// Numbas version: exam_results_page_options {"name": "Trigonometry: Sine Rule 2", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry: Sine Rule 2", "tags": [], "metadata": {"description": "

Given two side-lengths and an angle of a triangle, use the sine rule to calculate an unknown angle.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Find the angle $A$:

{geogebra_applet('https://www.geogebra.org/m/d4yzdkwg',defs)}

", "advice": "

Since we know two side-lengths and one of the their opposite angles, we can use the sine rule to find the unknown angle, $A$. The sine rules states:

\\[ \\frac{\\sin(A)}{a} = \\frac{\\sin(B)}{b} = \\frac{\\sin(C)}{c}, \\]

where $a$, $b$, and $c$ are the side-lengths, and $A$, $B$, and $C$ are the angles opposite the side with the same letter:

{geogebra_applet('https://www.geogebra.org/m/fjh4qce5')}

So, for the triangle

{geogebra_applet('https://www.geogebra.org/m/d4yzdkwg',defs)}

we know that side $a=\\var{sidear}$, side $b=\\var{sidebr}$, and angle $B=\\var{anglebr}^\\circ$. Substituting these values into the sine rule we can calculate the angle of $A$:

\\[ \\begin{split} \\frac{\\sin(A)}{\\var{sidear}}&\\,=\\frac{\\sin(\\var{anglebr})}{\\var{sidebr}}\\\\\\\\ \\sin(A) &\\,= \\frac{\\var{sidear}\\times\\sin(\\var{anglebr})}{\\var{sidebr}}\\\\\\\\ A &\\,= \\sin^{-1}\\left(\\frac{\\var{sidear}\\times\\sin(\\var{anglebr})}{\\var{sidebr}}\\right) \\\\ \\\\&\\,=\\var{aaar}\\,^\\circ \\text{ (2 d.p.).}\\end{split} \\]

", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n ['C',C]\n]", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "vector(8,0)", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "vector(random(-2..11 except [0,B[0]]),random(6..9))", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "vector(0,0)", "description": "", "templateType": "anything", "can_override": false}, "sidea": {"name": "sidea", "group": "Ungrouped variables", "definition": "abs(C-B)", "description": "", "templateType": "anything", "can_override": false}, "sideb": {"name": "sideb", "group": "Ungrouped variables", "definition": "abs(C)", "description": "", "templateType": "anything", "can_override": false}, "anglea": {"name": "anglea", "group": "Ungrouped variables", "definition": "arccos(dot(B,C)/(len(B)*len(C)))*(180/pi)", "description": "", "templateType": "anything", "can_override": false}, "angleb": {"name": "angleb", "group": "Ungrouped variables", "definition": "arccos(dot(B-C,B)/(len(B-C)*len(B)))*(180/pi)", "description": "", "templateType": "anything", "can_override": false}, "sidear": {"name": "sidear", "group": "Ungrouped variables", "definition": "precround(sidea,2)", "description": "", "templateType": "anything", "can_override": false}, "sidebr": {"name": "sidebr", "group": "Ungrouped variables", "definition": "precround(sideb,2)", "description": "", "templateType": "anything", "can_override": false}, "anglear": {"name": "anglear", "group": "Ungrouped variables", "definition": "precround(anglea,2)", "description": "", "templateType": "anything", "can_override": false}, "anglebr": {"name": "anglebr", "group": "Ungrouped variables", "definition": "precround(angleb,2)", "description": "", "templateType": "anything", "can_override": false}, "aaa": {"name": "aaa", "group": "Ungrouped variables", "definition": "arcsin(sidear*sin(anglebr*pi/180)/sidebr)*180/pi", "description": "", "templateType": "anything", "can_override": false}, "aaar": {"name": "aaar", "group": "Ungrouped variables", "definition": "precround(aaa,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["defs", "A", "B", "C", "sidea", "sideb", "anglea", "angleb", "sidear", "sidebr", "anglear", "anglebr", "aaa", "aaar"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$A=$[[0]] $^\\circ$

(Give your answer to 2 decimal places when necessary)

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{aaar}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}