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>\(ABD\) is a triangle, \(ADC\) is a straight line, \(AB=8\text{ cm}\), \(AD=9\text{ cm}\), and \(BD=10\text{ cm}\). Without using a calculator, find the exact value of \(\cos\angle BDC\).</p>
<div class="graph-block" data-graph-config="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" data-width="369" data-height="190" style="width:369px;height:190px;max-width:100%;display:block;margin:0 auto;"></div>
<p>Find the exact value of \(\cos\angle BDC\).<span class="inline-marks">[2]</span></p>

\(ABD\) is a triangle, \(ADC\) is a straight line, \(AB=8\text{ cm}\), \(AD=9\text{ cm}\), and \(BD=10\text{ cm}\). Without using a calculator, find the exact value of \(\cos\angle BDC\).

Find the exact value of \(\cos\angle BDC\).

Cosine Rule

2025 SJI PRELIMS P1 Q3

Trigonometry

<p>The boundary of a wildlife reserve is quadrilateral \(KLMN\).</p>
<div class="graph-block" data-graph-config="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" data-width="420" data-height="260" style="width:420px;height:260px;max-width:100%;display:block;margin:0 auto;"></div>
<p>\(KL=6.1\text{ km}\), \(LM=8.4\text{ km}\), \(MN=7.5\text{ km}\), \(\angle LMN=68^\circ\) and \(\angle KNL=36^\circ\).</p>
<p>The angle \(\angle LKN\) is acute. Give non-exact numerical answers correct to \(3\) significant figures.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q06-q-a" data-list-item-path="emath-aot-q06-q-a"><div class="list-item-paragraph">Find diagonal \(LN\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q06-q-b" data-list-item-path="emath-aot-q06-q-b"><div class="list-item-paragraph">Find \(\angle LKN\).<span class="inline-marks">[2]</span></div></li></ol>

The boundary of a wildlife reserve is quadrilateral \(KLMN\).

\(KL=6.1\text{ km}\), \(LM=8.4\text{ km}\), \(MN=7.5\text{ km}\), \(\angle LMN=68^\circ\) and \(\angle KNL=36^\circ\).

The angle \(\angle LKN\) is acute. Give non-exact numerical answers correct to \(3\) significant figures.

Sine Rule

Boundary of a wildlife reserve

<p>The diagram shows a coastal reserve bounded by four survey markers \(P\), \(Q\), \(R\) and \(S\) on level ground.</p>
<div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div>
<p>The bearing of \(P\) from \(S\) is \(342^\circ\) and \(SP=176\text{ m}\). Marker \(Q\) is \(80\text{ m}\) due west of \(P\).</p>
<p>The angle \(PQR\) is \(117^\circ\) and the bearing of \(P\) from \(R\) is \(052^\circ\).</p>
<p>Give non-exact numerical answers correct to \(3\) significant figures unless a different degree of accuracy is stated.</p>
<ol class="abc-list"><li data-list-item-id="emath-aot-q-a" data-list-item-path="emath-aot-q-a"><div class="list-item-paragraph">Find the bearing of \(R\) from \(P\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-b" data-list-item-path="emath-aot-q-b"><div class="list-item-paragraph">Find \(\angle QPR\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-c" data-list-item-path="emath-aot-q-c"><div class="list-item-paragraph">Show that the distance \(PR\) is \(168.66\text{ m}\), correct to two decimal places.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-d" data-list-item-path="emath-aot-q-d"><div class="list-item-paragraph">Find the area of triangle \(PQR\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="emath-aot-q-e" data-list-item-path="emath-aot-q-e"><div class="list-item-paragraph">Find \(\angle RPS\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="emath-aot-q-f" data-list-item-path="emath-aot-q-f"><div class="list-item-paragraph">Hence, calculate the distance \(SR\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="emath-aot-q-g" data-list-item-path="emath-aot-q-g"><div class="list-item-paragraph">An observation tower, \(24\text{ m}\) high, stands vertically at \(P\).</div><div class="list-item-paragraph">Find the angle of elevation of the top of the tower from \(S\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a coastal reserve bounded by four survey markers \(P\), \(Q\), \(R\) and \(S\) on level ground.

