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E Math (IGCSE)

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N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>In the diagram, $AB = 8$ cm, $AC = 10$ cm, $BC = 6$ cm, $BC = CD$ and $ACD$ is a straight line.</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">Explain why $\angle ABC$ is a right angle?</div></li><li><div class="list-item-paragraph">Find the exact value of $\cos \angle BCD$.</div></li><li><div class="list-item-paragraph">Hence find the length of $BD$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1770529461389-1yqtu5ewno1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1Mjk0NjEzODktMXlxdHU1ZXdubzEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.BpwYMJksFrWJhh3pGdCCbc4piu9Id5jYXOJyp-x1hxw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $AB = 8$ cm, $AC = 10$ cm, $BC = 6$ cm, $BC = CD$ and $ACD$ is a straight line.

Basics of Trigonometric Ratios

Cosine Rule

Relationship between Trigonometric Ratios of Acute and Obtuse Angles

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q6

Trigonometry

<p>In the diagram, $ABC$ is a straight line, $AD = 12$ cm, $BD = 5$ cm and $\sin \angle CBD = \frac{3}{5}$.</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;"><p>Find</p><ol class="i-list"><li><div class="list-item-paragraph">$\sin \angle ABD$ ,</div></li><li><div class="list-item-paragraph">$\sin \angle BAD$ .</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1770527323621-v69yyfeie5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjczMjM2MjEtdjY5eXlmZWllNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODM2LCJleHAiOjE3OTM1MDUxMzZ9.SPIOKlV_jkWJ_UIso1cMMn05KrdhNBxFbZ-0lyfIckk" style="max-width: min(260px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $ABC$ is a straight line, $AD = 12$ cm, $BD = 5$ cm and $\sin \angle CBD = \frac{3}{5}$.

Sine Rule

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q5

<p>In the diagram, $PQR$ is a straight line, $\angle QRS = 90^\circ$, $QS = 13$ cm and $QR = 5$ cm.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write $\cos x$ as a fraction.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $y$.</span></div></li></ol>
<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/1770525984395-hkgf5o3u5y.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjU5ODQzOTUtaGtnZjVvM3U1eS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODM2LCJleHAiOjE3OTM1MDUxMzZ9.vfh6VIAgzdfsBUYyCX6ChiTChFpyd-wsrScO5ebbLzw" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

In the diagram, $PQR$ is a straight line, $\angle QRS = 90^\circ$, $QS = 13$ cm and $QR = 5$ cm.

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q3

<p>$ABC$ is a triangle in which $B\hat{A}C = 90^\circ$, $AC = 40$ cm and $BC = 41$ cm.</p>
<p>$AB$ is produced to $P$ and $AC$ is produced to $Q$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Showing your workings clearly, explain why $AB = 9$ cm.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Express as a fraction in its simplest form,</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\tan B\hat{C}A$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\sin P\hat{B}C$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\cos B\hat{C}Q$.</span></div></li></ol></li></ol>
<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/1770523213533-8rfc0xev4ni.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjMyMTM1MzMtOHJmYzB4ZXY0bmkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ._6C3A8UBV95jZBI8GPvd3S5gOL4updZYCRLzeO0XBdM" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

$ABC$ is a triangle in which $B\hat{A}C = 90^\circ$, $AC = 40$ cm and $BC = 41$ cm.

$AB$ is produced to $P$ and $AC$ is produced to $Q$.

Showing your workings clearly, explain why $AB = 9$ cm.

Express as a fraction in its simplest form,

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q2

<p>If&nbsp;$0^\circ &lt; x &lt; 180^\circ$, determine the possible values of $x$ for each of the following, giving your answers correct to $1$ decimal place.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\cos x = -0.629$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\sin x = 0.478$</span></div></li></ol>

If $0^\circ < x < 180^\circ$, determine the possible values of $x$ for each of the following, giving your answers correct to $1$ decimal place.

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q1

<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/1769518939767-rz5dgqo1yi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MTg5Mzk3Njctcno1ZGdxbzF5aS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODM2LCJleHAiOjE3OTM1MDUxMzZ9.NFLTXbJjzrogS2TyBtAhdFpq0mVjIK6czu96Zx1wGOY" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>In the diagram, $AB = 16$ cm, $CD = 18$ cm, $\angle ABC = 90^\circ$, $\cos \angle BAC = \frac{4}{5}$ and $BCDE$ is a straight line.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $AC$,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\tan \angle ACB$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $AD$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find $\angle ADE$,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the shortest distance from $C$ to $AD$.</span><span class="inline-marks">[3]</span></div></li></ol>

In the diagram, $AB = 16$ cm, $CD = 18$ cm, $\angle ABC = 90^\circ$, $\cos \angle BAC = \frac{4}{5}$ and $BCDE$ is a straight line.

