Reciprocal & Inverse Trigonometric Functions (OCR A Level Maths A: Pure): Exam Questions

Use the definitions of the secant, cosecant and cotangent functions to show that

sec $\theta$ cot $\theta \equiv$cosec $\theta$.  

Hence solve, in the range $0\le \theta \le 2\mathrm{\pi }$, the equation

sec $\theta$ cot $\theta =-2$

Show that the equation

$3-$sec $\theta =\frac{2}{\sec \theta }$

can be rewritten in the form

 $\left(\sec \theta -2\right)$$\left(\sec \theta -1\right)=0$

Hence solve, in the range $0\le \theta \le 2\mathrm{\pi }$, the equation

$3-$sec $\theta =\frac{2}{\sec \theta }$

Using the double angle formula sin $2A\equiv 2$ sin $A$ cos $A$, show that the equation

sec $x$ cosec $x-5=$cosec $2x$

can be rewritten in the form

coesec $2x=5$.

Hence solve, in the range $0\le x\le 2\mathrm{\pi }$, the equation

sec $x$ cosec $x$$-5=$cosec $2x$

giving your answers correct to 3 significant figures.

Show that the equation

tan² $x=6$ sec $x-10$

can be rewritten in the form

 ${\left(\sec x-3\right)}^2=0$

Hence solve, in the range $0\le x\le 2\mathrm{\pi }$, the equation

tan² $x=6$ sec $x-10$

giving your answers correct to 3 significant figures.

Given that $x$ satisfies the equation $\arccos x=k$ , where   $0<k<\frac{\mathrm{\pi }}{2}$

(i) state the range of possible values of $x$,

(ii) express both sin $k$ and tan $k$ in terms of $x$.

 ,

Prove that for $0\le x\le 1$, $\mathrm{arc}\sin$ $x=\mathrm{arc}\cos$$\sqrt{1-x^2}$.

Explain why this is not true for $-1\le x<0$.

(i) Sketch, in the interval $-2\mathrm{\pi }\le \theta \le 2\mathrm{\pi }$, the graph of $y=3+2$ cosec $\theta$ , include asymptotes and label the coordinates of all maximum and minimum points.

(ii) Hence, deduce the number of solutions to the equation $3+2$ cosec $\theta =\frac{1}{2}$ in the interval  $-2\mathrm{\pi }\le \theta \le 2\mathrm{\pi }$.

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

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

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

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

Rewrite tan $\theta$ cosec $\theta$ as a single trigonometric function.

Hence solve, in the range $-\mathrm{\pi }<\theta \le \mathrm{\pi }$, the equation

tan $\theta$ cosec $\theta$$=-\frac{2\sqrt{3}}{3}$

Solve, in the range $0\le \theta \le 2\mathrm{\pi }$, the equation

$\frac{2}{\mathrm{cosec}\theta }-\mathrm{cosec}\theta =1$

Using the double angle formula sin $2A\equiv 2$ sin $A$ cos $A$,  find the solutions to the equation

sec $x$ cosec $x$$-75=5$ cosec $2x$

in the range $-\mathrm{\pi }<x\mathit{\le }\mathrm{\pi }$.   Give your answers correct to 3 significant figures.

Show that the equation

$2$ cot² $x=1-5$cosec $x$

can be rewritten in the form

$\left(2\mathrm{cosec}x-1\right)\left(\mathrm{cosec}x+3\right)=0$

Hence solve, in the range $0\le x\le 2\mathrm{\pi }$, the equation

$2$ cot² $x=1-5$cosec $x$

giving your answers correct to 3 significant figures.

Given that $x$ satisfies the equation  $\mathrm{arc}\sin$ $x=k$, where  $-\frac{\mathrm{\pi }}{2}<k<0$  ,

(i) state the range of possible values of $x$,

(ii) express both cos $k$ and tan $k$ in terms of $x$.

Prove that for $-1\le x\le 0$,$\mathrm{arc}\cos$ $x=\mathrm{\pi }-\arcsin \sqrt{1-x^{\mathit{2}}}$ .

