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PSLE Foundation Mathematics

<ol class="abc-list"><li data-list-item-id="tkss-p2-q11-q-0-content-items-0" data-list-item-path="tkss-p2-q11-q-0-content-items-0"><div class="list-item-paragraph">Prove \(\dfrac{\cos 2\theta}{\tan \theta}+\sin 2\theta=\cot \theta\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="tkss-p2-q11-q-0-content-items-1" data-list-item-path="tkss-p2-q11-q-0-content-items-1"><div class="list-item-paragraph">Solve \(3\sin x\tan x-\sec x=1\) for&nbsp;\(-180^\circ\leq x\leq 180^\circ\).<span class="inline-marks">[6]</span></div></li></ol>

Prove \(\dfrac{\cos 2\theta}{\tan \theta}+\sin 2\theta=\cot \theta\).

Solve \(3\sin x\tan x-\sec x=1\) for \(-180^\circ\leq x\leq 180^\circ\).

Trigonometric Equations

Compound and Double-Angle Identities

Proving and Simplifying Trigonometric Identities

2025 TKSS PRELIMS P2 Q11

Trigonometry

<p>Solve the equation \(9 \sin x = -2 \sec x\) for&nbsp;\(-\pi \le x \le 0\).<span class="inline-marks">[4]</span></p>

Solve the equation \(9 \sin x = -2 \sec x\) for \(-\pi \le x \le 0\).

2025 BVSS PRELIMS P1 Q2

<ol class="abc-list"><li data-list-item-id="t06-p1-q06-a" data-list-item-path="t06-p1-q06-a"><div class="list-item-paragraph">Show that&nbsp;\(\cos2nx=\cos((2n+1)x)\cos x+\sin((2n+1)x)\sin x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="t06-p1-q06-b" data-list-item-path="t06-p1-q06-b"><div class="list-item-paragraph">Hence prove by mathematical induction that&nbsp;\(\sin x+\sin3x+\sin5x+\cdots+\sin((2n-1)x)=\frac{1-\cos2nx}{2\sin x}\) for all \(n\in\mathbb Z^+\), \(\sin x\ne0\).<span class="inline-marks">[7]</span></div></li><li data-list-item-id="t06-p1-q06-c" data-list-item-path="t06-p1-q06-c"><div class="list-item-paragraph">Hence solve \(\sin x+\sin3x=\cos x\) for&nbsp;\(0&lt;x&lt;\pi\).<span class="inline-marks">[5]</span></div></li></ol>

Show that \(\cos2nx=\cos((2n+1)x)\cos x+\sin((2n+1)x)\sin x\).

Hence prove by mathematical induction that \(\sin x+\sin3x+\sin5x+\cdots+\sin((2n-1)x)=\frac{1-\cos2nx}{2\sin x}\) for all \(n\in\mathbb Z^+\), \(\sin x\ne0\).

Hence solve \(\sin x+\sin3x=\cos x\) for \(0<x<\pi\).

Proof by Mathematical Induction

Mathematical Induction

<p>Two supplementary angles \(x\) and \(y\) are such that&nbsp;\(\sin(x-y)=\dfrac{1}{2}\). Find the possible values of \(x\) and \(y\) in radians, given that&nbsp;\(x\) is an obtuse angle whereas \(y\) is an acute angle.</p>

Two supplementary angles \(x\) and \(y\) are such that \(\sin(x-y)=\dfrac{1}{2}\). Find the possible values of \(x\) and \(y\) in radians, given that \(x\) is an obtuse angle whereas \(y\) is an acute angle.

Supplementary angles from a sine difference

<p>It is given that&nbsp;\(\cos(3x)=\sin(2x)\). Find the value of \(x\) in degrees if&nbsp;\(3x\) is an acute angle.</p>

It is given that \(\cos(3x)=\sin(2x)\). Find the value of \(x\) in degrees if \(3x\) is an acute angle.

Symmetry of Trigonometric Functions

Solving a cofunction equation with acute angles

<p>It is given that&nbsp;\(\cos(7x)=\sin(2x)\). Find the value of \(x\) in degrees if&nbsp;\(7x\) is an acute angle.</p>

It is given that \(\cos(7x)=\sin(2x)\). Find the value of \(x\) in degrees if \(7x\) is an acute angle.

Cofunction equation with a larger multiple

Solve \(\operatorname{cosec}x=-2\) for \(0^\circ\le x\le360^\circ\).

Equation with a reciprocal ratio

<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Prove the identity \(\ \operatorname{cosec} 4x + \cot 4x = \cot 2x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Hence, find, for&nbsp;\(-180^\circ \le \theta \le 180^\circ\), the values of \(\theta\) for which \(\ \operatorname{cosec} 2\theta + \cot 2\theta = -\sqrt{3}\).<span class="inline-marks">[4]</span></div></li></ol>

2024 GESS PRELIMS P2 Q7

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODE1MDgxNDIyODItbWF4bTBnbnl1ajgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwODY5MCwiZXhwIjoxNzkzNTA4OTkwfQ.wnilX6Vddljqr_uoqLHAvS4rGvzPZUePUgvYEdfjblI" 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