Find the shortest distance from $C$ to $AD$.

Area of Triangle

Applications of Trigonometric Ratios

IP2 Class Test (May 2025) Q13

<p>Triangle $PQR$ is a right-angled triangle with $PQ = 10$cm, $QR = 6$cm and $\angle PQR = 90^\circ$. Given that&nbsp;$M$ is a point on $PR$ such that&nbsp;$\frac{MR}{PM} = \frac{5}{11}$, find, giving your answer as a fraction in its simplest form, the value of</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="abc-list"><li><div class="list-item-paragraph">$\tan \angle MQR$,</div></li><li><div class="list-item-paragraph">$\cos \angle PMQ$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1768028274008-x6oc1ksq4bg.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgwMjgyNzQwMDgteDZvYzFrc3E0YmcuanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.VI5GMxMxpHFzsNxozMm7l_rNYqGz3Mu4ZPdoQWasqUs" style="max-width: min(330px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Triangle $PQR$ is a right-angled triangle with $PQ = 10$cm, $QR = 6$cm and $\angle PRQ = 90^\circ$. Given that $M$ is a point on $PR$ such that $\frac{MR}{PM} = \frac{5}{11}$, find, giving your answer as a fraction in its simplest form, the value of

<p>In the diagram below, $ABC$ is a right-angled triangle in which $AB = 4$ cm, $BC = 6$ cm and $\angle BAC = 90^\circ$. $BCD$ is a straight line. Find the <strong>exact form,</strong></p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">$\sin \angle ABC$,</div></li><li><div class="list-item-paragraph">$\cos \angle ACD$.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><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/1768468690128-c6vjf00skbs.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg0Njg2OTAxMjgtYzZ2amYwMHNrYnMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.wiV5btg0QtJy5S9Mgh-4FcveDaW4EYHHEiMMLZ1wbw4" style="max-width: min(350px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram below, $ABC$ is a right-angled triangle in which $AB = 4$ cm, $BC = 6$ cm and $\angle BAC = 90^\circ$. $BCD$ is a straight line. Find the **exact form,**

<p>In the diagram, $BCD$ is a straight line. $BC=10$ cm, $AC=6$ cm and $\cos \angle ACD = -\frac{11}{24}$.</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;"><p>Find</p><ol class="i-list"><li><div class="list-item-paragraph">the value of $\cos \angle ACB$,</div></li><li><div class="list-item-paragraph">the value of $x$,</div></li><li><div class="list-item-paragraph">$\angle BAC$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767407582850-eherlzyecvn.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MDc1ODI4NTAtZWhlcmx6eWVjdm4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.KxZNtgB0ZsziFkhkawSHnCzNxrCmCSE1J_kjxeUQycM" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram, $BCD$ is a straight line. $BC=10$ cm, $AC=6$ cm and $\cos \angle ACD = -\frac{11}{24}$.

Fundamental Principles of the Cosine Rule

<p>In the diagram, $PRS$ is a straight line, $PQ=12$ cm, $PR=7$ cm and $\sin \angle QRS=\frac{4}{9}$.</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;"><p>Calculate the value of</p><ol class="i-list"><li><div class="list-item-paragraph">$\sin \angle PRQ$,</div></li><li><div class="list-item-paragraph">$\sin \angle PQR$, </div></li></ol><p>giving your answers as a fraction in its lowest term.</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767358910455-325hiiod3nu.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTg5MTA0NTUtMzI1aGlpb2QzbnUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.yspCzDomnrs0NUUWQzs4sSiFTkU5Ugf5xeNei4U7Yu4" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram, $PRS$ is a straight line, $PQ=12$ cm, $PR=7$ cm and $\sin \angle QRS=\frac{4}{9}$.

Calculating Sine Values in Scalene Triangles

<p>In the diagram, $PCBQ$ is a straight line.  $AB=15$ cm, $\sin \ \angle ABC=\frac{1}{2}$ and $\sin \ \angle ACP=\frac{1}{3}$.</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">Write down the value of $\sin \ \angle ACB$.</div></li><li><div class="list-item-paragraph">Calculate the length of $AC$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767358747455-wd5ygwibcc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTg3NDc0NTUtd2Q1eWd3aWJjYy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODM2LCJleHAiOjE3OTM1MDUxMzZ9.oOaTgY5SFJRN1mAtn2qGs9-KpUShl48KOQOvkkaMf9c" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $PCBQ$ is a straight line.  $AB=15$ cm, $\sin \ \angle ABC=\frac{1}{2}$ and $\sin \ \angle ACP=\frac{1}{3}$.

