// Numbas version: exam_results_page_options {"name": "Trigonometric functions of a triangle", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"prompt": "

Use $\\sin^{-1}$.

", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "prompt": "

Given $y=\\var{y1}, r=\\var{r1}$, find $\\theta$.

$\\theta=$ [[0]]

Use $\\cos^{-1}$.

Given $x=\\var{x2}, r=\\var{r2}$, find $\\theta$.

$\\theta=$ [[0]]

Use $\\tan^{-1}$.

Given $x=\\var{x3}, y=\\var{y3}$, find $\\theta$.

$\\theta=$ [[0]]

", "stepsPenalty": 1, "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "{t3}-0.001", "maxValue": "{t3}+0.001", "precision": 3, "unitTests": [], "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 2, "mustBeReducedPC": 0}], "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arcsin(y1/r1),3)", "name": "t1", "description": ""}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(y1..20)", "name": "r1", "description": ""}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arctan(y3/x3),3)", "name": "t3", "description": ""}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(4..10 except y1)", "name": "x2", "description": ""}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(x2..20)", "name": "r2", "description": ""}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(x2/r2),3)", "name": "t2", "description": ""}, "y1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(4..10)", "name": "y1", "description": ""}, "y3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..20)", "name": "y3", "description": ""}, "x3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..20)", "name": "x3", "description": ""}}, "ungrouped_variables": ["r1", "r2", "t2", "t3", "t1", "y1", "x2", "x3", "y3"], "name": "Trigonometric functions of a triangle", "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

The following parts refer to a right-angled triangle with hypotenuse length denoted by $r$ and horizontal and vertical side lengths denoted by $x$ and $y$. The angle $\\theta$ is as indicated in the diagram below. Each part gives two side lengths and you are asked to deduce the size of the angle $\\theta$ using appropriate inverse trigonometrical functions. Express your answers in radians, written as decimals to 3dp.

", "tags": ["arccos", "arcsin", "arctan", "checked2015", "Inverse trigonometrical functions", "Right-angled triangle", "triangle", "Triangle"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Questions on right-angled triangles asking for the calculation of angles using inverse-trigonometrical functions.

"}, "extensions": [], "advice": "

(a) $\\sin \\theta =\\dfrac{\\var{y1}}{\\var{r1}}$ so $\\theta= \\sin^{-1} \\left( \\dfrac{\\var{y1}}{\\var{r1}}\\right) = \\var{t1}$.

(b) $\\cos \\theta =\\dfrac{\\var{x2}}{\\var{r2}}$ so $\\theta= \\cos^{-1} \\left( \\dfrac{\\var{x2}}{\\var{r2}}\\right) = \\var{t2}$.

(c) $\\tan \\theta =\\dfrac{\\var{y3}}{\\var{x3}}$ so $\\theta= \\tan^{-1} \\left( \\dfrac{\\var{y3}}{\\var{x3}}\\right) = \\var{t3}$.

", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}