Trigonometric Equations & Identities (DP IB Analysis & Approaches (AA): HL): Exam Questions

It is given that $\sin \alpha =\frac{3}{7}$, where $0\le \alpha \le \frac{\pi }{2}$. Find

(i) $\cos \alpha$

(ii) $\sin 2\alpha$

(iii) $\cos 2\alpha$

(iv) $\tan 2\alpha$.

It is given that $\cos B=\frac{1}{5}$, where $\frac{3\pi }{2}\le B\le 2\pi$. Find

(i) $\cos 2B$

(ii) $\sin 2B$

(iii) $\tan 2B$.

Solve $2\sin 2\theta =1$ for $0°\le \theta \le 360°$.

Solve the equation $2\sin x=\frac{1}{\sin x}$ for  $0°\le x\le 360°$.

Show that $(x+1)(x-2)(x-3)=x^3-4x^2+x+6$.

Hence, solve the equation ${\tan }^{3}x-4{\tan }^{2}x+\tan x+6=0$ for $0°\le x\le 360°$.

Show that the equation $2{\sin }^{2}x+3\cos x=0$ can be written in the form $2{\cos }^{2}x-3\cos x-2=0$.

Hence, solve the equation $2{\sin }^{2}x+3\cos x=0$ for $-180°\le x\le 180°$.

Show that the equation

$2{\cos }^{2}x-\sin x=1$

can be written in the form

$2{\sin }^{2}x+\sin x-1=0$.

Hence, solve the equation $2{\cos }^{2}x-\sin x=1$ for $0\le x\le 4\pi$.

The following diagram shows the graph of $y=f\left(x\right)$, where $f\left(x\right)=\cos x$, for $-\pi \le x\le \pi$.

The graph of $g$ is obtained from the graph of $f$ by a vertical translation of 1 unit downwards.

The graph of $h$ is obtained from the graph of $f$ by a horizontal stretch with scale factor $\frac{1}{2}$. The domain of $h$ is the same as the domain of $f$.

(i) On the same set of axes, sketch the graphs of $y=h\left(x\right)$ and $y=g\left(x\right)$.

(ii) Write down the number of roots of $g\left(x\right)=0$.

Find the solutions of the equation $\cos 2x=\cos x-1$ for $-\pi \le x\le \pi$, and label the points of intersection on your sketch from part (a)(i).

Solve the equation $\sqrt{3}\cos 2\theta =2{\cos }^{2}2\theta$ for $0°\le \theta \le 360°$.

Two functions, $f$ and $g$, are defined by $f\left(x\right)=\sin x$ and $g\left(x\right)=\cos 2x$.

Describe a single transformation that maps the graph of $y=\cos x$ onto the graph of $y=g\left(x\right)$.

On the same set of axes, sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ for $-\pi \le x\le 3\pi$.

Use an appropriate trigonometric identity to solve the equation $\sin x=\cos 2x$ for $-\pi \le x\le 3\pi$.

Hence label the points of intersection of the two graphs on your sketch from part (b).

Solve the equation $\sqrt{3}\tan x=\frac{1}{\sqrt{3}\tan x}$ for $0°\le x\le 540°$.

The following diagram shows the triangle $\mathrm{ABC}$, where $\mathrm{AB}=\sqrt{2}$, $\mathrm{AC}=\sqrt{3}$ and $\mathrm{B}\hat{\mathrm{A}}\mathrm{C}$ = 75°.

By writing 75° as 30° + 45°, find the exact value of sin 75°.

Find the area of the triangle, giving your answer in the form $\frac{a+\sqrt{b}}{c},$ where $a,b,c\in \mathrm{\mathbb{Z}}.$

Given that $\cos x=\frac{\sqrt{2}}{5}$ and  $\pi \le x\le 2\pi$, find the exact value of $\tan \left(2x\right)$.

The following diagram shows the triangle $\mathrm{ABC}$, where $\mathrm{AC}=\sqrt{3}$, $\mathrm{A}\hat{\mathrm{B}}\mathrm{C}$ = 135° and $\mathrm{B}\hat{\mathrm{C}}\mathrm{A}$ = 15°.

Find the length $\mathrm{AB}$, giving your answer in the form $\frac{a-\sqrt{b}}{c}$, where $a,b,c\in {\mathrm{\mathbb{Z}}}^{+}$.

Given that $\sin A=-\frac{1}{\sqrt{3}}$ for $\pi \le A\le \frac{3\pi }{2}$, and $\cos B=-\frac{\sqrt{2}}{3}$ for $\frac{\pi }{2}\le B\le \pi$, find the value of $\tan \left(A+B\right)$.

Give your answer in the form $\sqrt{p}\left(\sqrt{q}-r\right)$, where $p$, $q$ and $r$ are prime numbers.

It is given that $\tan \alpha =-\frac{3}{2}$, where $\frac{\pi }{2}<\alpha <\pi$.

Find

(i) $\sin \alpha$

(ii) $\cos \alpha$

(iii) $\sin 2\alpha$

(iv) $\cos 2\alpha$

(v) $\tan 2\alpha$.