The bearing of \(P\) from \(S\) is \(342^\circ\) and \(SP=176\text{ m}\). Marker \(Q\) is \(80\text{ m}\) due west of \(P\).

The angle \(PQR\) is \(117^\circ\) and the bearing of \(P\) from \(R\) is \(052^\circ\).

Give non-exact numerical answers correct to \(3\) significant figures unless a different degree of accuracy is stated.

Area of Triangle

Angles of Elevation and Depression

Bearings

Surveying a coastal reserve

<p>\(AB\), \(AC\) and \(AD\) are chords of a circle such that&nbsp;\(AB = 4\) cm, \(AC = 10\) cm and \(AD = 11\) cm. Given that&nbsp;\(AC\) bisects \(\angle BAD\), find the area of the circle.</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/1770728619550-uu6wi9exw8l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3Mjg2MTk1NTAtdXU2d2k5ZXh3OGwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.HxpsatHGdGFhzewrwf6eqHbee_qebOQmOhAzv3WjtkE" style="max-width: min(160px, 100%); height: auto; display: block;" alt="Chords in a circle" class="alignright custom-width" /></figure></div>

\(AB\), \(AC\) and \(AD\) are chords of a circle such that \(AB = 4\) cm, \(AC = 10\) cm and \(AD = 11\) cm. Given that \(AC\) bisects \(\angle BAD\), find the area of the circle.

Angles in the Same Segment

Equal Chords and Subtended Angles

Angles in a Semicircle and the Converse

Circumference and Area of Circles

IP2U14 HOT Q5

Properties of Circle

Perimeter and Area of Plane Figures

<p>In triangle \(ABC\), \(AB = 4\) cm, \(AC = 3\) cm and \(BC = 6\) cm. Triangle \(A'B'C\) is obtained by rotating triangle \(ABC\) in clockwise by \(\frac{\pi}{2}\) rad with \(C\) as the centre. Find the exact area of the shaded region.</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/1770721740037-13c3fputrpa.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MjE3NDAwMzctMTNjM2ZwdXRycGEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.1uxvpmrS6oQ4iYZ7wKtfMFpQaFO3JqNK4eItXojAl0w" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div>

In triangle \(ABC\), \(AB = 4\) cm, \(AC = 3\) cm and \(BC = 6\) cm. Triangle \(A'B'C\) is obtained by rotating triangle \(ABC\) in clockwise by \(\frac{\pi}{2}\) rad with \(C\) as the centre. Find the exact area of the shaded region.

IP2U13 HOT Q2

<p>In the diagram below, \(O\) is the centre of the circle with radius \(4\) cm and passing through the points \(A\), \(B\), \(C\) and \(D\). \(E\) is the midpoint of \(OC\). \(AOEC\) is a straight line. \(A\) is the centre of a circle through \(B\), \(E\) and \(D\).</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">Show that&nbsp;\(\angle AOB\) is approximately \(1.70\) rad.</div></li><li><div class="list-item-paragraph">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">the area of sector \(OBCD\),</div></li><li><div class="list-item-paragraph">the perimeter of the shaded region,</div></li><li><div class="list-item-paragraph">the area of the shaded region.</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/1770717368151-kxff6a7eiqb.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MTczNjgxNTEta3hmZjZhN2VpcWIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.QLZbkUTWYFGYQBOkRT9hBLAyztHhSNWU4TulDPVAlpw" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram below, \(O\) is the centre of the circle with radius \(4\) cm and passing through the points \(A\), \(B\), \(C\) and \(D\). \(E\) is the midpoint of \(OC\). \(AOEC\) is a straight line. \(A\) is the centre of a circle through \(B\), \(E\) and \(D\).

