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

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 $\cos \left(\mathrm{arc}\sin x\right)$.

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 $\cot \theta$.

Show that the equation $33+\frac{6\cos x}{{\cot }^{2}x}=\sqrt{2}\left(11\sec x+2\right)$   can be rewritten in the form $a{\cos }^{2}x+b\cos x+c=0$ .

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 $-\mathrm{\pi }\le \theta \mathit{\le }\frac{\mathit{\pi }}{\mathit{2}}$.

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 $f^{-1}\left(x\right)$ .

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

State whether the function is $f$ 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.

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  $\frac{\sqrt{3}\cot \theta {\sec }^{\mathit{2}}\theta }{\mathrm{cosec}\theta }=2$  .

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.

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 $-270°\le x\le 90°$. Give your answers correct to 1 decimal place.

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.

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 $\frac{\sec \left({\displaystyle \frac{2\pi }{3}}\right)}{\cot \left({\displaystyle \frac{\pi }{4}}\right)}$ .

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 $5\sin x-9\cos x=\frac{2\cos ecx}{1+{\tan }^{2}x},-\pi \le x\le \pi .$

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

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

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

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

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 $a,b,c,d\in \mathrm{\mathbb{Z}}$.

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 $\frac{1-{\cos }^2\theta }{\cot \theta }=-\frac{1}{\cos e{c}^2\theta },\mathrm{for}-\pi \le \theta \le 2\pi$.

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

Solve the equation $2\sin x+\frac{\cot x}{\sec x}=3\cos x$for the interval $-90°<x<90°$.