ""}, "aa1": {"definition": "\n    //Angle A calculated from B,C\n    pi-bb0-cc0\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "aa1", "description": ""}, "aa0": {"definition": "\n    //The angle A\n    precround(arccos(p0),4)\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "aa0", "description": ""}, "aa2": {"definition": "precround(aa1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "aa2", "description": ""}, "cc01": {"definition": "precround(pi-arcsin(u1),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc01", "description": ""}, "bb21": {"definition": "precround(bb11,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "bb21", "description": ""}, "a0": {"definition": "random(10..25)", "templateType": "anything", "group": "Ungrouped variables", "name": "a0", "description": ""}, "c0": {"definition": "random(c01..c02)", "templateType": "anything", "group": "Ungrouped variables", "name": "c0", "description": ""}, "p0": {"definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p0", "description": ""}, "r0": {"definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "templateType": "anything", "group": "Ungrouped variables", "name": "r0", "description": ""}, "t2": {"definition": "sin(bb2)", "templateType": "anything", "group": "Ungrouped variables", "name": "t2", "description": ""}, "t0": {"definition": "sin(bb0)", "templateType": "anything", "group": "Ungrouped variables", "name": "t0", "description": ""}, "t21": {"definition": "sin(bb21)", "templateType": "anything", "group": "Ungrouped variables", "name": "t21", "description": ""}, "c01": {"definition": "ceil(sqrt(a0^2+b0^2))+1", "templateType": "anything", "group": "Ungrouped variables", "name": "c01", "description": ""}, "c02": {"definition": "floor(max(a0+0.9*b0,b0+0.9*a0))", "templateType": "anything", "group": "Ungrouped variables", "name": "c02", "description": ""}, "bb2": {"definition": "precround(bb1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "bb2", "description": ""}, "bb1": {"definition": "\n    //Angle B calculated from A,C\n    pi-aa0-cc0\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "bb1", "description": ""}, "bb0": {"definition": "\n    //The angle B\n    precround(arccos(q0),4)\n    ", "templateType": "anything", "group": "Ungrouped variables", "name": "bb0", "description": ""}}, "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", "description": "<p>A question testing the application of the Sine Rule when&nbsp;given two sides and an angle.&nbsp; In this question the triangle is obtuse and the first angle to be found is obtuse.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Robert Pullan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/766/"}]}]}], "contributors": [{"name": "Robert Pullan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/766/"}]}