// Numbas version: exam_results_page_options {"name": "Sine, cosine, and area rules", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Sine, cosine, and area rules", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The following questions refer to this triangle diagram.

{plotgraph(a,b,c,angleA,angleB,angleC)}

", "advice": "

For part (a), the sine rule tells us that $\\frac{\\sin(A)}{a} = \\frac{\\sin(B)}{b}$,

so here $B = \\sin^{-1}\\left(\\frac{\\var{b}\\times\\sin(\\var{angleA}^{\\circ})}{\\var{a}}\\right)$.

For part (b), the angle sum of a triangle tells us that angle $C$ is given by $180^{\\circ}-A-B = 180^{\\circ} - \\var{angleA}^{\\circ} - \\var{precround(angleB,1)}^{\\circ}$.

Then we can use the area formula $|ABC| = \\frac{1}{2} a b \\sin(C)$, which in this case is $\\frac{1}{2} \\times \\var{a} \\times \\var{b} \\times \\sin(\\var{precround(angleC,1)}^{\\circ})$.

For part (c), the cosine rule tells us that $c^2 = a^2 + b^2 - 2ab\\cos(C)$,

so here $c = \\sqrt{\\var{a}^2+\\var{b}^2 - 2 \\times \\var{a} \\times \\var{b} \\times \\cos(\\var{precround(angleC,1)}^{\\circ})}$.

It's also possible to use the sine rule or area rule to find $AB$ here.

", "rulesets": {}, "extensions": ["jsxgraph"], "variables": {"asinA": {"name": "asinA", "group": "Ungrouped variables", "definition": "a/(sin(radians(angleA)))\n", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "sqrt(a^2+b^2-2*a*b*cos(radians(angleC)))", "description": "

asinA*sin(radians(angleC))

", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(5..a-1)", "description": "

asinA*sin(radians(angleB))

", "templateType": "anything"}, "angleC": {"name": "angleC", "group": "Ungrouped variables", "definition": "180-angleA-angleB", "description": "

random(30..60 except angleB)

", "templateType": "anything"}, "angleA": {"name": "angleA", "group": "Ungrouped variables", "definition": "random(75..120 except 89..91)", "description": "

180-(angleB+angleC)

", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(6..30)", "description": "", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "0.5*a*b*sin(radians(angleC))", "description": "", "templateType": "anything"}, "angleB": {"name": "angleB", "group": "Ungrouped variables", "definition": "180*arcsin(b/asinA)/pi\n", "description": "

anglerandom(30..60)

