A Math

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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

<ol class="i-list"><li data-list-item-id="atan-30" data-list-item-path="atan-30"><div class="list-item-paragraph">Using a right-angled triangle, show that, for&nbsp;$a&gt;0$, \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\]<span class="inline-marks">[2]</span></div></li></ol>
<p>The curves $D_1$ and $D_2$ have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where&nbsp;$k&gt;0$.</p>
<ol class="i-list" start="2"><li data-list-item-id="atan-33" data-list-item-path="atan-33"><div class="list-item-paragraph">Determine the values of $k$ for which the curves have at least one point of intersection.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts, take $k=5$.</p>
<ol class="i-list" start="3"><li data-list-item-id="atan-36" data-list-item-path="atan-36"><div class="list-item-paragraph">Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of $D_1$, and all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="atan-37" data-list-item-path="atan-37"><div class="list-item-paragraph">Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part <strong>(i)</strong>.<span class="inline-marks">[5]</span></div></li></ol>

Using a right-angled triangle, show that, for $a>0$, \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\]

The curves $D_1$ and $D_2$ have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where $k>0$.

Determine the values of $k$ for which the curves have at least one point of intersection.

Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of $D_1$, and all asymptotes.

Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part **(i)**.

Sketching Rational Functions

Inverse Trigonometric Functions

Parameter Conditions for Graphs and Equations

Asymptotes and Discontinuities

Area bounded by 2 curves

A complementary-angle identity and a shifted-curve area

Applications of Integration

Graphing Techniques

Functions

<ol class="i-list"><li data-list-item-id="atan-21" data-list-item-path="atan-21"><div class="list-item-paragraph">By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\]<span class="inline-marks">[1]</span></div></li></ol>
<p>The curves $C_1$ and $C_2$ have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where&nbsp;$k&gt;0$.</p>
<ol class="i-list" start="2"><li data-list-item-id="atan-24" data-list-item-path="atan-24"><div class="list-item-paragraph">Find the set of values of $k$ for which the curves have at least one point of intersection.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts, take $k=\frac32$.</p>
<ol class="i-list" start="3"><li data-list-item-id="atan-27" data-list-item-path="atan-27"><div class="list-item-paragraph">Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="atan-28" data-list-item-path="atan-28"><div class="list-item-paragraph">Find the exact area of the finite region enclosed by the curves. Use the identity in part <strong>(i)</strong> to simplify your answer.<span class="inline-marks">[5]</span></div></li></ol>

By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\]

The curves $C_1$ and $C_2$ have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where $k>0$.

Find the set of values of $k$ for which the curves have at least one point of intersection.

For the remaining parts, take $k=\frac32$.

Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.

Find the exact area of the finite region enclosed by the curves. Use the identity in part **(i)** to simplify your answer.

An inverse-tangent identity and an enclosed rational-curve area

<p>The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0\leq x\leq k,\] where&nbsp;$k&gt;0$.</p>
<ol class="abc-list"><li data-list-item-id="rf-65" data-list-item-path="rf-65"><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)$, where&nbsp;$R&gt;0$ and $0&lt;\alpha&lt;\frac{\pi}{2}$, giving the exact values of $R$ and $\alpha$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-66" data-list-item-path="rf-66"><div class="list-item-paragraph">Find the largest value of $k$ for which $\mathrm{f}^{-1}$ exists. For this value of $k$, find $\mathrm{f}^{-1}(x)$ and state its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="rf-67" data-list-item-path="rf-67"><div class="list-item-paragraph"><p>Using the value of $k$ found in part <strong>(b)</strong>, sketch $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram. Label the endpoints of both curves and indicate their relationship to the line $y=x$.</p><span class="inline-marks">[3]</span></div></li></ol>
<p>The functions $\mathrm{g}$ and $\mathrm{h}$ are defined by \[\mathrm{g}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0&lt;x\leq k,\] \[\mathrm{h}:x\mapsto\frac{1}{x-\sqrt{3}},\quad x&gt;\sqrt{3},\]</p>
<p>where&nbsp;$k$ has the value found in part <strong>(b)</strong>.</p>
<ol class="abc-list" start="4"><li data-list-item-id="rf-70" data-list-item-path="rf-70"><div class="list-item-paragraph">Explain why $\mathrm{hg}$ exists. Find $\mathrm{hg}(x)$ and state its range in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0\leq x\leq k,\] where $k>0$.

Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$, giving the exact values of $R$ and $\alpha$.

Find the largest value of $k$ for which $\mathrm{f}^{-1}$ exists. For this value of $k$, find $\mathrm{f}^{-1}(x)$ and state its domain.

The functions $\mathrm{g}$ and $\mathrm{h}$ are defined by \[\mathrm{g}:x\mapsto\sin 2x+\sqrt{3}\cos 2x,\quad 0<x\leq k,\] \[\mathrm{h}:x\mapsto\frac{1}{x-\sqrt{3}},\quad x>\sqrt{3},\]

where $k$ has the value found in part **(b)**.

Explain why $\mathrm{hg}$ exists. Find $\mathrm{hg}(x)$ and state its range in exact form.

Composite Functions

Sketching a Function and Its Inverse

Existence of Composite Functions

Restricting a Domain to Obtain an Inverse

Range of a Composite Function

Inverse Functions

Inverting a shifted sine and forming a reciprocal composite

<p>The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\pi\leq x\leq\pi.\]</p>
<ol class="abc-list"><li data-list-item-id="rf-33" data-list-item-path="rf-33"><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the form $R\cos(x+\alpha)$, where&nbsp;$R&gt;0$ and $0&lt;\alpha&lt;\frac{\pi}{2}$, giving the exact values of $R$ and $\alpha$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-34" data-list-item-path="rf-34"><div class="list-item-paragraph">Hence state the range of $\mathrm{f}$ and sketch the graph of $y=\mathrm{f}(x)$, indicating the coordinates of the turning points and the intersections with the coordinate axes.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{g}$ is defined by \[\mathrm{g}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\alpha\leq x\leq b,\] where&nbsp;$b&gt;-\alpha$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="rf-37" data-list-item-path="rf-37"><div class="list-item-paragraph">Find the largest value of $b$ for which $\mathrm{g}^{-1}$ exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="rf-38" data-list-item-path="rf-38"><div class="list-item-paragraph">Using this value of $b$, find $\mathrm{g}^{-1}(x)$, stating its domain and range.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by \[\mathrm{f}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\pi\leq x\leq\pi.\]

Express $\mathrm{f}(x)$ in the form $R\cos(x+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$, giving the exact values of $R$ and $\alpha$.

Hence state the range of $\mathrm{f}$ and sketch the graph of $y=\mathrm{f}(x)$, indicating the coordinates of the turning points and the intersections with the coordinate axes.

The function $\mathrm{g}$ is defined by \[\mathrm{g}:x\mapsto\sqrt{3}\cos x-\sin x,\quad -\alpha\leq x\leq b,\] where $b>-\alpha$.

Find the largest value of $b$ for which $\mathrm{g}^{-1}$ exists.

Using this value of $b$, find $\mathrm{g}^{-1}(x)$, stating its domain and range.

Range of a Function

Restricting a shifted cosine for an inverse

<p>The function $\mathrm{f}$ and $\mathrm{g}$ are defined as</p>
<p style="margin-left: 11rem">$\mathrm{f}: x \mapsto \pi x, \quad x \in \mathbb{R}.$<br>$\mathrm{g}: x \mapsto \begin{cases} \quad \sin x & \text{for}\hspace{0.5em} 0 \leq x \leq \frac{\pi}{2}, \\ \cos x + 1 & \text{for}\hspace{0.5em} \frac{\pi}{2} &lt; x \leq \pi. \end{cases}$</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-3-item-0" data-list-item-path="imported-item-block-3-item-0"><div class="list-item-paragraph">Find $\mathrm{f}^{25}(x)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="imported-item-block-3-item-1" data-list-item-path="imported-item-block-3-item-1"><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{g}(x)$ for&nbsp;$0 \le x \le \pi$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-3-item-2" data-list-item-path="imported-item-block-3-item-2"><div class="list-item-paragraph">If the domain of $\mathrm{g}$ is restricted to $D = (a, \pi]$ such that&nbsp;$\mathrm{g}^{-1}$ exists, state the smallest exact value of $a$. Hence find $\mathrm{g}^{-1}(x)$ and state the domain of $\mathrm{g}^{-1}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-3" data-list-item-path="imported-item-block-3-item-3"><div class="list-item-paragraph"><p>For the domain $D$ found in part <strong>(c)</strong>, show that the composite function $\mathrm{fg}$ exists and find the exact range of $\mathrm{fg}$.</p><span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-3-item-4" data-list-item-path="imported-item-block-3-item-4"><div class="list-item-paragraph">Without finding $\mathrm{f}^{-1}$ and $\mathrm{fg}$, find the exact solution of $\mathrm{f}^{-1}(x) = \mathrm{g}\left(\frac{3\pi}{4}\right)$.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ and $\mathrm{g}$ are defined as

