// Numbas version: finer_feedback_settings
{"name": "5.3 Right-angled triangles", "extensions": [], "custom_part_types": [], "resources": [["question-resources/PictureTriangle_0.png", "PictureTriangle_0.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "5.3 Right-angled triangles", "tags": [], "metadata": {"description": "<p>Finding the lengths and angles within a right-angled triangle using: pythagoras theorem, SOHCAHTOA and principle of angles adding up to 180 degrees.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Using the right-angled triangle pictured below (not to scale), find the specified side lengths or angles using trigonometric ratios and the given values.&nbsp;</p>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/PictureTriangle_0.png\" width=\"387\" height=\"265\"/></p>\n<p style=\"text-align: left;\">Before starting, check that your calculator is in degrees mode by verifying that $\\cos{60} = \\frac{1}{2}$.</p>", "advice": "<p>Each angle on a triangle is formed between two sides.</p>\n<p>The longest side of a right-angled triangle is called the <strong>hypotenuse</strong>, and is&nbsp;<em>opposite</em> the right angle.</p>\n<p>For each of the angles formed at the hypotenuse, we refer to the other side where the angle is formed as the <strong>adjacent</strong> side, and the remaining side of the triangle as the&nbsp;<strong>opposite</strong>.</p>\n<p>Note that we&nbsp;<strong>cannot</strong> apply SOHCAHTOA or Pythagoras' Theorem to triangles which are not right-angled.</p>\n<p>&nbsp;</p>\n<p><strong>a)</strong></p>\n<p>One of the ways you can approach this style of question is by using SOHCAHTOA.</p>\n<p>This can be written more visually as</p>\n<h4>\\[\\text{S}^\\text{O}_\\text{H}\\space\\text{C}^\\text{A}_\\text{H}\\space\\text{T}^\\text{O}_\\text{A}\\]</h4>\n<p>It represents each trigonometric function and what they are equivalent to.</p>\n<p>Written out in full, we would have:</p>\n<p><b>SIN:&nbsp;</b>opposite / hypotenuse</p>\n<p><b>COS:&nbsp;</b>adjacent /&nbsp;hypotenuse</p>\n<p><b>TAN:&nbsp;</b>opposite / adjacent</p>\n<p>&nbsp;</p>\n<p>You will need to first use Pythagoras's theorem to evaluate the hypotenuse, which will be&nbsp;</p>\n<p>\\[ \\sqrt{\\var{b[6]}^2 + \\var{a[6]}^2} = \\sqrt{\\var{b[6]^2 + a[6]^2}}\\]</p>\n<div data-jme-visible=\"(a[6]^2+b[6]^2) = 25 or (a[6]^2+b[6]^2) = 169\">\n<p>which simplifies to $\\var{sqrt(a[6]^2+b[6]^2)}$</p>\n</div>\n<p>and now we can write out our ratios for each of sin, cos, and tan - noting that <span data-jme-visible=\"a_xory=&quot;x&quot;\">for $x$, the opposite side is $\\var{b[6]}$ and the adjacent side is $\\var{a[6]}$.</span> <span data-jme-visible=\"a_xory=&quot;y&quot;\">for $y$, the opposite side is $\\var{a[6]}$ and the adjacent side is $\\var{b[6]}$.</span></p>\n<p></p>\n<p><strong>b)</strong>&nbsp;</p>\n<p>In this part we can use the fact that the two angles other than the right angle will add together to 90 degrees.</p>\n<p>&nbsp;</p>\n<p><strong>c)</strong></p>\n<p>In this question we have been given the values of <span data-jme-visible=\"c_xory=&quot;x&quot;\">$x$</span> <span data-jme-visible=\"c_xory=&quot;y&quot;\">$y$</span> and <span data-jme-visible=\"c_aorb=&quot;a&quot;\">$a$</span> <span data-jme-visible=\"c_aorb=&quot;b&quot;\">$b$</span> as well as $c$, the hypotenuse.&nbsp;&nbsp;</p>\n<p>Hence we can calculate</p>\n<p><span data-jme-visible=\"c_aorb=&quot;b&quot; and c_xory=&quot;x&quot;\">\\[ \\sin(x) = \\frac{b}{c} \\to c \\sin (x) = b \\]</span></p>\n<p><span data-jme-visible=\"c_aorb=&quot;a&quot; and c_xory=&quot;y&quot;\">\\[ \\sin(y) = \\frac{a}{c} \\to c \\sin (y) = a \\]</span></p>\n<p><span data-jme-visible=\"c_aorb=&quot;a&quot; and c_xory=&quot;x&quot;\">\\[ \\cos(x) = \\frac{a}{c} \\to c 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"a_cos", "a_tan"]}], "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": "<p>Find sin/cos/tan of angle <strong><span data-jme-visible=\"a_xory=&quot;x&quot;\">x</span> <span data-jme-visible=\"a_xory=&quot;y&quot;\">y</span></strong> in the triangle given the following values of $a$ and $b$.</p>\n<p>$a=\\var{a[6]}$</p>\n<p>$b=\\var{b[6]}$</p>\n<p><strong>Note</strong>: write your ratios exactly, using fractions and \"sqrt(..)\" for square roots if required.</p>\n<p><span data-jme-visible=\"a_xory=&quot;y&quot;\">$\\sin(y) = $</span> <span 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