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PSLE

PSLE Foundation Mathematics

It is given that \(\sin A=-\frac{1}{\sqrt{5}}\) and \(\tan B=-\frac{3}{4}\), where \(A\) and \(B\) are in the same quadrant. Without using a calculator, find the exact value of

Given that \(\sin A=q\), where \(A\) is an obtuse angle, express \(\tan^2 A\) in terms of \(q\).

Compound and Double-Angle Identities

Symmetry of Trigonometric Functions

Exact Trigonometric Values and the Unit Circle

2025 NASS PRELIMS P1 Q6

Trigonometry

<p>Given that&nbsp;\(\sin x=-\dfrac{5}{13}\) and \(180{}^\circ&lt;x&lt;270{}^\circ\), find the exact value of each of the following.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(\cos x\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(\tan x\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">\(\operatorname{cosec} x\)</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">\(\cos(-x)\)</div></li></ol>

Given that \(\sin x=-\dfrac{5}{13}\) and \(180{}^\circ<x<270{}^\circ\), find the exact value of each of the following.

Reciprocal Trigonometric Functions

Exact ratios from sine in the third quadrant

<p>Given that&nbsp;\(\tan x=-\dfrac{20}{21}\) and that&nbsp;\(\cos x\) and \(\tan x\) have opposite signs, find the exact value of each of the following.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(\sin(-x)\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(\tan(-x)\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">\(\cos(90{}^\circ-x)\)</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">\(\cot(90{}^\circ-x)\)</div></li></ol>

Given that \(\tan x=-\dfrac{20}{21}\) and that \(\cos x\) and \(\tan x\) have opposite signs, find the exact value of each of the following.

Negative-angle and complementary ratios from tangent

<p>Given that&nbsp;\(\sec x=\dfrac{5}{2}\) and that&nbsp;\(\sin x\) and \(\cos x\) have opposite signs, find the exact value of each of the following.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(\cos(\pi-x)\)</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(\sin(-x)\)</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">\(\sec\left(\dfrac{\pi}{2}-x\right)\)</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">\(\cot\left(\dfrac{\pi}{2}-x\right)\)</div></li></ol>

Given that \(\sec x=\dfrac{5}{2}\) and that \(\sin x\) and \(\cos x\) have opposite signs, find the exact value of each of the following.

Exact ratios from secant with a surd

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">\(A\) and \(B\) are acute angles such that&nbsp;\(\sin{(A - B)} = \frac{3}{8}\) and \(\sin{A}\cos{B} = \frac{5}{8} \).<br>Without using a calculator, find the value of \(\cos A \sin B\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Express \(2\sin 2\theta(\sec \theta - \tan \theta)\) as a quadratic expression in \(\sin \theta\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Use your answer to part <strong>(b)</strong> to find, for&nbsp;\(0 \le \theta \le 2\pi\), the exact solutions of the equation \(2\sin 2\theta(\sec \theta - \tan \theta) + 3 = 0\).<span class="inline-marks">[3]</span></div></li></ol>

Trigonometric Equations

Proving and Simplifying Trigonometric Identities

2024 FMSS PRELIMS P2 Q5

<p>If&nbsp;\(\mathrm{A}\) is an obtuse angle in a triangle and \( \sin{\mathrm{A}} = \frac{5}{13}\), calculate the exact value of \( \sin{2\mathrm{A}}\).</p>

If \(\mathrm{A}\) is an obtuse angle in a triangle and \( \sin{\mathrm{A}} = \frac{5}{13}\), calculate the exact value of \( \sin{2\mathrm{A}}\).

<p>Given that&nbsp;\( \sin{x} = \frac{1}{3}\),  where&nbsp;\(x\) is an acute angle, find the <strong>exact</strong> value of</p>
<ol class="abc-list"><li><div class="list-item-paragraph">\(\cos{x}\)</div></li><li><div class="list-item-paragraph">\( \cos{2x}\)</div></li></ol>

Given that \( \sin{x} = \frac{1}{3}\),  where \(x\) is an acute angle, find the **exact** value of

<p>The following diagram shows a triagle \(\mathrm{ABC}\), where&nbsp;\(\mathrm{ABC}\), where&nbsp;\(\angle \mathrm{ACB}\) is \(90^{\circ}\), \(\mathrm{AB} = 3, \mathrm{AC} = 2\) and \(\angle \mathrm{BAC}\) is \(\theta\)</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1778486973348-rpyy0xmwwur.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Nzg0ODY5NzMzNDgtcnB5eTB4bXd3dXIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwODUzMSwiZXhwIjoxNzkzNTA4ODMxfQ.WpvvaXg9gRurc55uKAmIjum6l-YXAlrUfo7fMhAX6vo" style="max-width: min(150px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(\sin{\theta} = \frac{\sqrt{5}}{3}\).</div></li><li><div class="list-item-paragraph">Show that&nbsp;\(\sin {2\theta} = \frac{4\sqrt{5}}{9}\).</div></li><li><div class="list-item-paragraph">Find the <strong>exact</strong> value of \(\cos {2\theta}\).</div></li></ol>