$\mathrm{f}: x \mapsto \pi x, \quad x \in \mathbb{R}.$
$\mathrm{g}: x \mapsto \begin{cases} \quad \sin x & \text{for } 0 \leq x \leq \frac{\pi}{2}, \\ \cos x + 1 & \text{for } \frac{\pi}{2} < x \leq \pi. \end{cases}$

Sketch the graph of $y = \mathrm{g}(x)$ for $0 \le x \le \pi$.

If the domain of $\mathrm{g}$ is restricted to $D = (a, \pi]$ such that $\mathrm{g}^{-1}$ exists, state the smallest exact value of $a$. Hence find $\mathrm{g}^{-1}(x)$ and state the domain of $\mathrm{g}^{-1}$.

Without finding $\mathrm{f}^{-1}$ and $\mathrm{fg}$, find the exact solution of $\mathrm{f}^{-1}(x) = \mathrm{g}\left(\frac{3\pi}{4}\right)$.

Piecewise-Defined Functions

Iterates $f^n$ for Large $n$

2025 HCI Promo Q8

<p>Find the exact value of each of the following, giving your answers in degrees where angles appear.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\dfrac{\cos^{-1}\left(-\dfrac{1}{2}\right)}{\sin^{-1}\dfrac{1}{2}}$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\dfrac{\tan^{-1}\left(-\sqrt{3}\right)}{\tan^{-1}1}$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$\dfrac{\sin^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)}{\cos^{-1}0}$</div></li></ol>

Find the exact value of each of the following, giving your answers in degrees where angles appear.

$\dfrac{\cos^{-1}\left(-\dfrac{1}{2}\right)}{\sin^{-1}\dfrac{1}{2}}$

$\dfrac{\tan^{-1}\left(-\sqrt{3}\right)}{\tan^{-1}1}$

$\dfrac{\sin^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)}{\cos^{-1}0}$

Ratios of inverse trigonometric values in degrees

Trigonometry

<p>Find the exact value of each of the following, giving your answers in radians.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\sin^{-1}\dfrac{\sqrt{3}}{2}+\cos^{-1}\dfrac{\sqrt{3}}{2}$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\tan^{-1}\sqrt{3}-\tan^{-1}1$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$\sin^{-1}\left(-\dfrac{1}{2}\right)+\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$</div></li></ol>

Find the exact value of each of the following, giving your answers in radians.

$\sin^{-1}\dfrac{\sqrt{3}}{2}+\cos^{-1}\dfrac{\sqrt{3}}{2}$

$\sin^{-1}\left(-\dfrac{1}{2}\right)+\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$

Exact values of inverse trigonometric expressions in radians

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le k\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that the largest exact value of $k$ such that&nbsp;$\mathrm{f}^{-1}$ exists is $\frac{1}{3}\pi$. Find $\mathrm{f}^{-1}(x)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">It is given that&nbsp;$k = \frac{1}{3}\pi$. In a single diagram, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}^{-1}(x)$, labelling your graphs clearly.<span class="inline-marks">[3]</span></div></li></ol>
<p>The function $\mathrm{h}$ is defined by $\mathrm{h} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le \frac{1}{3}\pi, \, x \neq \frac{1}{12}\pi$.</p>
<p>Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left| 3 - \frac{1}{x} \right|, \, -3 \le x \le 3, \, x \neq 0$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Explain clearly why $\mathrm{gh}$ exists. Find $\mathrm{gh}(x)$ and its range.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Supposing $(\mathrm{gh})^{-1}$ exists for a restricted domain. Find the exact value of $x$ for which $(\mathrm{gh})^{-1}(x) = 0$. Show your working clearly.<span class="inline-marks">[2]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

\[\mathrm{f} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le k\]

Show that the largest exact value of $k$ such that $\mathrm{f}^{-1}$ exists is $\frac{1}{3}\pi$. Find $\mathrm{f}^{-1}(x)$.