IP2U13 13.4 PN1 Q10

<p>The diagram shows a sector of a circle with centre \(O\) and radius \(8\) cm.</p>
<p>\(P\) is a point on \(OA\) such that&nbsp;\(4OP = 3OA\) and \(Q\) is a point on \(OB\) such that&nbsp;\(5OQ = 2OB\).</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>Given that&nbsp;\(\angle AOB = 0.8\) radians, calculate</p><ol class="i-list"><li><div class="list-item-paragraph">the length of \(PQ\),</div></li><li><div class="list-item-paragraph">the arc length \(AB\),</div></li><li><div class="list-item-paragraph">the area of the shaded region.</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/1770715613188-j6lcrix7ouc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA3MTU2MTMxODgtajZsY3JpeDdvdWMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.deZ_6PmzoJ7MJpMJn-fDQzlgPI9ZzfGBAYi1gqAzjgQ" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

The diagram shows a sector of a circle with centre \(O\) and radius \(8\) cm.

\(P\) is a point on \(OA\) such that \(4OP = 3OA\) and \(Q\) is a point on \(OB\) such that \(5OQ = 2OB\).

IP2U13 13.4 PN1 Q8

<p>In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively.</p>
<p>Let&nbsp;\(s = \frac{a + b + c}{2}\), show that the area of triangle is given by \(\sqrt{s(s - a)(s - b)(s - c)}\).</p>

In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively.

Let \(s = \frac{a + b + c}{2}\), show that the area of triangle is given by \(\sqrt{s(s - a)(s - b)(s - c)}\).

Heron's Formula

<p>In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively. It is known that&nbsp;\(a = \sqrt{3} \sin A\).</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;\(\frac{abc}{2c^2 + 2ab \cos C} \leq \frac{\sqrt{3} \sin C}{4 - 2 \cos C}.\)</div></li></ol>
<p>You may find the famous <strong>Inequality of Arithmetic and Geometric Means (AM-GM Inequality)</strong><br>useful: For any non-negative real numbers \(x_1\) and \(x_2\), it holds that \[AM = \frac{x_1 + x_2}{2} \geq \sqrt{x_1 x_2} = GM,\]where&nbsp;\(AM\) and \(GM\) denotes the arithmetic mean and geometric mean of \(x_1\) and \(x_2\), respectively.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, show that the maximum value of \(\frac{abc}{2c^2 + 2ab \cos C}\) is \(\frac{1}{2}\) and this maximum value is achieved when&nbsp;\(\Delta ABC\) is an equilateral triangle.</span></div></li></ol>

In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively. It is known that \(a = \sqrt{3} \sin A\).

Show that \[\frac{abc}{2c^2 + 2ab \cos C} \leq \frac{\sqrt{3} \sin C}{4 - 2 \cos C}.\]

You may find the famous **Inequality of Arithmetic and Geometric Means (AM-GM Inequality)**
useful: For any non-negative real numbers \(x_1\) and \(x_2\), it holds that \[AM = \frac{x_1 + x_2}{2} \geq \sqrt{x_1 x_2} = GM,\]where \(AM\) and \(GM\) denotes the arithmetic mean and geometric mean of \(x_1\) and \(x_2\), respectively.

Hence, or otherwise, show that the maximum value of \(\frac{abc}{2c^2 + 2ab \cos C}\) is \(\frac{1}{2}\) and this maximum value is achieved when \(\Delta ABC\) is an equilateral triangle.

R-Formula and Trigonometric Extrema

Application of AM-GM Inequality in Trigonometry

<p>In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively.</p>
<p>It is given that&nbsp;\(b^2 - a^2 = ac\), show that&nbsp;\(\frac{a}{b} = \frac{\cos A}{1 + \cos B}\).</p>

It is given that \(b^2 - a^2 = ac\), show that \(\frac{a}{b} = \frac{\cos A}{1 + \cos B}\).

IP2 U12 HOT Q4

<p>In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively.</p>
<p>It is given that&nbsp;\(c = 2\) and \(b = \sqrt{2}a\), show that the maximum area of triangle \(ABC\) is \(2\sqrt{2}\).</p>
<p>Hence find the exact value of \(a\) when the area of triangle is maximised.</p>

It is given that \(c = 2\) and \(b = \sqrt{2}a\), show that the maximum area of triangle \(ABC\) is \(2\sqrt{2}\).