Principles of Plane Geometry: Triangle Properties and Proofs

<p>In the diagram, $ABD$ is a straight line, $AB=10$ cm, $BC=6$ cm and angle $BCD$ is a right angle.  <br>Given that&nbsp;$\angle CBD=x$, where&nbsp;$\sin x=0.6$, $\cos x=0.8$and $\tan x=0.75$, calculate</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1780210381095-qq7r2k0iq" data-list-item-path="item-1780210381095-qq7r2k0iq"><div class="list-item-paragraph">the length of $BD$,</div></li><li data-list-item-id="item-1780210381921-wualsl7pw" data-list-item-path="item-1780210381921-wualsl7pw"><div class="list-item-paragraph">the value of $\sin \angle ABC$,</div></li><li data-list-item-id="item-1780210382313-z6wmxdugu" data-list-item-path="item-1780210382313-z6wmxdugu"><div class="list-item-paragraph">the area of triangle $ABC$.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><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/1767355031886-rv7lcyd8hbh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTUwMzE4ODYtcnY3bGN5ZDhoYmgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.9RYthsRYwHqVXl68tYoEWybHuMtjpx5ZS6hpYn539yw" style="max-width: min(400px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>
<p style="margin-left: 3rem"><strong>Tip: </strong></p>
<p style="margin-left: 3rem">1.	For intermediate working, correct them to $5$ significant figures to maintain accuracy of answers</p>
<p style="margin-left: 3rem">2.	For answers in degree, correct to $1$ decimal places.</p>
<p style="margin-left: 3rem">3.	For other answers, correct to $3$ significant figures.</p>

In the diagram, $ABD$ is a straight line, $AB=10$ cm, $BC=6$ cm and angle $BCD$ is a right angle.  
Given that $\angle CBD=x$, where $\sin x=0.6$, $\cos x=0.8$and $\tan x=0.75$, calculate

1.	For intermediate working, correct them to $5$ significant figures to maintain accuracy of answers

2.	For answers in degree, correct to $1$ decimal places.

3.	For other answers, correct to $3$ significant figures.

Problem involving the Area of a Triangle

<p>In the diagram, $SACT$is a straight line and $\angle ABC=90{}^\circ $.<br>Given that&nbsp;$BC=12$ cm and $AB=16$ cm.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Calculate the length of $AC$.</div></li><li><div class="list-item-paragraph">Express, as a fraction in its lowest terms, the value of</div><ol class="i-list"><li><div class="list-item-paragraph">$\sin x{}^\circ $,</div></li><li><div class="list-item-paragraph">$\cos y{}^\circ $.</div></li></ol></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><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/1767353848170-iz626oh54q.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTM4NDgxNzAtaXo2MjZvaDU0cS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODM2LCJleHAiOjE3OTM1MDUxMzZ9.74cV2fKnKVYvyQglI8uLkBcaPULPkCMlDkq4FmztYho" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, $SACT$is a straight line and $\angle ABC=90{}^\circ $.
Given that $BC=12$ cm and $AB=16$ cm.

Trigonometric Ratios in Right-Angled and Extended Triangles

<p>In the diagram, $ABC$ is a right-angled triangle in which $\angle BAC=90{}^\circ $.<br>$AC=16$ cm, $BC=20$ cm, $AB$ is produced to $P$ and $AC$ is produced to $Q$.</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: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Calculate $\angle ABC$, correct to $1$ decimal place.</div></li><li><div class="list-item-paragraph">Express each of the following as a fraction, in its simplest form.</div><ol class="i-list"><li><div class="list-item-paragraph">$\sin \angle BCA$,</div></li><li><div class="list-item-paragraph">$\cos \angle BCQ$.</div></li></ol></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767353465433-e66r3oynu5c.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTM0NjU0MzMtZTY2cjNveW51NWMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.TI_ZEMGo_Dl1MPXYMt9ctzYPUKRloUdSBRNwuUduIPg" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram, $ABC$ is a right-angled triangle in which $\angle BAC=90{}^\circ $.
$AC=16$ cm, $BC=20$ cm, $AB$ is produced to $P$ and $AC$ is produced to $Q$.

Supplementary Angle Identities: Analyzing Trigonometry Beyond $90^\circ$

<p>In the diagram, $PQR$ is a straight line, $\angle QPS=90{}^\circ $, $PS=5$ cm and $PQ=12$ cm.</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="abc-list"><li><div class="list-item-paragraph">Find the length of $QS$.</div></li><li><div class="list-item-paragraph">Write down as a fraction, in the simplest form,</div><ol class="i-list"><li><div class="list-item-paragraph">$\tan \angle PQS$,</div></li><li><div class="list-item-paragraph">$\cos x{}^\circ $.</div></li></ol></li><li><div class="list-item-paragraph">Find the value of $y$, correct to $1$ decimal place.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767353154375-5454a25h7hj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTMxNTQzNzUtNTQ1NGEyNWg3aGoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.QJhM-O2_Tap9XGYa0eF0VggEuBOjZRVGnFSkbnRfd4A" style="max-width: min(400px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the diagram, $PQR$ is a straight line, $\angle QPS=90{}^\circ $, $PS=5$ cm and $PQ=12$ cm.