It is given that $k = \frac{1}{3}\pi$. In a single diagram, sketch the graphs of $y = \mathrm{f}(x)$ and $y = \mathrm{f}^{-1}(x)$, labelling your graphs clearly.

The function $\mathrm{h}$ is defined by $\mathrm{h} : x \mapsto 3\sin \left( 2x - \frac{1}{6}\pi \right), \, 0 \le x \le \frac{1}{3}\pi, \, x \neq \frac{1}{12}\pi$.

Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left| 3 - \frac{1}{x} \right|, \, -3 \le x \le 3, \, x \neq 0$.

Explain clearly why $\mathrm{gh}$ exists. Find $\mathrm{gh}(x)$ and its range.

Supposing $(\mathrm{gh})^{-1}$ exists for a restricted domain. Find the exact value of $x$ for which $(\mathrm{gh})^{-1}(x) = 0$. Show your working clearly.

Modulus Functions

2024 DHS Promo Q10

<p>Without using a calculator, express the principal value of</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-content"><span class="list-item-text">$\cos^{-1} \frac{\sqrt{2}}{2}$ in degrees,</span></div></li><li data-list-item-id="item-ii"><div class="list-item-content"><span class="list-item-text">$\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ in radians in terms of $\pi$,</span></div></li><li data-list-item-id="item-iii"><div class="list-item-content"><span class="list-item-text">$\tan^{-1}(-\sqrt{3})$ in degrees,</span></div></li><li data-list-item-id="item-iv"><div class="list-item-content"><span class="list-item-text">$\cos^{-1} 1$ in radians in terms of $\pi$,</span></div></li><li data-list-item-id="item-v"><div class="list-item-content"><span class="list-item-text">$\sin^{-1} \frac{1}{2}$ in degrees, and</span></div></li><li data-list-item-id="item-vi"><div class="list-item-content"><span class="list-item-text">$\tan^{-1}(-1)$ in radians in terms of $\pi$.</span></div></li></ol>

Without using a calculator, express the principal value of

$\cos^{-1} \frac{\sqrt{2}}{2}$ in degrees,

$\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$ in radians in terms of $\pi$,

$\cos^{-1} 1$ in radians in terms of $\pi$,

$\sin^{-1} \frac{1}{2}$ in degrees, and

$\tan^{-1}(-1)$ in radians in terms of $\pi$.

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down the principal value of $\sin^{-1} \left( -\frac{1}{2} \right)$ in radian as a multiple of $\pi$.</span><span class="inline-marks">[1]</span></div></li><li><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768299348495-pkvrgsewu5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyOTkzNDg0OTUtcGt2cmdzZXd1NS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTA0NzI3LCJleHAiOjE3OTM1MDUwMjd9.moKO7ovLyc0buQTXlC9DaVA0-c-cHh9PUryfk0Czmwc" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">The diagram shows the curve $y = a \sin bx + c$ for $0 \le x \le \pi$ radians. The curve has a maximum point at $\left( \frac{\pi}{4}, 3 \right)$ and a minimum point at $\left( \frac{3\pi}{4}, -7 \right)$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why $c = -2$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why $b = 2$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the equation of the curve.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

Write down the principal value of $\sin^{-1} \left( -\frac{1}{2} \right)$ in radian as a multiple of $\pi$.

Trigonometric Graphs and Parameters

2023 Ahmad Ibrahim Sec AM S3 EOY Q5 

<p>Write down the principal value, in radians as a multiple of $\pi$, of $\cos^{-1} \left(- \frac{\sqrt{3}}{2} \right)$.</p>

Write down the principal value, in radians as a multiple of $\pi$, of $\cos^{-1} \left(- \frac{\sqrt{3}}{2} \right)$.

<p>Giving your answer in radians as a multiple of $\pi$, state the principal value of $\tan^{-1} \left(\frac{1}{\sqrt{3}}\right)$.</p>

Giving your answer in radians as a multiple of $\pi$, state the principal value of $\tan^{-1} \left(\frac{1}{\sqrt{3}}\right)$.

<p>Giving your answer in radians as a multiple of $\pi$, state the principal value of $\sin^{-1} \left(- \frac{\sqrt{3}}{2} \right)$ .</p>

Giving your answer in radians as a multiple of $\pi$, state the principal value of $\sin^{-1} \left(- \frac{\sqrt{3}}{2} \right)$ .