Hence find the exact value of \(a\) when the area of triangle is maximised.

IP2 U12 HOT Q3

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MzMwMjkyOTEtdTJ5cHg1ZW51MS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0ODMxLCJleHAiOjE3OTM1MDUxMzF9.AbbY-Os5fMOdomyjwqwesw6OTBZJKVnCXnMCK11YHCg" 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 \(\angle ADC = 90^\circ\).

Applications of Trigonometric Ratios

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

<p>In the diagram, \(BCD\) is a straight line, \(BC = 6\) cm, \(AC = 7.9\) cm, \(AD = 6.4\) cm and \(\angle ACB = 130^\circ\).</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;"><p>Calculate</p><ol class="i-list"><li><div class="list-item-paragraph">\(AB\),</div></li><li><div class="list-item-paragraph">the area of triangle \(ABC\),</div></li><li><div class="list-item-paragraph">the shortest distance from \(B\) to \(AC\) produced,</div></li><li><div class="list-item-paragraph">\(\angle ADC\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: center;"><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/1770532306077-rvbo3a2qk29.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MzIzMDYwNzctcnZibzNhMnFrMjkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.qy7vneIhvChm8t_V9HmPo0RmPFz8nyKeeFrVGr1w_BQ" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, \(BCD\) is a straight line, \(BC = 6\) cm, \(AC = 7.9\) cm, \(AD = 6.4\) cm and \(\angle ACB = 130^\circ\).

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q10

<p>In the diagram, \(AB = 4\) cm, \(BC = 7.5\) cm, \(AD = 5\) cm, \(BD = 6.2\) cm and \(\angle BDC = 55^\circ\).</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;"><p>Calculate</p><ol class="i-list"><li><div class="list-item-paragraph">\(\angle ADB\),</div></li><li><div class="list-item-paragraph">\(\angle BCD\),</div></li><li><div class="list-item-paragraph">the area of \(ABCD\).</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/1770531648311-ftgm9c3c6f5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MzE2NDgzMTEtZnRnbTljM2M2ZjUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.x83IAHEoEWK-nZipdCxoebm_MKJR-gRSmRayhAjaEmQ" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, \(AB = 4\) cm, \(BC = 7.5\) cm, \(AD = 5\) cm, \(BD = 6.2\) cm and \(\angle BDC = 55^\circ\).

<p>The diagram shows a quadrilateral \(ABCD\).</p>
<p>The length of \(AB\) is \(5\) m, the length \(BC\) is \(8\) m and the length of \(AD\) is \(4\) m.</p>
<p>Angle \(ABC\) is \(60^\circ\) and angle \(ADC\) is \(100^\circ\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Calculate the length of \(AC\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Calculate angle \(ACD\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the quadrilateral \(ABCD\), correct to the nearest whole number.</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/1770530950473-g12h1z2uknv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MzA5NTA0NzMtZzEyaDF6MnVrbnYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.GWd4O72n9ubK7dFe7dij94qABIHIjnLO07Okxc-k3yQ" style="max-width: min(220px, 100%); height: auto; display: block;" alt="Diagram of quadrilateral ABCD" class="aligncenter custom-width" /></figure></div>

The diagram shows a quadrilateral \(ABCD\).

The length of \(AB\) is \(5\) m, the length \(BC\) is \(8\) m and the length of \(AD\) is \(4\) m.

Angle \(ABC\) is \(60^\circ\) and angle \(ADC\) is \(100^\circ\).

Find the area of the quadrilateral \(ABCD\), correct to the nearest whole number.

