Inverse & Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA): HL): Exam Questions

Sketch the graph of $y=\cot x$ for $-\pi \le x\le \pi .$

Given that $\cot \mathrm{\theta }=\frac{9}{7}$ and $\pi \le \theta \le \frac{3\mathrm{\pi }}{2},$ find the values of $\cos \theta ,\sin \theta$ and $\tan \theta .$

By using an appropriate identity, solve ${\tan }^{2}x=\sec x+11$ for $0\le x\le \pi$.

Show that the equation

$\sec \mathrm{\theta }-5\cos \mathrm{\theta }=2\sqrt{2}$

can be rewritten as 

$5{\cos }^2\theta +2\sqrt{2}\cos \theta -1=0$

Hence, solve the equation $\sec \theta -5\cos \theta =2\sqrt{2}$ for all values of $\theta$ in the interval $-\pi \le \theta \le \frac{\mathrm{\pi }}{2}.$

A function $f$ can be defined by $f\left(x\right)=3x-5x\arcsin \left(x\right)$, where $-1\le x\le 1$.

Sketch the graph of $f$, indicating clearly any intercepts with the coordinate axes and the coordinates of any local maximum or minimum points.

State the domain and range of $f$.

Solve the inequality $3x-5x\arcsin \left(x\right)>-2$.

The function $f$ is defined as $f\left(x\right)=\arccos x$, $-1\le x\le 1$, and the function $g$ is such that $g\left(x\right)=f\left(3x\right)$.

Sketch the graph of $y=f\left(x\right)$ and state the range of $f$.

Sketch the graph of $y=g\left(x\right)$ and state the domain of $g$.

Find the inverse function $g^{-1}\left(x\right)$ and state its domain.

Show that  $\sec \theta \cot \theta \equiv \mathrm{cosec}\theta$.

Hence solve in the range $0\le \theta \le 2\pi$,  the equation $\sec \theta \cot \theta =-2$ 

Show that the equation

${\tan }^{2}x=6\sec x-10$

can be rewritten in the form

${\left(\sec x-3\right)}^2=0$

Hence, solve the equation ${\tan }^{2}x=6\sec x-10$  in the range $0\le x\le 2\pi$.

Show that the equation 

${\cot }^{2}x=9-3\mathrm{cosec}x$

can be rewritten in the form 

$(\mathrm{cosec}x-2)(\mathrm{cosec}x+5)=0.$

Hence, solve the equation ${\cot }^{2}x=9-3\mathrm{cosec}x$ in the interval $-180°\le x\le 180°$. Give your answers correct to 1 decimal place.

Find the exact values for the following expressions

(i) $\sec \left(\frac{\pi }{4}\right)\cos \left(\frac{\pi }{6}\right)$

(ii) $\frac{2{\tan }^2\left({\displaystyle \frac{\pi }{3}}\right)}{\mathrm{cosec}\left({\displaystyle \frac{\pi }{6}}\right)}$.

Sketch the graph of $y=\mathrm{cosec}x$ for $-\pi \le x\le 2\pi$

Given that $2\mathrm{cosec}\theta =-\frac{9}{4}$ and $-\pi <\mathrm{\theta }<-\frac{\pi }{2}$,  find the exact value of $\cot \theta$.

Consider the function $f\left(x\right)=\frac{1}{2}\arccos x+1$,  where $-1\le x\le 1$.

Sketch the graph of $f$ indicating clearly any intercepts with the coordinate axes and any maximum and minimum values.

Write down the domain of $f^{-1}\left(x\right)$ .

Find an expression for $f^{-1}\left(x\right)$ .

Show that $\frac{\cot \theta {\sec }^2\theta }{\mathrm{cosec}\theta }\equiv \sec \theta$ .

Hence solve in the range $-\pi \le \theta \le 2\pi$ ,  the equation  $\frac{\sqrt{3}\cot \theta {\sec }^{\mathit{2}}\theta }{\mathrm{cosec}\theta }=2$  .

Show that the equation

$3\left(\frac{1}{{\cot }^{2}x}-\frac{3}{\cos x}\right)=8\sec x+25$

can be written in the form

$\left(a\sec x+b\right)\left(\sec x+c\right)=0$.

Hence solve the equation $3\left(\frac{1}{{\cot }^{2}x}-\frac{3}{\cos x}\right)=8\sec x+25$ in the interval −270° ≤ x ≤ 90°. Give your answers correct to 1 decimal place.

Find the exact value for the expression ${\cot }^2\left(\frac{\pi }{6}\right)+\mathrm{cosec}\left(\frac{2\pi }{3}\right)$.

Give your answer in the form $a+\frac{b\sqrt{c}}{d}$ where $a,b,c,d\in \mathrm{\mathbb{Z}}$.