<p>Write down the principal value of $\tan^{-1} \left(-\sqrt{3} \right)$, leaving your answer in radian as a multiple of $\pi$.<br><br></p>

Write down the principal value of $\tan^{-1} \left(-\sqrt{3} \right)$, leaving your answer in radian as a multiple of $\pi$.



<p>Giving your answer in radians as a multiple of $\pi$, state the principal value of</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\cos^{-1} \left( \frac{\sqrt{3}}{2} \right)$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\sin^{-1} \left( -\frac{1}{2} \right)$.</span></div></li></ol>
<p>Use the results of part <strong>(a)</strong> and <strong>(b)</strong> to find the exact value of $\cos^{-1} \left( \frac{\sqrt{3}}{2} \right) - 2\sin^{-1} \left( -\frac{1}{2} \right)$.</p>

Giving your answer in radians as a multiple of $\pi$, state the principal value of

$\cos^{-1} \left( \frac{\sqrt{3}}{2} \right)$.

Use the results of part **(a)** and **(b)** to find the exact value of $\cos^{-1} \left( \frac{\sqrt{3}}{2} \right) - 2\sin^{-1} \left( -\frac{1}{2} \right)$.

<ol class="abc-list"><li><div class="list-item-paragraph">Write down the principal value of ${\sin^{-1}\left(-\frac{1}{2}\right)}$ in radians as a multiple of ${\pi}$.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1760869837214-parqvro3xsf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjA4Njk4MzcyMTQtcGFycXZybzN4c2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MzUwNDcyNywiZXhwIjoxNzkzNTA1MDI3fQ.dZG36kBIsijHwWMPQHobri5jC0Y1kfAtyVOuWIuHlGI" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><div class="list-item-paragraph">The diagram shows the curve ${y = a\sin{bx} + c}$ for ${0 \leq x \leq \pi}$ radians.</div><div class="list-item-paragraph">The curve has a maximum point at ${\left(\frac{\pi}{4}, 3\right)}$ and a minimum point at ${\left(\frac{3\pi}{4}, -7\right)}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why ${c = -2}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Explain why ${b = 2}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Hence find the equation of the curve.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

2023 AISS AM S3 EOY Q5

<p>A curve $C$ is defined by the parametric equations</p>
<p style="text-align: center">$x={{t}^{2}}$, $y={{t}^{3}}$.</p>
<ol class="i-list"><li data-list-item-id="item-1760682212352-b77b35t1z" data-list-item-path="item-1760682212352-b77b35t1z"><div class="list-item-paragraph">Prove that the equation of the tangent at the point $\left( {{t}^{2}},{{t}^{3}} \right)$ on the curve is $2y-3tx+{{t}^{3}}=0$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760682213162-dkcmy6j24" data-list-item-path="item-1760682213162-dkcmy6j24"><div class="list-item-paragraph">This tangent passes through a fixed point $\left( a,b \right)$. Explain why there cannot be more than $3$ tangents through $\left( a,b \right)$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-1760682214085-3o6hgbq9x" data-list-item-path="item-1760682214085-3o6hgbq9x"><div class="list-item-paragraph">The tangent at the point $P$ when&nbsp;$t=2$ meets the curve again at the point $Q$ where&nbsp;$t=k$. Find the value of $k$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760682214932-4sblr9qnl" data-list-item-path="item-1760682214932-4sblr9qnl"><div class="list-item-paragraph">Find the equation of the normal at the point $P$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760682216073-fsc9m6rib" data-list-item-path="item-1760682216073-fsc9m6rib"><div class="list-item-paragraph">Sketch the curve $C$, indicating the intersection point(s) with the axes.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1760682217370-h23woyt5i" data-list-item-path="item-1760682217370-h23woyt5i"><div class="list-item-paragraph">By sketching the tangent and normal at the point $P$ on the sketch in (v), explain why ${{\tan }^{-1}}\left( 3 \right)+{{\tan }^{-1}}\left( \frac{1}{3} \right)=\frac{\pi }{2}$.<span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ is defined by the parametric equations

