// Numbas version: exam_results_page_options {"name": "Trigonometry I: Right-angled triangles", "extensions": [], "custom_part_types": [], "resources": [["question-resources/triangle_6nKlln9.png", "/srv/numbas/media/question-resources/triangle_6nKlln9.png"], ["question-resources/triangle_2.png", "/srv/numbas/media/question-resources/triangle_2.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "x", "fa", "fh", "fa_1"], "name": "Trigonometry I: Right-angled triangles", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "

Each angle on a triangle is connected to two sides and is facing another.

\n

The longest side of the triangle is always the hypotenuse

\n

The other side that makes the angle is called the adjacent.

\n

The final side not connected in any way to the angle is called the opposite.

\n

(Note that we only call the longest side the hypotenuse if we have a right-angled triangle. We cannot apply SOHCAHTOA or Pythagoras' Theorem to triangles which are not right-angled, so we would use the sine or cosine rules instead.)

\n

For example, using the image below, you can see which side is denoted by each term from the highlighted angle's perspective.

\n

\n

\n

\n

\n

One of the ways you can approach this style of question is by using SOHCAHTOA.

\n

This can be written more visually as

\n

#### \$\\text{S}^\\text{O}_\\text{H}\\space\\text{C}^\\text{A}_\\text{H}\\space\\text{T}^\\text{O}_\\text{A}\$

\n

It represents each trigonometric function and what they are equivalent to.

\n

Written out in full, we would have:

\n

SIN: opposite / hypotenuse

\n

\n

\n

\n

\n

\n

\n

For example, $\\sin$ is represented by the first S.

\n

If we were given an angle, say of $30^\\circ$,

\n

$\\sin(30^\\circ)=\\frac{\\text{opposite}}{\\text{hypotenuse}}$

\n

Evaluating $\\sin(30^\\circ)=\\frac{1}{2}$, we now know that $\\frac{\\text{opposite}}{\\text{hypotenuse}}=\\frac{1}{2}$

\n

If we were given one of these sides, we would then be able to work out the other one by multiplying accordingly.

\n

\n

\n

\n

Similarly if we were given two sides, and told to work out a specific angle, we could.

\n

Referring to the image above, suppose we want to find the highlighted angle and we are given that the hypotenuse is equal to $5$ units, and the adjacent is $4$ units.

\n

We would determine from SOHCAHTOA that we need to use cos since we have the values for A and H.

\n

So, $\\cos(x)=\\frac{4}{5}$

\n

Hence, $x=\\cos^{-1}(\\frac{4}{5})=36.87^\\circ$

", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "

$b=\\var{a[5]}$

\n

$c=\\var{fa_1}$

\n

$a=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Since we have a right-angled triangle, use your knowledge of Pythagoras' theorem $a^2+b^2=c^2$ to work out the side length.

\n

Remember: c is always the hypotenuse (so in this case side a)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "sqrt(({a[5]}^2)+({fa_1}^2))", "strictPrecision": true, "minValue": "sqrt(({a[5]}^2)+({fa_1}^2))", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$x=\\var{x[4]}^\\circ$

\n

$y=$ [[0]]$^\\circ$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

For this question, use the principle that the interior angles in any triangle always add up to $180^\\circ$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "90-{x[4]}", "minValue": "90-{x[4]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$a=\\var{a[0]}$

\n

$x=\\var{x[0]}^\\circ$

\n

$c=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use CAH to answer this question as you know the values of the side adjacent to the angle and the hypotenuse.

\n

Cos(x)=A/H

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "{a[0]}cos({x[0]}pi/180)", "strictPrecision": false, "minValue": "{a[0]}cos({x[0]}pi/180)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$b=\\var{a[1]}$

\n

$y=\\var{x[1]}^\\circ$

\n

$c=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use TOA to answer this question as you know the values of the side opposite and adjacent to the angle.

\n

tan(x)=O/A

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "{a[1]}tan({x[1]}pi/180)", "strictPrecision": false, "minValue": "{a[1]}tan({x[1]}pi/180)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$a=\\var{a[2]}$

\n

$y=\\var{x[2]}^\\circ$

\n

$c=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use SOH to answer this question as you know the values of the side opposite to the angle and the hypotenuse.

\n

Sin(x)=O/H

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "{a[2]}sin({x[2]}pi/180)", "strictPrecision": false, "minValue": "{a[2]}sin({x[2]}pi/180)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$b=\\var{a[3]}$

\n

$x=\\var{x[3]}^\\circ$

\n

$a=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use SOH to answer this question as you know the values of the side opposite to the angle and the hypotenuse.

\n

Sin(x)=O/H

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "{a[3]}/sin({x[3]}pi/180)", "strictPrecision": false, "minValue": "{a[3]}/sin({x[3]}pi/180)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$b=\\var{a[4]}$

\n

$x=\\var{x[4]}^\\circ$

\n

$c=$ [[0]]

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

Use TOA to answer this question as you know the values of the side opposite and adjacent to the angle.

\n

tan(x)=O/A

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "{a[4]}/tan({x[4]}pi/180)", "strictPrecision": false, "minValue": "{a[4]}/tan({x[4]}pi/180)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$a=\\var{fh}$

\n

$c=\\var{fa}$

\n

$x=$ [[0]]$^\\circ$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

You need to use CAH for this question as you have the adjacent and the hypotenuse

\n

Cos(x)=A/H

\n

However because you are trying to find the angle this time, you must use the $cos^{-1}$ button on your calculator to convert your answer into an angle.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "

", "allowFractions": false, "variableReplacements": [], "maxValue": "arccos({fa}/{fh})*(180/pi)", "strictPrecision": false, "minValue": "arccos({fa}/{fh})*(180/pi)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "

$b=\\var{fa_1}$

\n

$c=\\var{fa}$

\n

$x=$ [[0]]$^\\circ$

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

You need to use TOA for this question as you have the opposite and the adjacent

\n

Tan(x)=O/A

\n

However because you are trying to find the angle this time, you must use the $tan^{-1}$ button on your calculator to convert your answer into an angle.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "arctan({fa_1}/{fa})*(180/pi)", "strictPrecision": true, "minValue": "arctan({fa_1}/{fa})*(180/pi)", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "

Using the right-angled triangle pictured below (not to scale), find the specified side lengths or angles using trigonometry and the given values. If this topic is new to you, clike Reveal answers for the first time and click Advice.

\n

\n

\n

Before starting, check that your calculator is in degrees mode by verifying that $\\cos{60} = \\frac{1}{2}$.

\n

(If your calculator does not give this answer, press SHIFT $\\rightarrow$ SETUP $\\rightarrow$ 3 to enter degrees mode if using a Casio $fx$ model calculator, otherwise refer to your calculator's manual.)

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "repeat(random(3..10),7)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "fa": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "fa", "description": ""}, "fa_1": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "fa_1", "description": ""}, "fh": {"definition": "random(6..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "fh", "description": ""}, "x": {"definition": "shuffle([25,30,40,50,60,70])", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}}, "metadata": {"description": "

Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}]}]}], "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}]}