The following diagram shows a triagle \(\mathrm{ABC}\), where \(\mathrm{ABC}\), where \(\angle \mathrm{ACB}\) is \(90^{\circ}\), \(\mathrm{AB} = 3, \mathrm{AC} = 2\) and \(\angle \mathrm{BAC}\) is \(\theta\)

Basics of Trigonometric Ratios

Let \( f(x) = \sin^3{x} + \cos^3{x}\tan{x}\), \(\frac{\pi}{2} < x < \pi\).

<p>In the diagram, \(\triangle ABC\) and \(\triangle ACE\) are right-angled triangles. \(AC = 12\) cm, \(BC = 13\) cm, \(EC = 2AB\) and \(\angle BCE = x\) in degrees.</p>
<div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjQwMCwiaGVpZ2h0IjoyMDAsImVsZW1lbnRzIjpbeyJpZCI6ImJmODhlYTgyLThiMjItNDNkNC05NDU2LTQ5ZDU0ZWJhOWU5MyIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbLTIuNTMsMi40MV0sInBvaW50MiI6Wy0zLjMzLDAuNF0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiIwYWIxZWRkYS01YjJkLTRlMDQtYWU0Mi1lZjgzMTU3NTMyODYiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiMTNjbSIsInBvaW50MSI6Wy0zLjMzLDAuNDJdLCJwb2ludDIiOlsyLjk5LC0wLjIxXSwibGFiZWxPZmZzZXQiOlstMjAsMTBdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiMmVhNGYwMTMtOWJlYS00M2UzLTliM2YtMDBkOThjNjM2NGQ3IiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IjEyY20iLCJwb2ludDEiOlstMi41MywyLjQxXSwicG9pbnQyIjpbMi45OSwtMC4yMV0sImxhYmVsT2Zmc2V0IjpbMCwxMF0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJmYjg1Yzc4Ny1mODIyLTQwODUtYWJkNC1hNTI2MGY2NmNjOTYiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6Wy0yLjUzLDIuNDFdLCJwb2ludDIiOlstMi40NywtMC4yNF0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJkZWZhNDc4Ni0yOWQwLTQyNzYtYWM0Yi1jMTFiMzc1YTg0MDAiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6Wy0yLjQ3LC0wLjI3XSwicG9pbnQyIjpbMi45OSwtMC4yMV0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiJmZWViMjg4Ny02NGU4LTRjMjQtOGU1ZS0xOTY2ZTI0MDdiMzMiLCJ0eXBlIjoic2VnbWVudCIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50MSI6Wy0yLjY0LDIuMTJdLCJwb2ludDIiOlstMi4zMywxLjk1XSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6IjM4YjFiOGYzLWM0MWUtNDJkMi1iNjBmLTU0NTgzZWU4YTk1MiIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbLTIuMzMsMS45NV0sInBvaW50MiI6Wy0yLjE4LDIuMjNdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiMjRlYTgyZDctMWZhNi00ZDAxLTgyMTQtMDI1NzUyYWQyNTUyIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlstMi40NywtMC4yNF0sInBvaW50MiI6Wy0yLjQ3LC0wLjI0XSwic3Ryb2tlV2lkdGgiOjEuNX0seyJpZCI6IjcwZDY2NjRjLWZjZTItNGNkYi1hN2I1LTM5OTAzZmZmMDEyNyIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwicG9pbnQxIjpbLTIuNDcsMC4wNV0sInBvaW50MiI6Wy0yLjE4LDAuMDVdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiOWFjMjdlOGItMDRmZC00MzBiLWFhZmEtMzg4MzBjNDY1YTQxIiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJwb2ludDEiOlstMi4xNiwwLjA4XSwicG9pbnQyIjpbLTIuMTYsLTAuMjZdLCJzdHJva2VXaWR0aCI6MS41fSx7IngiOi0yLjY0LCJ5IjoyLjc1LCJpZCI6IjRlMDhlNGZiLWYzYWMtNDBjYS04ZGVmLTJmMmZlNmYwNDZmMSIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJBIn0seyJ4IjotMi44MSwieSI6LTAuNDQsImlkIjoiOWU0OGIzMWQtM2UyYy00NDJhLWFlOGEtNDQ5YzFkODljMTcyIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IkUifSx7IngiOi0yLjg0LCJ5IjowLjEsImlkIjoiZDg1NzgyMDQtM2JmMS00MzNhLWEwNTAtMDgxZTcwNjQzOWE1IiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IkQifSx7IngiOi0zLjcsInkiOjAuMjgsImlkIjoiNGRhY2RmY2EtNzRkMC00OTY3LTk1ZDUtOTJlOWE3NGVjMTYzIiwidHlwZSI6InRleHQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IkIifSx7IngiOjMuMTksInkiOi0wLjI5LCJpZCI6IjBjNzA1ZWVkLWJiMzktNDg5Yy1iYzAwLTBiZjljZDVmZGI2MiIsInR5cGUiOiJ0ZXh0IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJDIn1dLCJzaG93WEF4aXMiOmZhbHNlLCJzaG93WUF4aXMiOmZhbHNlLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstNSwyLDUsMF0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsInNob3dBeGlzTGFiZWxzIjpmYWxzZSwic2hvd05hdmlnYXRpb24iOmZhbHNlLCJrZWVwQXNwZWN0UmF0aW8iOnRydWUsInhBeGlzTGFiZWxPZmZzZXQiOlsyLC0xMV0sInlBeGlzTGFiZWxPZmZzZXQiOlstMTQsNV0sIm9yaWdpbkxhYmVsT2Zmc2V0IjpbLTUsLTJdfQ==" data-width="400" data-height="200" style="width:400px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact value of \(\cos x\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, show that the length of \(BD\) is \(\frac{13\sqrt{11}}{12 + \sqrt{11}}\) cm.</span><span class="inline-marks">[2]</span></div></li></ol>

