A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The diagram above shows two rods $AB$ and $BC$ of length $4$ m and $9$ m respectively and $\angle ABC = 90^{\circ} $. Rollers are fixed at points $A$ and $B$ such that&nbsp;$A$ is able to move along the wall and $B$ is able to move along the celling. The horizontal distance of $C$ from the vertical wall is $d$ m.</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/1781508142282-maxm0gnyuj8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODE1MDgxNDIyODItbWF4bTBnbnl1ajgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.fGTiXaeSEGO7D4FfCWnCgtf4M2-S_j1-aKEDMy7quMU" style="max-width: min(320px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show clearly that&nbsp;$d = 4\sin \theta + 9\cos \theta$.<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 $d$ in the form $R\cos(\theta - \alpha)$ where&nbsp;$R &gt; 0$ and $0^{\circ} &lt; \alpha &lt; 90^{\circ}$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the value of $\theta$ for which $d = 6\text{ m}$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4" data-list-item-path="item-4"><div class="list-item-paragraph">Find the maximum value of $d$ and the corresponding value of $\theta$.<span class="inline-marks">[2]</span></div></li></ol>

The diagram above shows two rods $AB$ and $BC$ of length $4$ m and $9$ m respectively and $\angle ABC = 90^{\circ} $. Rollers are fixed at points $A$ and $B$ such that $A$ is able to move along the wall and $B$ is able to move along the celling. The horizontal distance of $C$ from the vertical wall is $d$ m.

Trigonometric Modelling

Trigonometric Equations

R-Formula and Trigonometric Extrema

Applications of Trigonometric Ratios

2024 FMSS PRELIMS P2 Q9

Trigonometry

<p>A drone is hovering at a fixed altitude directly above the edge of a skyscraper. The drone’s camera tracks two delivery robots, Robot A and Robot B, which are moving along a straight road extending from the base of the skyscraper.</p>
<p>The angle of depression from the drone to Robot A is $28.5°$. The angle of depression from the drone to Robot B is $14.2°$. Robot A is known to be $320$ m away from the base of the skyscraper.<br></p>
<ol class="abc-list"><li><div class="list-item-paragraph">Calculate the altitude at which the drone is hovering.</div></li><li><div class="list-item-paragraph">Find the distance between Robot A and Robot B. </div></li></ol>

A drone is hovering at a fixed altitude directly above the edge of a skyscraper. The drone’s camera tracks two delivery robots, Robot A and Robot B, which are moving along a straight road extending from the base of the skyscraper.

The angle of depression from the drone to Robot A is $28.5°$. The angle of depression from the drone to Robot B is $14.2°$. Robot A is known to be $320$ m away from the base of the skyscraper.


Angles of Elevation and Depression

<p>The figure shows a triangle $ABC$ where&nbsp;$BC = 23$ cm, $K$ lies on $BC$ such that the length of $BK$ is $135\%$ of the length of $KC$ and $\angle AKC = 90^\circ$. Given that the area of $\Delta ABK$ is $68\text{ cm}^2$, find $\angle BAC$.</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/1770787730690-57r7qj595po.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3ODc3MzA2OTAtNTdyN3FqNTk1cG8ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.v40vEPtX6bypnSHJTE_rrzBGcIZ66Zdt2Bf3ZPOzjTc" style="max-width: min(320px, 100%); height: auto; display: block;" alt="Diagram of triangle ABC with altitude AK to base BC" class="alignright custom-width" /></figure></div>

The figure shows a triangle $ABC$ where $BC = 23$ cm, $K$ lies on $BC$ such that the length of $BK$ is $135\%$ of the length of $KC$ and $\angle AKC = 90^\circ$. Given that the area of $\Delta ABK$ is $68\text{ cm}^2$, find $\angle BAC$.