(i) Sketch, in the interval $-2\mathrm{\pi }\le \theta \le 2\mathrm{\pi }$, the graph of  $y=-5+\frac{1}{2}$sec $\theta$  , include asymptotes and label the coordinates of all maximum and minimum points.

(ii) Hence deduce the range of values for $k$ for which the equation   $-5+\frac{1}{2}$sec $\theta =\mathrm{k}$   has no solutions.

The function  $\mathrm{f}$  is defined as $\mathrm{f}\left(x\right)=\arctan$ $x$ ,  $x\in \mathrm{ℝ}$  , and the function  $\mathrm{g}$  is such that  $\mathrm{g}\left(x\right)=\frac{2}{\mathrm{\pi }}\mathrm{f}\left(x\right)-1$  .

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

Sketch the graph of $y=\mathrm{g}\left(x\right)$and state the range of  $\mathrm{g}$.

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

Solve, in the range $-\mathrm{\pi }<\theta \le \mathrm{\pi }$, the equation

$\frac{\sec \theta \cot \theta }{\mathrm{cosec}\theta \tan \theta }=-\sqrt{3}$

Solve, in the range $0\le \theta \le 2\mathrm{\pi }$, the equation

$6$ sec$\theta$$+\frac{2\sqrt{3}}{\sec \theta }=-3-4\sqrt{3}$

Leaving your answers as exact values.

Using the double angle formulae $\sin 2A\equiv 2\sin A\cos A$ and $\cos 2A\equiv {\cos }^{2}A-{\sin }^{2}A$,  find the solutions to the equation

$\left(\mathrm{cosec}x\mathit{-}\sec x\right)\left(\frac{1}{\sec x}+\frac{1}{\mathrm{cosec}x}\right)=\cot 2x+3$

in the range $-\mathrm{\pi }<x\mathit{\le }\mathrm{\pi }$.   Give your answers correct to 3 significant figures.

Solve, in the range $0\le x\le 2\mathrm{\pi }$, the equation

$3{\cot }^{2}x-4\sqrt{3}=\left(6-2\sqrt{3}\right)\mathrm{cosec}x-3$

Leaving your answers as exact values.

Given that $x$ satisfies the equation  $\mathrm{ar}c\tan$ $x=k$, where    $-\frac{\mathrm{\pi }}{2}\le k\le 0$

(i) state the range of possible values of $x$,

(ii) express both sin $k$ and cos $k$ in terms of $x$.

Prove that for $x\le -1$,

$\arcsin \frac{1}{x}=-\arccos \left(-\frac{\sqrt{x^2-1}}{x}\right)$

Sketch, in the interval $-2\mathrm{\pi }\le \theta \le \mathrm{\pi }$, the graph of  $y=2+3\sec \left(\theta +\frac{\mathrm{\pi }}{2}\right)$ , include asymptotes and label the coordinates of all maximum and minimum points.

Deduce the maximum and minimum values of  $\frac{1}{2+3\sec \left(\theta +{\displaystyle \frac{\mathrm{\pi }}{2}}\right)}$.

The function  $\mathrm{f}$  is defined as  $\mathrm{f}:x\mapsto \arcsin x$,  $-1\le x\le 1$ , and the function  $\mathrm{g}$  is such that 

$\mathrm{g}\left(x\right)=\frac{4\mathrm{f}\left({\displaystyle \frac{x}{3}}\right)}{\mathrm{\pi }}+2$

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

Define the inverse function ${\mathrm{g}}^{-1}$ in the form  ${\mathrm{g}}^{-1}:x\mathit{\mapsto }$… 

Over the same domain as  $\mathrm{g}$, the function $\mathrm{h}$ is defined as  $\mathrm{h}:x\mathit{\mapsto }p\arccos \left(\mathrm{q}x\right)$.

Given that  $\mathrm{h}\left(x\right)=-\mathrm{g}\left(x\right)$ for all $x$ in the two functions’ common domain, determine the values of $p$ and $q$.