In the diagram, \(\triangle ABC\) and \(\triangle ACE\) are right-angled triangles. \(AC = 12\) cm, \(BC = 13\) cm, \(EC = 2AB\) and \(\angle BCE = x\) in degrees.

Hence, show that the length of \(BD\) is \(\frac{13\sqrt{11}}{12 + \sqrt{11}}\) cm.

NJC IP4 Further Trigonometric Identities Tutorial Q2

<p>Given that&nbsp;\(\sin A\) and \(\cos B\) are in the same quadrant such that&nbsp;\(\sin A = \frac{3}{5}\) and \(\cos B = -\frac{5}{13}\).</p>
<p>Find, without using a calculator, the value of</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">\(\cos(A - B)\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">\(\cos \frac{A}{2}\).<span class="inline-marks">[3]</span></div></li></ol>

Given that \(\sin A\) and \(\cos B\) are in the same quadrant such that \(\sin A=\frac{3}{5}\) and \(\cos B=-\frac{5}{13}\).

Find, without using a calculator, the value of

2025 AHS PRELIMS P1 Q10

<p>In the diagram, \(\angle ABC = 45^\circ\), \(\angle ACD = 60^\circ\), \(BC = 1\) cm, \(CD = 2\) cm. Find \(\angle BAD\).</p>
<div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770788238553-jvhma27prh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3ODgyMzg1NTMtanZobWEyN3ByaC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA4NTMxLCJleHAiOjE3OTM1MDg4MzF9.rJBl3PBsxf6oWGBH7WPWXuOI06ZkqFaLM-UN_EMHBG8" style="max-width: min(150px, 100%); height: auto; display: block;" alt="Diagram for the geometry problem showing a triangle with internal line segments and angles." class="alignright custom-width" /></figure></div>

In the diagram, \(\angle ABC = 45^\circ\), \(\angle ACD = 60^\circ\), \(BC = 1\) cm, \(CD = 2\) cm. Find \(\angle BAD\).

IP2U7 HOT Q1

<p>Given that&nbsp;\(\cos(A - B) = \frac{4}{5}\), \(\sin A \sin B = \frac{1}{10}\) and that&nbsp;\(A\) and \(B\) are both acute angles, calculate without using a calculator, the value of each of the following.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1770095809590-qtp4g8tpr"><div class="list-item-paragraph">\(\cos A \cos B\)</div></li></ol><ol class="i-list" start="3"><li data-list-item-id="item-1770095811885-dprpfw0fk"><div class="list-item-paragraph">\(\cos(A + B)\)</div></li></ol><ol class="i-list" start="5"><li data-list-item-id="item-1770095814352-2ehql8293"><div class="list-item-paragraph">\(\sin\left(\frac{A + B}{2}\right)\)</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li data-list-item-id="item-1770095835495-e7njpfhlh"><div class="list-item-paragraph">\(\tan A \tan B\)</div></li></ol><ol class="i-list" start="4"><li data-list-item-id="item-1770095838105-4ockoksez"><div class="list-item-paragraph">\(\sin 2A \sin 2B\)</div></li></ol><ol class="i-list" start="6"><li data-list-item-id="item-1770095840287-5wjm8zx5m"><div class="list-item-paragraph">\(\sec\left(\frac{A + B}{2}\right)\)</div></li></ol></div></div></div>

