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"bb4", "check2", "bb2", "bb1", "check1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Apply the sine rule", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb01}-0.001", "maxValue": "{bb01}+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": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{CC21}-0.001", "maxValue": "{CC21}+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": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{c0}", "maxValue": "{c0}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$A=\\var{AA0}$, $a=\\var{a0}$, $b=\\var{b0}$</p>\n        <p>Angle $B=$ [[0]]</p>\n        <p>Angle $C=$ [[1]]</p>\n        <p>Side length $c=$ [[2]]</p>", "steps": [{"type": "information", "prompt": "<p>Use the Sine Rule to find $\\sin B$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$, and then find $B$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{aa31}-0.001", "maxValue": "{aa31}+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": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb51}-0.001", "maxValue": "{bb51}+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": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$a=\\var{a3}$, $C=\\var{CC3}$, $c=\\var{c3}$</p>\n      <p>Angle $A=$ [[0]]</p>\n      <p>Angle $B=$ [[1]]</p>\n      <p>Side length $b=$ [[2]]</p>", "showCorrectAnswer": true, "marks": 0}], "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 as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).</p>\n    <p style=\"margin: 0px;\">Given the following angle and two side lengths, use the Sine Rule to determine the other side length and two angles. Write down the side length as a whole number and the angles (in radians) as decimals 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_1.PNG\"/></p>", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "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 acute triangles with side lengths $a,b,c$. I need $|a^2-b^2|&lt;c^2&lt;a^2+b^2$ along with corresponding conditions on $a,b$. In fact the conditions $a^2-b^2&lt;c^2&lt;a^2+b^2$ and $b^2-a^2&lt;c^2&lt;a^2+b^2$ imply also the corresponding conditions on $a,b$. Thus the design of the question involves choosing $a,b$ and then choosing $c$ to meet the required condition. The integer $c$ is chosen randomly between the ceiling of $\\sqrt{|a^2-b^2|}$ and the floor of $\\sqrt{a^2+b^2}$. The first is no greater than the second because $\\max\\{a,b\\}$ lies between them; if $a=b$, then $\\sqrt{a^2+b^2} &gt; 1$. The range of values for $a$ and $b$ may be changed according to taste without invalidating the question, but questions arise about accuracy. My calculations suggest that values of $a,b,c$ between 5 and 100 are safe, but I have been more conservative than that.</p>\n                \t\t                <p>The second part tests the ability to apply the same principles as the first part but with a different orientation to the triangle: the first part seeks $b,C,c$ whereas the second seeks $b,A,a$.</p>\n                \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Two questions testing the application of the Sine Rule when given two sides and an angle. In this question, the triangle is always acute and one of the given side lengths is opposite the given angle.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a) We use the Sine Rule to find $B$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $\\sin B=\\dfrac{b \\sin A}{a}=\\dfrac{\\var{b0}* \\var{s0}}{\\var{a0}}=\\var{b0*s0/a0}$. To find $B$ we need to calculate $\\sin^{-1} (\\var{b0*s0/a0})$, calculating the angle between $0$ and $\\dfrac{\\pi}{2}$, so $B=\\var{bb01}$ (to 3 decimal places).</p>\n  <p>Since $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC11}$. To 3dp, this gives $\\var{CC21}$.</p>\n  <p>We use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u21}}{\\var{s0}}=\\var{a0*u21/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC11}$ 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 $A$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $\\sin A=\\dfrac{a \\sin C}{c}=\\dfrac{\\var{a3}* \\var{u3}}{\\var{c3}}=\\var{a3*u3/c3}$. To find $A$ we need to calculate $\\sin^{-1} (\\var{a3*u3/c3})$, calculating the angle between $0$ and $\\dfrac{\\pi}{2}$, so $A=\\var{aa31}$ (to 3 decimal places).</p>\n  <p>Since $A+B+C=\\pi$, we calculate $B=\\pi-A-C=\\var{bb41}$. To 3dp, this gives $\\var{bb51}$.</p>\n  <p>We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t51}}{\\var{u3}}=\\var{c3*t51/u3}$. The closest integer is then $\\var{b3}$. Note that this solution uses the 3dp value of $B$; the answer using $\\var{bb41}$ would give a slightly different long decimal value of $b$, but the integer value would be the same.</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/"}]}