Area of Triangle

IP2U7 Exercise Q10

Perimeter and Area of Plane Figures

<p>Two points $P$ and $Q$, $10$ m apart on level ground, are due West of the foot $B$ of a tree $TB$. Given that&nbsp;$\angle TPB = 23^\circ$ and $\angle TQB = 32^\circ$, find the height of the tree.</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/1770787060352-92o0o1x4xfw.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3ODcwNjAzNTItOTJvMG8xeDR4ZncucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.ddY3I4IdLXVUraGwJqlRpujuGxrNougSZzRotMdoBOI" style="max-width: min(350px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

Two points $P$ and $Q$, $10$ m apart on level ground, are due West of the foot $B$ of a tree $TB$. Given that $\angle TPB = 23^\circ$ and $\angle TQB = 32^\circ$, find the height of the tree.

IP2U7 7.2 PN3 Q1

<p>Benjamin and Caitlyn stand at points $B$ and $C$ on the ground respectively. A tower $AH$ is such that&nbsp;$\angle ABH = 35^\circ$ and the horizontal distance between Caitlyn and the tower is twice that between Benjamin and the tower. Find $\angle ACH$.</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/1770786616681-zaquheuegca.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3ODY2MTY2ODEtemFxdWhldWVnY2EucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.idccLyJWQAXVOI9WnVtfNPP_fgmf2D68B9LB9f55uDo" style="max-width: min(350px, 100%); height: auto; display: block;" alt="Diagram showing a tower AH with Benjamin at B and Caitlyn at C on level ground." class="alignright custom-width" /></figure></div>

Benjamin and Caitlyn stand at points $B$ and $C$ on the ground respectively. A tower $AH$ is such that $\angle ABH = 35^\circ$ and the horizontal distance between Caitlyn and the tower is twice that between Benjamin and the tower. Find $\angle ACH$.

Applications of Trigonometric Ratios in Real Life

<p>In $\Delta ABC$, $AB=7.5$m and $\angle ACB=90{}^\circ $. $X$ lies on $CA$ such that&nbsp;$XB=3$m and $\angle XBC=43{}^\circ $. Find </p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">$\angle BAC$,</div></li><li><div class="list-item-paragraph">the length of $XA$.</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/1770786166865-wny8tw52taj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3ODYxNjY4NjUtd255OHR3NTJ0YWoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.b3wgSkRPb6aEXine3MBBKRIpbvQbz_UmN0387oRNg-E" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In $\Delta ABC$, $AB=7.5$m and $\angle ACB=90{}^\circ $. $X$ lies on $CA$ such that $XB=3$m and $\angle XBC=43{}^\circ $. Find 

Basics of Trigonometric Ratios

IP2U7 7.2 Worked Example 2 

<p>The diagram shows a sheet of cardboard $ABCD$.</p>
<p>$AD = 10$ cm, $\angle ABC = 46^\circ$, $\angle BAC = 75^\circ$, $\angle ACD = 33^\circ$ and $\angle ADC = 90^\circ$.</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</div><ol class="i-list"><li><div class="list-item-paragraph">$AC$,</div></li><li><div class="list-item-paragraph">$BC$ and,</div></li><li><div class="list-item-paragraph">the area of the triangle $ABC$.</div></li></ol></li><li><div class="list-item-paragraph">Triangle $ADC$ is folded along $AC$ to form triangle $AEC$.<br>Calculate the length of</div><ol class="i-list"><li><div class="list-item-paragraph">$EC$ and,</div></li><li><div class="list-item-paragraph">$BE$.</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 aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770533029291-u2ypx5enu1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MzMwMjkyOTEtdTJ5cHg1ZW51MS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODI4LCJleHAiOjE3OTM1MDUxMjh9._NaP_tIZuFXavh4XBt6jScMM_hSjK0rB-c0StEBVp1o" style="max-width: min(280px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram shows a sheet of cardboard $ABCD$.

$AD = 10$ cm, $\angle ABC = 46^\circ$, $\angle BAC = 75^\circ$, $\angle ACD = 33^\circ$ and $\angle ADC = 90^\circ$.

