Reciprocal & Inverse Trigonometric Functions (Edexcel IAL Maths): Exam Questions

Sketch the graph of $y=$cosec $x$, for $-180^{\circ}\leq x\leq180^{\circ}$.

Write down the domain and range for the function $\mathrm{arc}$cos $θ$.

Hence sketch the graph of $y=\quad \mathrm{arc}$cos $θ$.

Solve the equation cot $x=3$, for $-π\leq x\leqπ$, giving your answers to three significant figures.

Sketch the graph of $y=$sec $θ$, for $-π\leqθ\leqπ$.

Label any points of intersection with the coordinate axes and state the equations of any asymptotes.

Starting with the identity

sin² $x+$cos² $x\equiv1$

show that

(i) $1+$cot² $x\equiv$cosec² $x$

(ii) tan² $x+1\equiv$sec² $x$

Show that

sec² $θ$ sin $θ\equiv$tan $θ$ sec $θ$

Write down the domain and range for the function $\mathrm{arcsin}θ$.

Hence sketch the graph of $y=\mathrm{arc} \mathrm{sin}$ $θ$.

Solve the equation cosec² $x-2$ cosec $x-8=0$ for $0^{\circ}\leq x\leq360^{\circ}$, giving your answers to one decimal place where appropriate.

Show that

cot $x$ cosec $x$ sec $x\equiv1+$cot² $x$

Solve the equation sec $θ$ tan $θ-$sec $θ=0$, for $0\leq x\leq2π$, giving your answers in exact form.

Write down the domain and range for the function $\mathrm{arc} \mathrm{tan}$ $θ$.

Hence sketch the graph of $y=\mathrm{arc} \mathrm{tan}$ $θ$.

Sketch the graph of $y=2$ sec $2x$, for $-π\leq x\leqπ$.

Draw a suitable line on your graph to show that the equation $2$ sec $2x=4$ has four solutions in the range $-π\leq x\leqπ$.

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

sec $θ$ cot $θ\equiv$cosec $θ$.

Hence solve, in the range $0\leqθ\leq2π$, the equation

sec $θ$ cot $θ=-2$

Show that the equation

$3-$sec $θ=\frac{2}{\mathrm{sec}θ}$

can be rewritten in the form

$( \sec \\theta - 2 )\quad ( \sec \\theta - 1 ) = 0$

Hence solve, in the range $0\leqθ\leq2π$, the equation

$3-$sec $θ=\frac{2}{\mathrm{sec}θ}$

Using the double angle formula sin $2A\equiv2$ 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 $0\leq x\leq2π$, the equation

sec $x$ cosec $x\quad -5=$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

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

Hence solve, in the range $0\leq x\leq2π$, the equation

tan² $x=6$ sec $x-10$

giving your answers correct to 3 significant figures.

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

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

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

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Prove that for $0\leq x\leq1$, $\mathrm{arc} \mathrm{sin}$ $x=\mathrm{arc} \mathrm{cos}\quad \sqrt{1-x^{2}}$.

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

(i) Sketch, in the interval $-2π\leqθ\leq2π$, the graph of $y=3+2$ cosec $θ$ , 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 $θ=\frac{1}{2}$ in the interval $-2π\leqθ\leq2π$.

The function $f$ is defined as $\mathrm{f} ( x ) = \text{arc} \cos$ $x$ , $-1\leq x\leq1$, and the function $g$ is such that $\mathrm{g} ( x ) = \mathrm{f} ( 3 x )$.

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

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

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

Rewrite tan $θ$ cosec $θ$ as a single trigonometric function.

Hence solve, in the range $-π<θ\leqπ$, the equation

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

Solve, in the range $0\leqθ\leq2π$, the equation

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

Using the double angle formula sin $2A\equiv2$ sin $A$ cos $A$, find the solutions to the equation

sec $x$ cosec $x\quad -75=5$ cosec $2x$

in the range $-π<x\leqπ$. 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

$( 2 \text{cosec} x - 1 ) ( \text{cosec} x + 3 ) = 0$

Hence solve, in the range $0\leq x\leq2π$, the equation

$2$ cot² $x=1-5$cosec $x$

giving your answers correct to 3 significant figures.

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

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

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

Prove that for $-1\leq x\leq0$,$\mathrm{arc} \mathrm{cos}$ $x=π-\mathrm{arcsin}\sqrt{1-x^{2}}$ .

(i) Sketch, in the interval $-2π\leqθ\leq2π$, the graph of $y=-5+\frac{1}{2}$sec $θ$ , 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 $θ=k$ has no solutions.

The function $f$ is defined as $\mathrm{f} ( x ) = \arctan$ $x$ , $x\inℝ$ , and the function $g$ is such that g ( x ) = 2 / pi f ( x ) - 1 .

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

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

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

Solve, in the range $-π<θ\leqπ$, the equation

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

Solve, in the range $0\leqθ\leq2π$, the equation

$6$ sec$θ\quad +\frac{2\sqrt{3}}{\mathrm{sec}θ}=-3-4\sqrt{3}$

Leaving your answers as exact values.

Using the double angle formulae $\mathrm{sin}2A\equiv2\mathrm{sin}A\mathrm{cos}A$ and $\mathrm{cos}2A\equiv\mathrm{cos}^{2}A-\mathrm{sin}^{2}A$, find the solutions to the equation

$( \text{cosec} x - \sec x ) ( \frac{1}{\sec x} + \frac{1}{cosec x} ) = \cot 2 x + 3$

in the range $-π<x\leqπ$. Give your answers correct to 3 significant figures.

Solve, in the range $0\leq x\leq2π$, the equation

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

Leaving your answers as exact values.

Given that $x$ satisfies the equation $\mathrm{ar}c\mathrm{tan}$ $x=k$, where $-\frac{π}{2}\leq k\leq0$

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

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

Prove that for $x\leq-1$,

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

Sketch, in the interval $-2π\leqθ\leqπ$, the graph of y = 2 + 3 sec ( theta + pi / 2 ) , include asymptotes and label the coordinates of all maximum and minimum points.

Deduce the maximum and minimum values of (1)/(2 + 3 sec ( theta + pi / 2 )).

The function $f$ is defined as $\mathrm{f} : x \rightarrow \text{from} \text{bar} \arcsin x$, $-1\leq x\leq1$ , and the function $g$ is such that

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

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

Define the inverse function $g^{-1}$ in the form $\mathrm{g}^{- 1 } : x \rightarrow \text{from} \text{bar}$…

Over the same domain as $g$, the function $h$ is defined as $\mathrm{h} : x \rightarrow \text{from} \text{bar} p \arccos ( \mathrm{q} x )$.

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