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q8

<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 the diagram, \(AB = 6\) cm, \(BC = 4\) cm and \(AC = 5\) cm.<br>Without finding any angle, calculate</p><ol class="i-list"><li><div class="list-item-paragraph">the value of \(\cos C\).</div></li><li><div class="list-item-paragraph">the value of ratio \(\frac{\sin A}{\sin C}\).</div></li><li><div class="list-item-paragraph">find \(\sin B\) if the area of \(\Delta ABC = 9.9\text{ cm}^2\).</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/1770530337592-qftt32i4f4f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1MzAzMzc1OTItcWZ0dDMyaTRmNGYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.gYGT3RTPB2oAosE81xPUzuLtamd9CJUt81UBiNZsnUc" style="max-width: min(220px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

IP2U12 Exercise Trigonometry II (also for 3EM U7 Worksheet) Q7

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA1Mjk0NjEzODktMXlxdHU1ZXdubzEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.F1PlwLtyyrP7K6j5F9Pxvwtym3bxLw6e1_TYf28C2bs" 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

Relationship between Trigonometric Ratios of Acute and Obtuse Angles

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

<p>The diagram shows a sector of a circle with centre \(O\) and radius \(8\) cm.<br>\(P\) is a point on \(OA\) such that&nbsp;\(4OP=3OA\) and \(Q\) is a point on \(OB\) such that&nbsp;\(5OQ=2OB\).<br>Given that&nbsp;\(\angle AOB=0.8\) radians, calculate</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">the length of \(PQ\).</div></li><li><div class="list-item-paragraph">the arc length \(AB\).</div></li><li><div class="list-item-paragraph">the area of the shaded region.</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/1767586896466-1iqg1zboofq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1ODY4OTY0NjYtMWlxZzF6Ym9vZnEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.yUGZ60mcaZaiM6W3hS6Pwfl09yYzzV_V_mCK57VFcGA" style="max-width: min(270px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows a sector of a circle with centre \(O\) and radius \(8\) cm.
\(P\) is a point on \(OA\) such that \(4OP=3OA\) and \(Q\) is a point on \(OB\) such that \(5OQ=2OB\).
Given that \(\angle AOB=0.8\) radians, calculate

Perimeter and Area of Composite Figures

Area of Sector

Arc Length

Linear and Angular Measure Relationships

Arc Length and Area of Sector

<p>The diagram shows an isosceles triangle \(ABC\) in which \(BC=AC=10\) cm, and \(\angle BAC=0.7\) radians.<br>\(DC\) is an arc of a circle with centre \(A\) and radius \(AC\).</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><div class="list-item-paragraph">Show that&nbsp;\(\angle ACB=1.74\) rad.</div></li><li><div class="list-item-paragraph">Find the area of the shaded region.</div></li><li><div class="list-item-paragraph">Find the perimeter of the shaded region.</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/1767586569442-i1z6c3u46xq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1ODY1Njk0NDItaTF6NmMzdTQ2eHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.JyftgJVBj7mWn-kjZVI_LXZpabilkZZub5l7ffRPs9I" style="max-width: min(400px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The diagram shows an isosceles triangle \(ABC\) in which \(BC=AC=10\) cm, and \(\angle BAC=0.7\) radians.
\(DC\) is an arc of a circle with centre \(A\) and radius \(AC\).

Properties of Special Triangles

Radian Measure in Isosceles Triangles

Polygons

<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 the diagram below, \(APB\) is an arc of a circle centre \(O\) and radius \(5\) cm.<br>\(AQB\) is semicircle with \(AB\) as its diameter and \(\angle AOB = \frac{4}{9}\pi\).</p><ol class="i-list"><li><div class="list-item-paragraph">Show that the length of \(AB = 6.43\) cm.</div></li><li><div class="list-item-paragraph">Calculate the length of the arc \(APB\).</div></li><li><div class="list-item-paragraph">Calculate the area of the semi-circle \(AQB\).</div></li><li><div class="list-item-paragraph">Calculate the area of the \(\Delta AOB\) and hence, find the area of shaded region.</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/1767585912665-lxge0hrp2js.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1ODU5MTI2NjUtbHhnZTBocnAyanMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDgzMSwiZXhwIjoxNzkzNTA1MTMxfQ.PGf1e8xMkAO-mjEUwuF2tfYZ6TQv_PBSdoaeTPAd40c" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Math Question Bank