Given that \(\cos(A - B) = \frac{4}{5}\), \(\sin A \sin B = \frac{1}{10}\) and that \(A\) and \(B\) are both acute angles, calculate without using a calculator, the value of each of the following.

<p>Given that&nbsp;\(\tan A = -\frac{5}{12}\), \(\cos B = \frac{3}{5}\) and that&nbsp;\(A\) and \(B\) are in the same quadrant, for&nbsp;\(0^\circ &lt; A &lt; 360^\circ\), \(0^\circ &lt; B &lt; 360^\circ\), calculate without using a calculator, the value of each of the following.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1770095562503-60qaimkdq"><div class="list-item-paragraph">\(\sin (A - B)\)</div></li></ol><ol class="i-list" start="3"><li data-list-item-id="item-1770095565173-zdj6e22ba"><div class="list-item-paragraph">\(\cos (2A + B)\)</div></li></ol><ol class="i-list" start="5"><li data-list-item-id="item-1770095567033-qwb9tnm0q"><div class="list-item-paragraph">\(\sin \frac{1}{2} A\)</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li data-list-item-id="item-1770095524764-njjcebs7y"><div class="list-item-paragraph">\(\tan (A + B)\)</div></li></ol><ol class="i-list" start="4"><li data-list-item-id="item-1770095527344-a45ftnrr3"><div class="list-item-paragraph">\(\cot (A + 2B)\)</div></li></ol><ol class="i-list" start="6"><li data-list-item-id="item-1770095530217-ubgu9vasy"><div class="list-item-paragraph">\(\cos \frac{1}{2} A\)</div></li></ol></div></div></div>

Given that \(\tan A = -\frac{5}{12}\), \(\cos B = \frac{3}{5}\) and that \(A\) and \(B\) are in the same quadrant, for \(0^\circ < A < 360^\circ\), \(0^\circ < B < 360^\circ\), calculate without using a calculator, the value of each of the following.

<p>Given that&nbsp;\(ABC\) is a triangle and \(\tan \frac{A}{2} = \frac{1}{4}\), without using a calculator, find</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1770095323419-7amj51vdw"><div class="list-item-paragraph">\(\tan A\)</div></li></ol><ol class="i-list" start="3"><li data-list-item-id="item-1770095326509-5sgbhvagg"><div class="list-item-paragraph">\(\sin (B + C)\)</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li data-list-item-id="item-1770095336350-5r6ttxh40"><div class="list-item-paragraph">\(\cos A\)</div></li></ol><ol class="i-list" start="4"><li data-list-item-id="item-1770095339425-h02q8wv7d"><div class="list-item-paragraph">\(\cos (B + C)\)</div></li></ol></div></div></div>

Given that \(ABC\) is a triangle and \(\tan \frac{A}{2} = \frac{1}{4}\), without using a calculator, find

<p>Without using a calculator, express each of the following in exact form.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1770094974529-mqvbw3fcs"><div class="list-item-paragraph">\(\sin 105^\circ\)</div></li></ol><ol class="i-list" start="3"><li data-list-item-id="item-1770094976839-s2kzo9ad8"><div class="list-item-paragraph">\(\tan 15^\circ\)</div></li></ol><ol class="i-list" start="5"><li data-list-item-id="item-1770094978924-ew5sv9rm5"><div class="list-item-paragraph">\(\sec 285^\circ\)</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li data-list-item-id="item-1770095001874-z7b792t06"><div class="list-item-paragraph">\(\cos 75^\circ\)</div></li></ol><ol class="i-list" start="4"><li data-list-item-id="item-1770095004319-7p055wils"><div class="list-item-paragraph">\(\cot 195^\circ\)</div></li></ol><ol class="i-list" start="6"><li data-list-item-id="item-1770095007274-nah5r740q"><div class="list-item-paragraph">\(\operatorname{cosec} 165^\circ\)</div></li></ol></div></div></div>

Without using a calculator, express each of the following in exact form.