", "templateType": "anything"}}, "variablesTest": {"condition": "area>0", "maxRuns": 100}, "ungrouped_variables": ["a", "angleA", "asinA", "b", "angleB", "angleC", "c", "area"], "variable_groups": [], "functions": {"plotgraph": {"parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["angleA", "number"], ["angleB", "number"], ["angleC", "number"]], "type": "html", "language": "javascript", "definition": "// This functions plots a triangle based on two angles and a length\n\n//Function to convert angles to radians\nfunction toRadians (angle) {\n return angle * (Math.PI / 180);\n}\n\n//Calculate height of triangle\nvar h = b*Math.sin(toRadians(angleC))\n\n//Set text and graph offsets to appear uniform\nvar xOffset = Math.ceil(a/10)\nvar yOffset = Math.ceil(a/2+xOffset-h/2)\n//Consider removing scale once all adjusted correctly\nvar offset = 1;\nvar textHeight =15;\nvar scale = xOffset/2;\nvar scaleOffset = offset*scale;\nvar scaleText = textHeight*scale;\n// This functions plots two dimensioned lines \n// Max and min x and y values for the axis.\nvar x_min = 0;\nvar x_max = a+2*xOffset;\nvar y_min = 0;\nvar y_max = a+2*xOffset;\n\n//Browser compatibility\nJXG.Options.text.display = 'internal';\n//Use MathJax for LaTeX display\nJXG.Options.text.useMathJax = true;\n\n// Make the JSXGraph board.\nvar div = Numbas.extensions.jsxgraph.makeBoard(\n '500px',\n '500px',\n {\n boundingBox: [0,y_max,x_max,0],\n//Change to false after testing\n axis: false,\n }\n);\n\n// div.board is the object created by JSXGraph, which you use to manipulate elements\nvar board = div.board; \n\n/*\n//Dummy text for testing variables, remove after testing\ntempText = board.create('text',[(xOffset),(h+yOffset),\nfunction () {return 'A is ' + angleA + ' B is ' + angleB + ' C is ' + angleC}]);\ntempText2 = board.create('text',[(xOffset),(yOffset/2),\nfunction () {return 'a is ' + a + ' b is ' + b + ' c is ' + c}]);\n*/\n\n//Draw three points \nvar pA = board.create('point', [(b*Math.cos(toRadians(angleC))+xOffset), (b*Math.sin(toRadians(angleC))+yOffset)], \n//Make fixed:true after testing\n {size:0, fixed:true,\n label:{offset: [0,10]}});\nvar pB = board.create('point', [(a+xOffset), (yOffset)],\n {size:0, fixed:true,\n label:{offset: [10,-10]}});\nvar pC = board.create('point', [(xOffset), (yOffset)],\n {size:0, fixed:true,\n label:{offset: [-10,-10]}});\n\n\n//Draw a line between them\nvar AB = board.create('line',[pA,pB],{fixed:false, straightFirst:false, straightLast:false, strokeWidth: 1});\nvar BC = board.create('line',[pC,pB],{fixed:false, straightFirst:false, straightLast:false, strokeWidth: 1});\nvar AC = board.create('line',[pC,pA],{fixed:false, straightFirst:false, straightLast:false, strokeWidth: 1});\n\n//Draw angle and label\n//Used in other versions of this questions\n/*\nvar ABC = board.create('nonreflexangle', [pA,pB,pC], {type:'sector', orthoType:'square', orthoSensitivity:0.4, \nradius:function() { return scale;}\n });\nvar ABCLabel = ABC.label.setText(function () {\n var angle = 180.0 * ABC.Value() / Math.PI;\n if ((angle > 90.4) || (angle < 89.6)) {\n return ''+angle.toFixed(0) + '\\u00B0';\n } else {\n return '';\n }\n});\nABCLabel.setAttribute({anchorX:'middle'});\n*/\n\nvar CAB = board.create('nonreflexangle', [pC,pA,pB], {type:'sector', orthoType:'square', orthoSensitivity:0.4, \nradius:function() { return scale;} \n});\nvar CABLabel = CAB.label.setText(function () {\n var angle = 180.0 * CAB.Value() / Math.PI;\n if ((angle > 90.4) || (angle < 89.6)) {\n return ''+angle.toFixed(0) + '\\u00B0';\n } else {\n return '';\n }\n});\nCABLabel.setAttribute({anchorX:'middle'});\n\n/*\nvar BCA = board.create('nonreflexangle', [pB,pC,pA], {type:'sector', orthoType:'square', orthoSensitivity:0.4, \nradius:function() { return scale;} \n});\nvar BCALabel = BCA.label.setText(function () {\n var angle = 180.0 * BCA.Value() / Math.PI;\n if ((angle > 90.4) || (angle < 89.6)) {\n return ''+angle.toFixed(0) + '\\u00B0';\n } else {\n return '';\n }\n});\nCABLabel.setAttribute({anchorX:'middle'});\n*/\n\n//Not used in this version\n/*\ntextAB = board.create('text', \n [function () {return (pA.X() + pB.X())/2},\n function () {return ((pA.Y() + pB.Y())/2)+(scaleOffset/2)},\n function () {return +pA.Dist(pB).toFixed(2) + ' '}],\n {fontSize:15, anchorX:'middle'});\n*/\n\n//Set up dimension labels to be properly aligned\ntextBC = board.create('text', \n [function () {return (pB.X() + pC.X())/2},\n function () {return ((pB.Y() + pC.Y())/2)-scaleOffset},\n function () {return +pB.Dist(pC).toFixed(0) + ' '}],\n {fontSize:15, anchorX:'middle'});\n\n\n//Not used in this version\ntextAC = board.create('text', \n [function () {return (pA.X() + pC.X())/2},\n function () {return ((pA.Y() + pC.Y())/2)+(scaleOffset/2)},\n function () {return +pA.Dist(pC).toFixed(2) + ' '}],\n {fontSize:15, anchorX:'middle'});\n\n/*\nvar tABRot = board.create('transform', \n [function () {return AB.getAngle()}, \n function () {return (pA.X() + pB.X())/2}, \n function () {return (pA.Y() + pB.Y())/2}],\n {type:'rotate'});\n*/\n\n//Set up rotation transform for label\n\nvar tBCRot = board.create('transform', \n [function () {return BC.getAngle()}, \n function () {return (pB.X() + pC.X())/2}, \n function () {return (pB.Y() + pC.Y())/2}],\n {type:'rotate'});\n\n\n//Hide this for final version\nvar tACRot = board.create('transform', \n [function () {return AC.getAngle()}, \n function () {return (pA.X() + pC.X())/2}, \n function () {return (pA.Y() + pC.Y())/2}],\n {type:'rotate'});\n\n\n//Perform text rotations and update\n//tABRot.bindTo(textAB);\n//Hide this for final version\ntBCRot.bindTo(textBC); \n//Hide this for final version\ntACRot.bindTo(textAC);\nboard.update();\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

What is the approximate value of angle $B$ in degrees?

", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Hint: Try using the sine rule.

"}], "minValue": "angleB", "maxValue": "angleB", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

What is the approximate area of the triangle $ABC$?

Hint: Before using the area formula, find angle $C$ by using your answer to part (a).

"}], "minValue": "area", "maxValue": "area", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

What is the approximate length of the side $AB$?

Hint: Try using the cosine rule at angle $C$. You can also check your answer by using the sine rule or the area formula.

"}], "minValue": "c", "maxValue": "c", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}]}], "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Mark Hodds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/510/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}