Trigonometry of Acute and Obtuse Angles: Evaluating Ratios in the $0^\circ$ to $180^\circ$ Range.

<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;"><p>In $\Delta ABC$, $AB=16$ cm, $BC=12$ cm and $AC=20$ cm.<br>$BA$ is produced to $D$ and $AD=8$ cm.</p><ol class="i-list"><li><div class="list-item-paragraph">Explain why $\angle ABC$ is a right angle.</div></li><li><div class="list-item-paragraph">Find the area of $\Delta DAC$.</div></li><li><div class="list-item-paragraph">Write down the value of $\cos \angle DAC$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767352684445-ddfkx7xxw3k.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTI2ODQ0NDUtZGRma3g3eHh3M2sucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.xq3nMXKvLe43NOf8APkCYsYy0VzMuiyI6cOJr0dxf-U" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The Geometry of Trigonometric Ratios: From Coordinate Points to Sine and Cosine

<p>The figure shows $\Delta PQR$ formed by the 3 points $P(1,\text{}\hspace{0.5em}2)$, $Q(4,\text{}\hspace{0.5em}6\text{)}$ and $R(-5,\text{ 2)}$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--1-2" data-layout="1-2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Find the value of each of the following.</p><ol class="i-list"><li><div class="list-item-paragraph">$\sin \angle QPR$,</div></li><li><div class="list-item-paragraph">$\cos \angle QPR$,</div></li><li><div class="list-item-paragraph">$\tan \angle PRQ$.</div></li></ol></div><div class="question-column" style="flex: 2; align-self: flex-start;"><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/1767352329828-4p1jnrhdhom.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTIzMjk4MjgtNHAxam5yaGRob20ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzNiwiZXhwIjoxNzkzNTA1MTM2fQ.P6VzIioIitAF9yfTVNylheZEsrlb2v6a7w8Gpc6QBH8" style="max-width: min(400px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The figure shows $\Delta PQR$ formed by the 3 points $P(1,\text{ }2)$, $Q(4,\text{ }6\text{)}$ and $R(-5,\text{ 2)}$.

Trigonometry Problem Set: Calculating Angles and Fractional Ratios.

<p>$\Delta ABC$ is a right-angled triangle where&nbsp;$\angle ABC=90{}^\circ $.<br>$BC$ is produced to $D$ and $\tan \angle ACB=\frac{4}{3}$.<br>Write down each of the following as a fraction, without the use of a calculator.</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">$\sin \angle ACB$,</div></li><li><div class="list-item-paragraph">$\cos \angle BAC$,</div></li><li><div class="list-item-paragraph">$\cos \angle ACD$.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><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/1767351953029-z7l9ywdfd3.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjczNTE5NTMwMjktejdsOXl3ZGZkMy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODM2LCJleHAiOjE3OTM1MDUxMzZ9.uwzmDtp6t9JxQIPZalPlvJmIcxnrdd4dLwtnZYDjWnQ" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

$\Delta ABC$ is a right-angled triangle where $\angle ABC=90{}^\circ $.
$BC$ is produced to $D$ and $\tan \angle ACB=\frac{4}{3}$.
Write down each of the following as a fraction, without the use of a calculator.

Trigonometry of Acute and Obtuse Angles: Evaluating Sines and Cosines.

<p>Find <strong>obtuse</strong> angles $A$ and $B$ such that</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\cos A=-0.658$,</div></li><li><div class="list-item-paragraph">$\sin B=0.486$.</div></li></ol>
<p>Correct your answers to $1$ decimal place.</p>

Find **obtuse** angles $A$ and $B$ such that

Correct your answers to $1$ decimal place.

Solving Sine and Cosine Equations within the $0^\circ < x < 180^\circ$ Domain

<p>Given that&nbsp;$0{}^\circ &lt;x&lt;180{}^\circ $, find the possible values of $x$ for each of the following, correcting your answer to $1$ decimal place.</p>
<ol class="i-list"><li><div class="list-item-paragraph">$\cos x=-0.811$,</div></li><li><div class="list-item-paragraph">$\sin x=0.215$.</div></li></ol>

Given that $0{}^\circ <x<180{}^\circ $, find the possible values of $x$ for each of the following, correcting your answer to $1$ decimal place.

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