<p>It is given that&nbsp;\(\cos \frac{9\pi}{8} = -\frac{m}{n}\). Express each of the following in terms of \(m\) and \(n\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(\cos \frac{9\pi}{8} + \cos \frac{5\pi}{4} + \cos \frac{11\pi}{8}\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\sin \frac{9\pi}{8} + \sin \frac{5\pi}{4} + \sin \frac{11\pi}{8}\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\cot\left(-\frac{\pi}{8}\right) + \cot\left(-\frac{\pi}{4}\right) + \cot\left(-\frac{3\pi}{8}\right)\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\text{cosec}\hspace{0.5em} \frac{17\pi}{8} + \text{cosec}\hspace{0.5em} \frac{9\pi}{4} + \text{cosec}\hspace{0.5em} \frac{19\pi}{8}\)</span></div></li></ol>

It is given that \(\cos \frac{9\pi}{8} = -\frac{m}{n}\). Express each of the following in terms of \(m\) and \(n\).

\(\cos \frac{9\pi}{8} + \cos \frac{5\pi}{4} + \cos \frac{11\pi}{8}\)

\(\sin \frac{9\pi}{8} + \sin \frac{5\pi}{4} + \sin \frac{11\pi}{8}\)

\(\cot\left(-\frac{\pi}{8}\right) + \cot\left(-\frac{\pi}{4}\right) + \cot\left(-\frac{3\pi}{8}\right)\)

\(\text{cosec } \frac{17\pi}{8} + \text{cosec } \frac{9\pi}{4} + \text{cosec } \frac{19\pi}{8}\)

<p>It is given that&nbsp;\(\sin 25^\circ = \frac{a}{b}\). Express each of the following in terms of \(a\) and \(b\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(\sin 25^\circ + \sin 45^\circ + \sin 65^\circ\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\cos 25^\circ + \cos 45^\circ + \cos 65^\circ\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\tan 205^\circ + \tan 225^\circ + \tan 245^\circ\)</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\sec (-25^\circ) + \sec (-45^\circ) + \sec (-65^\circ)\)</span></div></li></ol>

It is given that \(\sin 25^\circ = \frac{a}{b}\). Express each of the following in terms of \(a\) and \(b\).

\(\sin 25^\circ + \sin 45^\circ + \sin 65^\circ\)

\(\cos 25^\circ + \cos 45^\circ + \cos 65^\circ\)

\(\tan 205^\circ + \tan 225^\circ + \tan 245^\circ\)

\(\sec (-25^\circ) + \sec (-45^\circ) + \sec (-65^\circ)\)

<p>Given that&nbsp;\(\cot \gamma = -q\) and \(\frac{3 \pi}{2} &lt; \gamma &lt; 2 \pi\), express each of the following in terms of \(q\).</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: center;"><ol class="i-list"><li><div class="list-item-paragraph">\(\sin \gamma\)</div></li></ol><ol class="i-list" start="3"><li><div class="list-item-paragraph">\(\cot (\pi - \gamma)\)</div></li></ol><ol class="i-list" start="5"><li><div class="list-item-paragraph">\(\sec (-\gamma)\)</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">\(\cos \left(\frac{\pi}{2} - \gamma\right)\)</div></li></ol><ol class="i-list" start="4"><li><div class="list-item-paragraph">\(\tan (\pi + \gamma)\)</div></li></ol><ol class="i-list" start="6"><li><div class="list-item-paragraph">\(\operatorname{cosec} \left(\frac{\pi}{2} - \gamma\right)\)</div></li></ol></div></div></div>

Given that \(\cot \gamma = -q\) and \(\frac{3 \pi}{2} < \gamma < 2 \pi\), express each of the following in terms of \(q\).

<p>Given that&nbsp;\(\tan \beta = p\) and \(180^\circ &lt; \beta &lt; 270^\circ\), express each of the following in terms of \(p\).</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">\(\cot \beta\)</div></li></ol><ol class="i-list" start="3"><li><div class="list-item-paragraph">\(\sin (-\beta)\)</div></li></ol><ol class="i-list" start="5"><li><div class="list-item-paragraph">\(\sec (270^\circ - \beta)\)</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list" start="2"><li><div class="list-item-paragraph">\(\sin (90^\circ - \beta)\)</div></li></ol><ol class="i-list" start="4"><li><div class="list-item-paragraph">\(\sec (\beta + 180^\circ)\)</div></li></ol><ol class="i-list" start="6"><li><div class="list-item-paragraph">\(\operatorname{cosec} (\beta - 90^\circ)\)</div></li></ol></div></div></div>

Given that \(\tan \beta = p\) and \(180^\circ < \beta < 270^\circ\), express each of the following in terms of \(p\).

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