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"description": "", "name": "c32"}, "temp2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b0-temp1", "description": "", "name": "temp2"}}, "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"], "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": "{a0}", "maxValue": "{a0}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$A=\\var{AA0}$, $b=\\var{b0}$, $c=\\var{c0}$</p>\n        <p>Side length $a=$ [[0]]</p>", "steps": [{"type": "information", "prompt": "<p>Use the Cosine Rule to find $a$: $a^2=b^2+c^2-2bc \\cos A$.</p>", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$B=\\var{BB3}$, $a=\\var{a3}$, $c=\\var{c3}$</p>\n        <p>Side length $b=$ [[0]]</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 two sides and an angle, determine the third side length. Write down the side length as a whole number.</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", "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 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 Cosine Rule when given two sides and an angle. In these questions, the triangle is always acute and both of the given side lengths are adjacent to the given angle.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a) Use the Cosine Rule to find $a$: $a^2=b^2+c^2-2bc \\cos A$. &nbsp;&nbsp;</p>\n    <p>\\[a^2=\\var{b0}^2+\\var{c0}^2-2 \\times \\var{b0}\\times\\var{c0} \\times \\cos (\\var{aa0})=\\var{b0^2}+\\var{c0^2}-\\var{2*b0*c0} \\times \\var{cos (aa0)}\\]</p>\n    <p>\\[=\\var{b0^2+c0^2-2*b0*c0* cos (aa0)}.\\]</p>\n    <p>Hence $a=\\sqrt{\\var{b0^2+c0^2-2*b0*c0* cos (aa0)}}=\\var{sqrt(b0^2+c0^2-2*b0*c0* cos (aa0))}$. To the nearest integer, this is $\\var{a0}$.</p>\n    <p>b) Use the Cosine Rule to find $b$: $b^2=a^2+c^2-2ac \\cos B$. &nbsp;&nbsp;</p>\n    <p>\\[b^2=\\var{a3}^2+\\var{c3}^2-2 \\times \\var{a3}\\times\\var{c3} \\times \\cos (\\var{bb3})=\\var{a3^2}+\\var{c3^2}-\\var{2*a3*c3} \\times \\var{cos (bb3)}\\]</p>\n    <p>\\[=\\var{a3^2+c3^2-2*a3*c3* cos (bb3)}.\\]</p>\n    <p>Hence $b=\\sqrt{\\var{a3^2+c3^2-2*a3*c3* cos (bb3)}}=\\var{sqrt(a3^2+c3^2-2*a3*c3* cos (bb3))}$. To the nearest integer, this is $\\var{b3}$.</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/"}]}