Find the exact value for the expression $\frac{\sec \left({\displaystyle \frac{2\pi }{3}}\right)}{\cot \left({\displaystyle \frac{\pi }{4}}\right)}$ .

By first forming a quadratic equation in $\tan x$, solve the equation $2\sin x+\frac{\cot x}{\sec x}=3\cos x$ for the interval −90° < x < 90°.

If $x=\sin \left(\frac{2\pi }{3}\right)$ find

(i) the exact value of $\cos \left(3\arccos x\right)$.

(ii) the exact value of $\cos \left(\arcsin x\right)$.

Show that the equation

$33+\frac{6\cos x}{{\cot }^{2}x}=\sqrt{2}\left(11\sec x+2\right)$

can be written as

$6{\cos }^{2}x+\left(2\sqrt{2}-33\right)\cos x+11\sqrt{2}-6=0$

Hence solve $33+\frac{6\cos x}{{\cot }^{2}x}=\sqrt{2}\left(11\sec x+2\right)$ for the interval $-\frac{3\pi }{2}\le x\le \pi$.

Show that the equation

$\cot 2\theta \sin \theta +\frac{8}{\sec \theta }=\sin \theta$

can be rewritten as  

${\tan }^2\theta +2\tan \theta -17=0$.

Hence solve the equation $\cot 2\theta \sin \theta +\frac{8}{\sec \theta }=\sin \theta$ for all values of $\theta$ in the interval $-\pi \le \theta \le \frac{\pi }{2}$.

Consider the function $f\left(x\right)=\arccos x$, $-1\le x\le 1$.

State whether $f$ is even, odd or neither. Give a reason for your answer.

A second function $g$ is such that $g\left(x\right)=3f\left(x\right)+a$ , where $g\left(x\right)$ is an odd function.

Find the value of $a$.

Sketch the graph of $g\left(x\right)$  and state the range of $g$. 

State whether $g^{-1}\left(x\right)$ will also be odd. Give a reason for your answer.

Consider the function defined by $f\left(x\right)=\sec \left(2\arcsin \mathrm{x}\right)$. 

Find the domain of $f$.

Show that $f\left(x\right)$  can be written as $\frac{1}{1-2x^2}$  for all $x$ in its domain.

Sketch the graph of $y=\sec x$ for $-2\pi \le x\le \pi$

Given that $\sec 2\theta =\frac{8}{7}$ and $-2\pi \le \theta \le -\frac{3\pi }{2}$, find the value of $\sin \theta$.

Show that the equation

 $5\sin x-9\cos x=\frac{2\mathrm{cosec}x}{1+{\tan }^{2}x}$ 

can be rewritten as

$\left(a\sin x+\cos x\right)\left(\sin x-b\cos x\right)=0$

Hence solve the equation $5\sin x-9\cos x=\frac{2cosecx}{1+{\tan }^{2}x}$, $-\pi \le x\le \pi$.

Consider the equation $\frac{1-{\cos }^2\theta }{\cot \theta }=-\frac{1}{{\mathrm{cosec}}^2\theta }$.

Explain why the equation is undefined when $\theta =\frac{1}{2}k\pi$ for any $k\in \mathrm{\mathbb{Z}}$.

Solve the equation $\frac{1-{\cos }^2\theta }{\cot \theta }=-\frac{1}{{cosec}^2\theta }$, for $-\pi \le \theta \le 2\pi$.

Show that $\frac{\cot x}{\sec x\left(1-\sin x\right)}\equiv 1+\frac{1}{\sin x},x\ne \frac{n\pi }{2},n\in \mathrm{\mathbb{Z}}$.

Hence determine the set of values for $k$ for which $\frac{\cot x}{\sec x\left(1-\sin x\right)}=k$ has no real solutions.

Consider the function $f\left(x\right)={\cos }^2\left(2\arcsin x\right)$, $0\le x\le a$, where $a$ is a constant.

Show that ${\cos }^2\left(2\arcsin x\right)=4x^4-4x^2+1$ for $-1\le x\le 1$.

Given that $f$ has an inverse, find the maximum value of $a$.

Given that $f$ has an inverse, find an expression for $f^{-1}\left(x\right)$.

A function $f$ can be defined by $f\left(x\right)=\frac{1}{2}\tan \left(\arcsin \left(x\right)\right)+5$.

Sketch the graph of $f$ indicating clearly any intercepts with the coordinate axes and asymptotes.

Solve the inequality $\frac{1}{2}\tan \left(\arcsin \left(x\right)\right)+5>2$. 

Give your answer in the form  $\frac{a\sqrt{b}}{c}<x<d$ where $a,b,c,d\in \mathrm{\mathbb{Z}}$.

Show that ${\sin }^2\left(\frac{1}{2}\arctan \left(x\right)\right)=\frac{1+x^2-\sqrt{1+x^2}}{2\left(1+x^2\right)},x\in \mathrm{ℝ}.$