Sine Rule

Cosine Rule

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q11

<p>In the diagram, $BDCE$ is a straight line, $BD = 4$ cm, $AC = 10$ cm and $AB = AD$.</p>
<p>Given that the area of $\Delta ABD = 16$ cm$^2$, find the value of $\sin \angle ACD$.</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/1770526492553-v1727jwj35.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MjY0OTI1NTMtdjE3Mjdqd2ozNS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODI4LCJleHAiOjE3OTM1MDUxMjh9.aPtg0XPd8It5F2nfnzAsdJz7XqxmAhIECqSBnNb-4Tg" style="max-width: min(300px, 100%); height: auto; display: block;" alt="Geometry diagram showing triangle ABC with point D on BC." class="aligncenter custom-width" /></figure></div>

In the diagram, $BDCE$ is a straight line, $BD = 4$ cm, $AC = 10$ cm and $AB = AD$.

Given that the area of $\Delta ABD = 16$ cm$^2$, find the value of $\sin \angle ACD$.

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q4

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MTg5Mzk3Njctcno1ZGdxbzF5aS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODI4LCJleHAiOjE3OTM1MDUxMjh9.mFzxedmwaB1_Hd--gPSVwQ59G6hTS8RPgUAfDaqnJB4" 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 $AD$.

Relationship between Trigonometric Ratios of Acute and Obtuse Angles

IP2 Class Test (May 2025) Q13

<p>In a physics experiment, a formula is given by $F = mg \sin(\theta) - \mu mg \cos(\theta)$, where&nbsp;$\mu$, $g$ are constants and $m$, $\theta$ may vary. It is given that&nbsp;$g = 9.8$, $\mu = 0.5$, $m &gt; 0$ and $0 &lt; \theta &lt; 90{}^\circ$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Evaluate $F$ given that&nbsp;$m = 15$ and $\theta = 30{}^\circ$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Find the value of $\theta$ such that&nbsp;$F = 0$. Note that&nbsp;$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Another variable, $a$, is defined by $a = \frac{F}{m}$. Express $a$ in terms of $\mu$, $g$ and $\theta$.</div><div class="list-item-paragraph">Hence find the range of values of $\theta$ such that&nbsp;$a &gt; 0$.</div><div class="list-item-paragraph">You may assume the fact that&nbsp;$\tan(\theta) &gt; p \iff \theta &gt; \tan^{-1}(p)$ for&nbsp;$0 &lt; \theta &lt; 90{}^\circ$.<span class="inline-marks">[2]</span></div></li></ol>

In a physics experiment, a formula is given by $F = mg \sin(\theta) - \mu mg \cos(\theta)$, where $\mu$, $g$ are constants and $m$, $\theta$ may vary. It is given that $g = 9.8$, $\mu = 0.5$, $m > 0$ and $0 < \theta < 90{}^\circ$.

Evaluate $F$ given that $m = 15$ and $\theta = 30{}^\circ$.

Find the value of $\theta$ such that $F = 0$. Note that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Another variable, $a$, is defined by $a = \frac{F}{m}$. Express $a$ in terms of $\mu$, $g$ and $\theta$.

Trigonometric Inequalities

IP2 Class Test (May 2025) Q6

<p>The diagram shows an L-shaped rod $ABC$ leaning against a vertical wall. </p>
<p>$AB=8$ cm, $BC=15$ cm and angle $\angle BCO=\theta$.</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/1768293828392-aywndi308zh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyOTM4MjgzOTItYXl3bmRpMzA4emgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.IbvM92Y3knlSHGwzKs7lP2uFLTrP840mP9TBfYio_a8" style="max-width: min(200px, 100%); height: auto; display: block;" alt="L-shaped rod ABC leaning against a wall" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that $OC = 8 \sin \theta + 15 \cos \theta$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Express $OC$ in the form $R \sin(\theta + \alpha)$, where $R &gt; 0$ and $0{}^\circ &lt; \alpha &lt; 90{}^\circ$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the values of $\theta$ when $OC=16$ cm.</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an L-shaped rod $ABC$ leaning against a vertical wall. 

$AB=8$ cm, $BC=15$ cm and angle $\angle BCO=\theta$.

Show that $OC = 8 \sin \theta + 15 \cos \theta$.