R-Formula and Trigonometric Extrema

Applications of Trigonometric Ratios

2024 FMSS PRELIMS P2 Q9

<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>

Exact Trigonometric Values and the Unit Circle

2024 FMSS PRELIMS P2 Q5

<ol class="i-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that&nbsp;\(\frac{\tan x}{1 + \sec x} + \frac{1 + \sec x}{\tan x} = \frac{2}{\sin x}\).<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Hence solve the equation \(\frac{\tan x}{1 + \sec x} + \frac{1 + \sec x}{\tan x} = 1 + 3\sin x\) for&nbsp;\(0^{\circ} \le x \le 360^{\circ}\).<span class="inline-marks">[4]</span></div></li></ol>

2024 FMSS PRELIMS P1 Q10

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">If&nbsp;\(a \cos^2 \theta + b \sin^2 \theta = c\), show that&nbsp;\(\tan^2 \theta = \frac{a - c}{c - b}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-bi" data-list-item-path="item-bi"><ol class="i-list"><li data-list-item-id="subitem-1781085512961-qeke46jc7" data-list-item-path="item-bi/subitem-1781085512961-qeke46jc7"><div class="list-item-paragraph">Given that&nbsp;\(\sqrt{ (2 \cos^2 x - \sin x) } = 2 \sin x\), show that&nbsp;\(6 \sin^2 x + \sin x - 2 = 0\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1781085513516-spykf9uxq" data-list-item-path="item-bi/subitem-1781085513516-spykf9uxq"><div class="list-item-paragraph">Hence, solve the equation \(\sqrt{ (2 \cos^2 x - \sin x) } = 2 \sin x\) for&nbsp;\(0\le x\le2\pi\).<span class="inline-marks">[4]</span></div></li></ol></li></ol>

2024 CCHY P1 Q11

<ol class="abc-list"><li><div class="list-item-paragraph">Express \( 2\cos^2{x} + \sin{x}\) in terms of \(\sin{x}\) only.</div></li><li><div class="list-item-paragraph">Solve the equation \(2\cos^2{x} + \sin{x} = 2\) for&nbsp;\(x\) in the interval \(0 \leq x \leq \pi\), giving your answers exactly.</div></li></ol>

<ol class="abc-list"><li><div class="list-item-paragraph">Write the expression \(3\sin^2{x} + 4\cos{x}\) in the form \(a\cos^2{x} + b\cos{x} + c\).</div></li><li><div class="list-item-paragraph">Hence or otherwise, solve the equation \(3\sin^2{x} + 4\cos{x} - 4 = 0\), for&nbsp;\(0^{\circ} \leq x \leq 90^{\circ}\).</div></li></ol>

<p>Consider the trigonometric equation \(2\sin^2{x} = 1 + \cos{x}\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write this equation in the form \(f(x) = 0\), where&nbsp;\(f(x) = a\cos^2{x} + b\cos{x} + c\), and \(a, b, c \in \mathbb{Z}\).</div></li><li><div class="list-item-paragraph">Factorise \(f(x)\).</div></li><li><div class="list-item-paragraph">Solve \(f(x) = 0\) for&nbsp;\(0^{\circ} \leq x \leq 360^{\circ}\).</div></li></ol>

Consider the trigonometric equation \(2\sin^2{x} = 1 + \cos{x}\).

<p>Solve the equation \(2\cos^2{x} = \sin{2x}\) for&nbsp;\(0 \leq x \leq \pi\), giving your answers in terms of \(\pi\).</p>

Solve the equation \(2\cos^2{x} = \sin{2x}\) for \(0 \leq x \leq \pi\), giving your answers in terms of \(\pi\).

<p>The function \(f\) is defined by \(f:x \rightarrow 30\sin{3x}\cos{3x}\), \(0 \leq x \leq \frac{\pi}{3}\)</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Write down an expression for&nbsp;\(f(x)\) in the form \(a\sin{6x}\), where&nbsp;\(a\) is an integer.</div></li><li><div class="list-item-paragraph">Solve \(f(x) = 0\), giving your answers in terms of \(\pi\).</div></li></ol>

The function \(f\) is defined by \(f:x \rightarrow 30\sin{3x}\cos{3x}\), \(0 \leq x \leq \frac{\pi}{3}\)

<p>Solve \(\cos{2x} - 3\cos{x} - 3 - \cos^2{x} = \sin^2{x}\), for&nbsp;\(0 \leq x \leq 2\pi\).</p>

Solve \(\cos{2x} - 3\cos{x} - 3 - \cos^2{x} = \sin^2{x}\), for \(0 \leq x \leq 2\pi\).

<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(4 - \cos{2\theta} + 5\sin{\theta} = 2\sin^2{\theta} + 5\sin{\theta} +3\).</div></li><li><div class="list-item-paragraph"><strong>Hence</strong>, solve the equation \(4 - \cos{2\theta} + 5\sin{\theta} = 0\) for&nbsp;\(0 \leq \theta \leq 2\pi\).</div></li></ol>

<p>Solve the equation \(2\cos{x} = \sin{2x}\), for&nbsp;\(0 \leq x \leq 3\pi\).</p>

Solve the equation \(2\cos{x} = \sin{2x}\), for \(0 \leq x \leq 3\pi\).

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