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Cosine Rule when given three side lengths. In this question, the triangle is always acute. A secondary application is finding the area of a triangle.</p>"}, "name": "cormac's copy of Apply the cosine rule", "extensions": [], "variable_groups": [], "advice": "<p>(a) Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Therefore</p>\n  <p>\\[\\cos A =\\dfrac{\\var{b0}^2+\\var{c0}^2-\\var{a0}^2}{2 \\times \\var{b0} \\times \\var{c0}}=\\dfrac{\\var{b0^2+c0^2-a0^2}}{\\var{2 *b0*c0}}\\]</p>\n  <p>\\[=\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)}\\]</p>\n  <p>and so $A=\\cos^{-1}(\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)})=\\var{aa0}$.</p>\n  <p></p>\n  <p>Similarly $\\cos B =\\dfrac{a^2+c^2-b^2}{2ac}$ and $\\cos C =\\dfrac{a^2+b^2-c^2}{2ab}$. So</p>\n  <p>\\[\\cos B =\\dfrac{\\var{a0}^2+\\var{c0}^2-\\var{b0}^2}{2 \\times \\var{a0} \\times \\var{c0}}=\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)}\\]</p>\n  <p>and so $B=\\cos^{-1}(\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)})=\\var{bb0}$.</p>\n  <p>\\[\\cos C =\\dfrac{\\var{a0}^2+\\var{b0}^2-\\var{c0}^2}{2 \\times \\var{a0} \\times \\var{b0}}=\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)}\\]</p>\n  <p>and so $C=\\cos^{-1}(\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)})=\\var{cc0}$.</p>\n  <p>(b) We can use any of the formulae &nbsp;$\\dfrac{1}{2}ac \\sin B$, $\\dfrac{1}{2}bc \\sin A$ or $\\dfrac{1}{2}ab \\sin C$&nbsp;for the area. For example&nbsp;</p>\n  <p>\\[\\dfrac{1}{2}bc \\sin A = \\dfrac{1}{2} \\times \\var{b0} \\times \\var{c0} \\times \\sin \\var{aa0}\\]</p>\n  <p>\\[=\\dfrac{1}{2} \\times \\var{b0 * c0} \\times \\var{sin(aa0)}=\\var{Area}\\]</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "parts": [{"showCorrectAnswer": true, "prompt": "<p>$a=\\var{a0}$, $b=\\var{b0}$, $c=\\var{c0}$</p>\n      <p>Angle $A=$ [[0]]</p>\n      <p>Angle $B=$ [[1]]</p>\n      <p>Angle $C=$ [[2]]</p>", "customMarkingAlgorithm": "", "gaps": [{"strictPrecision": false, "customMarkingAlgorithm": "", "precisionType": "dp", "marks": 2, "variableReplacements": [], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "precision": "3", "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "allowFractions": false, "scripts": {}, "maxValue": "{aa0}+0.001", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "{aa0}-0.001", "mustBeReduced": false, "unitTests": [], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true}, {"strictPrecision": true, "customMarkingAlgorithm": "", "precisionType": "dp", "marks": 2, "variableReplacements": [], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "precision": 4, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "allowFractions": false, "scripts": {}, "maxValue": "{bb0}+0.0001", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "{bb0}-0.0001", "mustBeReduced": false, "unitTests": [], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true}, {"strictPrecision": true, "customMarkingAlgorithm": "", "precisionType": "dp", "marks": 2, "variableReplacements": [], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "precision": 4, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "allowFractions": false, "scripts": {}, "maxValue": "{cc0}+0.0001", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "{cc0}-0.0001", "mustBeReduced": false, "unitTests": [], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true}], "marks": 0, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "unitTests": [], "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "prompt": "<p>The area is [[0]]</p>", "customMarkingAlgorithm": "", "gaps": [{"strictPrecision": true, "customMarkingAlgorithm": "", "precisionType": "dp", "marks": 2, "variableReplacements": [], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "precision": 3, "correctAnswerStyle": "plain", "precisionPartialCredit": 0, "allowFractions": false, "scripts": {}, "maxValue": "{Area}+0.001", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "{Area}-0.001", "mustBeReduced": false, "unitTests": [], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true}], "marks": 0, "variableReplacements": [], "steps": [{"unitTests": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "prompt": "<p>The area uses any of the formulae $\\dfrac{1}{2}ac \\sin B$, $\\dfrac{1}{2}bc \\sin A$ or $\\dfrac{1}{2}ab \\sin C$.</p>", "scripts": {}, "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "type": "information", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "stepsPenalty": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true}], "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).</p>\n<p style=\"margin: 0px;\">Given the following three side lengths, determine the three angles using the Cosine Rule. Write down the angles (in radians) as decimals to 4dp. [Before submitting answers, you can check that the sum of the three angles is $\\pi$.]&nbsp;</p>\n<p style=\"margin: 0px;\"></p>\n<p style=\"margin: 0px;\">Also calculate the area of the triangle, giving your answer as a decimal to 3dp.</p>\n<p style=\"margin: 0px;\"></p>\n<p style=\"margin: 0px;\"></p>", "preamble": {"js": "", "css": ""}, "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/"}]}