Use your results from part (a) to explain why $\pi <2\alpha <\frac{3\pi }{2}$.

It is given that $\cos B=x$, where $\pi \le B\le \frac{3\pi }{2}$.

Explain why

(i) $x\le 0$

(ii) $\sin B=-\sqrt{1-x^2}$.

Hence find, in terms of $x$,

(i) $\cos 2B$

(ii) $\sin 2B$

(iii) $\tan 2B$.

An angle $M$ is such that $\sin M=p$ and $\cos M=q$. Show that

(i) $\sin 4M=4p{q}^3-4p^{3}q$

(ii) $\cos 4M=8q^4-8q^2+1$.

Use the fact that

 $16x^3-12x^2-4x+3=(4x-3)(4x^2-1)$

to fully factorise  $16x^3-12x^2-4x+3$.

Hence, solve the equation

$16{\sin }^{3}3\theta -12{\sin }^{2}3\theta -4\sin 3\theta +3=0$

for $0\le \theta \le \frac{\pi }{2}$. Give your answers as exact values where possible.

Solve the equation

${\sin }^{2}x=\frac{\cos x+4}{6}$

for $-\pi \le x\le \pi$. Give your answers as exact values where possible.

Solve the equation

$\frac{5{\sin }^{2}x-2}{\cos x}=3-\cos x$

for $-\pi \le x\le 2\pi$.

It is given that $\sin B=x$, where $\frac{\pi }{2}<B<\pi$.

Find the following in terms of $x$:

(i) $\sin 2B$

(ii) $\cos 2B$

(iii) $\tan 2B$.

Find the exact value of cos 55° cos 5° − cos 85° cos 35°.

Use the fact that

$3p^3-\left(7-3\sqrt{3}\right)p^2-\left(20+7\sqrt{3}\right)p-20\sqrt{3}=\left(3p+5\right)\left(p^2+\left(\sqrt{3}-4\right)p-4\sqrt{3}\right)$

to fully factorise $3p^3-\left(7-3\sqrt{3}\right)p^2-\left(20+7\sqrt{3}\right)p-20\sqrt{3}$.

Two functions, $f$ and $g$, are defined by

$f\left(x\right)=3{\tan }^{3}3x-\left(20+7\sqrt{3}\right)\tan 3x$

and

$g\left(x\right)=\left(7-3\sqrt{3}\right){\tan }^{2}3x+20\sqrt{3}$

for $-\frac{\pi }{6}<x<\frac{\pi }{3}$, $x\ne \frac{\pi }{6}$.

Use an algebraic method along with your result from part (a) to determine the $x$-coordinates of the points of intersection of the curves $y=f\left(x\right)$ and $y=g\left(x\right)$.

Your solution should show clear algebraic working, and your answers should be given as exact values where possible.

The following diagram shows the triangle $\mathrm{PQR}$ where $\mathrm{PQ}=3$, $\mathrm{PR}=\sqrt{2}$ and $\mathrm{R}\hat{\mathrm{P}}\mathrm{Q}=\frac{5\pi }{12}.$

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

Show that the length of $\mathrm{QR}$ is $\frac{3\sqrt{6}-\sqrt{2}}{2}$ units.

Solve the equation $\sin 2x\tan 2x=\frac{1}{2\cos 2x}$ for $-90°\le x\le 90°$.

It is given that $\tan 2\alpha =\frac{2}{5}$, where $\frac{\pi }{2}<2\alpha <\frac{3\pi }{2}$.

Find

(i) $\sin 2\alpha$

(ii) $\cos 2\alpha$.

Hence show that

$\sin \alpha =\sqrt{\frac{k+5\sqrt{k}}{2k}}$

and

$\cos \alpha =-\sqrt{\frac{k-5\sqrt{k}}{2k}}$

where $k$ is a positive integer to be determined, and use these results to find the exact value of $\tan \alpha$.

Find the exact value of $\arctan \left(\frac{7}{3}\right)-\arctan \left(\frac{2}{5}\right)$.

It is given that $\cos 6x=p$.

Show that

$\tan 3x=\pm \sqrt{\frac{1-p}{1+p}}$

For $0<x<\frac{\pi }{3}$, determine

(i) the range of values of $x$ for which the ‘plus’ version of the result in part (a) should be used

(ii) the range of values of $x$ for which the ‘minus’ version of the result in part (a) should be used

(iii) the value of $x$ for which $\tan 3x$ is not defined.

Solve the equation $\sin 2\theta -\sin \theta +\sqrt{3}\cos \theta =\frac{\sqrt{3}}{2}$ for $0°\le \theta \le 360°$.

Consider triangle OAB, where $\mathrm{OA}=\mathrm{OB}$ and $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\theta$. Let $\mathrm{AB}=p$ and let the area of triangle OAB be $q$.

Show that

$\cos \theta =\frac{1-m}{1+m}$

where

$m=\frac{p^4}{16q^2}$

The following diagram shows sector OAB of a circle with centre O, where $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\theta$.

Given that $\mathrm{AB}=2\sqrt{3}$ and the area of triangle OAB is $\sqrt{3}$, find the area of sector OAB and the length of arc AB.