Tangents and Normals to Parametric Curves

Differentiation of Parametric Equations

Sketching Parametric Curves

2023 SAJC Promo Q9

Applications of Differentiation

Techniques of Differentiation

<ol class="abc-list"><li><div class="list-item-paragraph">Two functions are defined on the domain $0\le x\le \pi $ by $\mathrm{f}:x\mapsto \sin x$ and $\mathrm{g}:x\mapsto \cos x$.</div><ol class="i-list"><li><div class="list-item-paragraph">Explain why one of these functions has an inverse while the other does not.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">If the domain is restricted to $0\le x\le k$, the inverse exists for both functions.</div></li><li><div class="list-item-paragraph">State the greatest possible value of $k$ in exact form.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Evaluate ${{\mathrm{f}}^{-1}}\mathrm{g}\left( \frac{\pi }{3} \right)$ in exact form.</div><p style="text-align: right;">[1]</p></li></ol></li><li><div class="list-item-paragraph">The function $\mathrm{h}$ is defined by:</div><div class="list-item-paragraph" style="text-align: center">$\mathrm{h}:x\mapsto \frac{x+a}{2x+b}$, for $x\in \mathbb{R}$, $x\ne k$.</div><ol class="i-list"><li><div class="list-item-paragraph">Give an expression for $k$ and explain why this value of $x$ has to be excluded from the domain of $\mathrm{h}$.</div><p style="text-align: right;">[2]</p><div class="list-item-paragraph">The function $\mathrm{h}$ is such that $\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\left( x \right)$ for all $x$ in the domain of $\mathrm{h}$.</div></li><li><div class="list-item-paragraph">Determine the possible values of $a$ and $b$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find an expression for ${{\mathrm{h}}^{-1}}\left( -4 \right)$.</div><p style="text-align: right;">[1]</p></li></ol></li></ol>

One-to-One Functions and Inverse Existence

Domain of a Function

Self-inverse Functions: $ \mathrm{f} = \mathrm{f}^{-1}$

2022 NYJC CT Q7

<p>The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows</p>
<p style="text-align: center">$\mathrm{f}:x\mapsto \sin x$, $x\in \mathbb{R}$, $-\pi \le x\le 2\pi $.</p>
<p style="text-align: center">$\mathrm{g}:x\mapsto \cos x$, $x\in \mathbb{R}$, $0\le x\le 2\pi $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}\left( x \right)$ and explain why ${{\mathrm{f}}^{-1}}$ does not exist.</div></li></ol>
<p>The function $\mathrm{h}$ is a restriction on $\mathrm{f}$ such that $\mathrm{f}$ and $\mathrm{h}$ share the same range, and both the functions ${{\mathrm{h}}^{-1}}$ and $\mathrm{g}{{\mathrm{h}}^{-1}}$ exists.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">Write down the domain of $\mathrm{h}$.</div></li><li><div class="list-item-paragraph">Determine the range of $\mathrm{g}{{\mathrm{h}}^{-1}}$.</div></li><li><div class="list-item-paragraph">Find ${{\mathrm{h}}^{-1}}$ in a similar form.</div></li></ol>

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows

$\mathrm{f}:x\mapsto \sin x$, $x\in \mathbb{R}$, $-\pi \le x\le 2\pi $.

$\mathrm{g}:x\mapsto \cos x$, $x\in \mathbb{R}$, $0\le x\le 2\pi $.

The function $\mathrm{h}$ is a restriction on $\mathrm{f}$ such that $\mathrm{f}$ and $\mathrm{h}$ share the same range, and both the functions ${{\mathrm{h}}^{-1}}$ and $\mathrm{g}{{\mathrm{h}}^{-1}}$ exists.

Horizontal Line Test

SAJC Functions Tutorial Q11

<p>The function $\mathrm{f}$ is defined by $\mathrm{f}:x\to \ln \left[ {{\left( x-1 \right)}^{2}}-k \right]$, where $x<-2$ and $k$ is a constant. Find the maximum value of $k$. Another function $\mathrm{g}$ is defined by $\mathrm{g}:x\to \tan x,-\frac{\pi }{2} < x < \frac{\pi }{2}.$ If $k=8$,</p>
<ol class="i-list"><li><div class="list-item-paragraph">define ${{\mathrm{f}}^{-1}}$, the inverse function of $\mathrm{f}$, in a similar form.</div></li><li><div class="list-item-paragraph">show that the composite function ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$ exists and find the range of this composite function.</div></li></ol>

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\to \ln \left[ {{\left( x-1 \right)}^{2}}-k \right]$, where $x<-2$ and $k$ is a constant. Find the maximum value of $k$. Another function $\mathrm{g}$ is defined by $\mathrm{g}:x\to \tan x,-\frac{\pi }{2} < x < \frac{\pi }{2}.$ If $k=8$,

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