Express $OC$ in the form $R \sin(\theta + \alpha)$, where $R > 0$ and $0{}^\circ < \alpha < 90{}^\circ$.

Find the values of $\theta$ when $OC=16$ cm.

2023 ACSBR AM S3 EOY Q11 

<p>The diagram below shows the chords $PQ$ and $RS$ and they intersect at the point $X$. Given that&nbsp;$PQ=7$ cm, $RS=24$ cm, $\angle RXQ=90{}^\circ$ and $\angle PRS=\theta$ where&nbsp;$\theta$ varies.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780548663457-r781zxsue"><div class="list-item-paragraph">Prove that&nbsp;$PR = 24 \cos \theta + 7 \sin \theta$.</div></li><li data-list-item-id="item-1780548664001-0ii36cfa5"><div class="list-item-paragraph">Express $PR$ in the form $R \cos(\theta - \alpha)$.</div></li><li data-list-item-id="item-1780548664352-k2zljpf4c"><div class="list-item-paragraph">Find the maximum length of $PR$ and the corresponding value of $\theta$.</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/1768102861843-0qiwbhfd6kk9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDI4NjE4NDMtMHFpd2JoZmQ2a2s5LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTM1MDQ4MjgsImV4cCI6MTc5MzUwNTEyOH0.bjXdX8Spy6Ll_BURHnGSB-b-sQ3mhVrfGD-YW482yGk" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram below shows the chords $PQ$ and $RS$ and they intersect at the point $X$. Given that $PQ=7$ cm, $RS=24$ cm, $\angle RXQ=90{}^\circ$ and $\angle PRS=\theta$ where $\theta$ varies.

<p>The diagram shows a trapezium $ORST$ inscribed in a circle with centre $O$ and diameter $16$ cm. $OR$ makes an angle $\theta$, $0{}^\circ &lt; \theta &lt; 90{}^\circ$, with the diameter of the circle. $ST$ is perpendicular to both&nbsp;$RS$ and $OT$. The perimeter of the trapezium is $P$ cm.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780548600520-ol4l6gd8k"><div class="list-item-paragraph">Show that&nbsp;$P = 8 + 24 \cos \theta + 8 \sin \theta$.</div></li><li data-list-item-id="item-1780548600968-lprtp57hi"><div class="list-item-paragraph">Express $24 \cos \theta + 8 \sin \theta$ in the form of $R \cos(\theta - \alpha)$. Hence, find the maximum value of $P$ and the corresponding value of $\theta$ at which this value occurs.</div></li><li data-list-item-id="item-1780548601568-pmbljj84d"><div class="list-item-paragraph">Find the value of $\theta$ when&nbsp;$P = 30$ cm.</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/1768102718306-7df8srwyum5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDI3MTgzMDYtN2RmOHNyd3l1bTUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.amo5k2zHZHLCLwVbw4dnonC0_pz3ZOAx9bWY9oJn0OQ" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows a trapezium $ORST$ inscribed in a circle with centre $O$ and diameter $16$ cm. $OR$ makes an angle $\theta$, $0{}^\circ < \theta < 90{}^\circ$, with the diameter of the circle. $ST$ is perpendicular to both $RS$ and $OT$. The perimeter of the trapezium is $P$ cm.

<p>The diagram shows a quadrilateral divided into two right-angled triangles and a rectangle.<br>Given that&nbsp;$\angle ADC = \theta{}^\circ$, $AB = 7$ cm, $AD = 15$ cm and $\angle BAD = \angle BCD = \angle AFB = 90{}^\circ$.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="abc-list"><li data-list-item-id="item-1780548515975-2vm0rfxc7"><div class="list-item-paragraph">Find the length of $BC$ in the form of $a \sin \theta - b \cos \theta$. Hence show that the perimeter, $P$ cm, of the quadrilateral $ABCD$ is given by $P = 22 + 22 \sin \theta + 8 \cos \theta$.</div></li><li data-list-item-id="item-1780548516847-6b0z8who0"><div class="list-item-paragraph">Expressing $P$ in the form $P = 22 + R \sin(\theta + \alpha)$.</div></li><li data-list-item-id="item-1780548517335-yo4pr6mso"><div class="list-item-paragraph">Hence state the maximum value of the perimeter and state the corresponding value of $\theta$.</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/1768102555320-swimt9mjcxh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDI1NTUzMjAtc3dpbXQ5bWpjeGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.Md3HsI9em5pXjKph8jTlLrHriM1AJFk7skeB30NQdWg" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows a quadrilateral divided into two right-angled triangles and a rectangle.
Given that $\angle ADC = \theta{}^\circ$, $AB = 7$ cm, $AD = 15$ cm and $\angle BAD = \angle BCD = \angle AFB = 90{}^\circ$.

<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><p>The diagram shows four points $A$, $B$, $C$ and $O$ such that that&nbsp;$AB = 2$ m, $BC = 10$ m,<br>$\angle AOC = \angle ABC = 90{}^\circ$ and $\angle OCB = \theta{}^\circ$.</p><ol class="abc-list"><li data-list-item-id="item-1780548397790-cpclwcpn6"><div class="list-item-paragraph">Show that&nbsp;$OC = 2\sin \theta + 10\cos \theta$.</div></li><li data-list-item-id="item-1780548398221-uzovu47tj"><div class="list-item-paragraph">Express $OC$ in the form $R\sin(\theta + \alpha)$.</div></li><li data-list-item-id="item-1780548398613-t3y5ob0ft"><div class="list-item-paragraph">Find the maximum value of $OC$ and the corresponding value of $\theta$.</div></li><li data-list-item-id="item-1780548399229-kusfvjqx4"><div class="list-item-paragraph">Find the value of $\theta$ for which $OC = 8$ m.</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/1768102467338-ny3m0f7qie8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDI0NjczMzgtbnkzbTBmN3FpZTgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.OQl-tdgsmKyGK5lEH0Vic5wsR2-FwQ85fQRC6ZvRHrE" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><p>The diagram shows an L-shaped dustpan $ABC$ leaning against a vertical wall.<br>It is given that&nbsp;$AB = 20$ cm, $BC = 50$ cm, $\angle AOC = \angle ABC = 90{}^\circ$ and $\angle OCB = \theta{}^\circ$.</p><ol class="abc-list"><li data-list-item-id="item-1780548281620-y6yqd6q9k"><div class="list-item-paragraph">Show that&nbsp;$AO + OC = (30 \cos \theta + 70 \sin \theta)$ cm.</div></li><li data-list-item-id="item-1780548282652-51t5lphme"><div class="list-item-paragraph">Find the value of $\theta$ for which $AO + OC = 65$ cm.</div></li><li data-list-item-id="item-1780548283244-2wzry829i"><div class="list-item-paragraph">State the maximum value of $(AO + OC)$ and the corresponding value of $\theta$.</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/1768102193710-f44e9h7yijm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDIxOTM3MTAtZjQ0ZTloN3lpam0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.KLR0mjrwasjXmdMPoY0ySdp5QtzjQ2MY02fLCaVBaI0" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

<p>The diagram shows a robotic arm $PQR$ such that&nbsp;$PQ$ makes an angle of $\theta$ with the $y$-axis, where&nbsp;$0{}^\circ &lt; \theta &lt; 90{}^\circ$. $QR$ makes an angle of $\theta$ with the horizontal. The lengths of $PQ$ and $QR$ are $6$ m and $4$ m respectively.</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 data-list-item-id="item-1780548132946-8829rjc6m"><div class="list-item-paragraph">Given that the vertical length of the robotic arm is $L$ m,</div><ol class="i-list"><li data-list-item-id="subitem-1780548136403-gr4kqztcb"><div class="list-item-paragraph">show that&nbsp;$L=4\sin \theta + 6\cos \theta$.</div></li><li data-list-item-id="subitem-1780548137458-k0wv8ir0e"><div class="list-item-paragraph">express $L$ in the form $R\sin(\theta + \alpha){}^\circ$, where&nbsp;$R&gt;0$ and $0{}^\circ &lt; \theta &lt; 90{}^\circ$.</div></li></ol></li><li data-list-item-id="item-1780548134034-hl9nsfoyd"><div class="list-item-paragraph">Hence,</div><ol class="i-list"><li data-list-item-id="subitem-1780548144203-fiw6t1v8q"><div class="list-item-paragraph">state the maximum length of $L$ and find the corresponding value of $\theta$.</div></li><li data-list-item-id="subitem-1780548144610-ukwa0l0py"><div class="list-item-paragraph">find the values of $\theta$ when&nbsp;$L=7$ m.</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/1768101995709-k9s03wd7r1.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDE5OTU3MDktazlzMDN3ZDdyMS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODI4LCJleHAiOjE3OTM1MDUxMjh9.-0q42PDethij0vDINrHAW3goVdLYIUdlpdycW6kusOY" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows a robotic arm $PQR$ such that $PQ$ makes an angle of $\theta$ with the $y$-axis, where $0{}^\circ < \theta < 90{}^\circ$. $QR$ makes an angle of $\theta$ with the horizontal. The lengths of $PQ$ and $QR$ are $6$ m and $4$ m respectively.

<p>In the figure, $\angle ADC = 90{}^\circ$, $\angle ACD = 42{}^\circ$, $AD = 6.5$ cm, $AB = 13.5$ 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;"><p>Find</p><ol class="i-list"><li><div class="list-item-paragraph">the length of $CD$,</div></li><li><div class="list-item-paragraph">the length of $BC$,</div></li><li><div class="list-item-paragraph">the area of triangle $ABC$.</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/1767421189006-4kt9bmpclkr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MjExODkwMDYtNGt0OWJtcGNsa3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.mjC2vEc5W59V8-aP4t71xkAVp56HVn34qSPYGbw3IL4" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, $\angle ADC = 90{}^\circ$, $\angle ACD = 42{}^\circ$, $AD = 6.5$ cm, $AB = 13.5$ cm.

The Cosine Code: Unlocking the Mystery of Side $x$

<p>The diagram shows two triangles, $\Delta ABC$ and $\Delta ACD$.<br>$\angle CAD = 90{}^\circ$, $\angle ABC = 25{}^\circ$, $AD = 4$ cm and $AB = 8$ cm.<br>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><div class="list-item-paragraph">$\angle ADC$,</div></li><li><div class="list-item-paragraph">the length of $CD$,</div></li><li><div class="list-item-paragraph">the length of $BC$.</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/1767401864792-6jtm2nola7r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc0MDE4NjQ3OTItNmp0bTJub2xhN3IucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.LAzvF1braGUFAIXVj2IT-Ohr29D8dp8o8XSBWlccwX8" style="max-width: min(300px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows two triangles, $\Delta ABC$ and $\Delta ACD$.
$\angle CAD = 90{}^\circ$, $\angle ABC = 25{}^\circ$, $AD = 4$ cm and $AB = 8$ cm.
Find

Application of Sine and Cosine Ratios in Multi-Triangle Systems

<p>The figure shows a triangle $ABC$ where&nbsp;$BC=23$ cm, $K$ lies on $BC$ such that the length of $BK$ is $135\%$ of the length of $KC$  and $\angle AKC=90{}^\circ $. Given that the area of $\Delta ABK$ is $68$ cm$^{2}$, find $\angle BAC$.</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/1765189381065-71ylq0x06oi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjUxODkzODEwNjUtNzF5bHEweDA2b2kucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgyOCwiZXhwIjoxNzkzNTA1MTI4fQ.8KdciM3yWmRecfL7HQxUjaW5DVtlBNc3SKmA3jWdvag" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

The figure shows a triangle $ABC$ where $BC=23$ cm, $K$ lies on $BC$ such that the length of $BK$ is $135\%$ of the length of $KC$  and $\angle AKC=90{}^\circ $. Given that the area of $\Delta ABK$ is $68$ cm$^{2}$, find $\angle BAC$.

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