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

The value of $\tan \alpha =-\frac{3}{2}$ for $\frac{\mathrm{\pi }}{2}\le \alpha \le \pi$.

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 $\pi <2\alpha <\frac{3\pi }{2}$ must be true.

The value of $\cos B=x$, for  $\pi \le B\le \frac{3\pi }{2}$. 

Explain why

(i) $x\le 0$

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

Hence find the following 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) $\cos 4M=8q^4-8q^2+1.$

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

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

Use the fact that

 $16x^3-12x^2-4x+3=(4x-3)(4x^2-1)$

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

Use your result from part (a) to solve the equation

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

in the interval  $0\le \theta \le \frac{\pi }{2}$.  You should give your answers as exact values where possible.

Solve the equation

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

in the interval $-\pi \le x\le \pi$. Give your answers as exact values where possible.

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

Describe the single transformation of the graph of $y=\cos x$ that will produce the graph of  $y=g\left(x\right)$.

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

By using an appropriate trigonometric identity to solve the equation $\sin x=\cos 2x$ in the interval  $-\pi \le x\le 3\pi$, determine the points of intersection of the two curves from your graph in part (b).  Label those points on your graph.

The value of $\tan 2\alpha =\frac{2}{5}$ for $\frac{\pi }{2}\le 2\alpha \le \frac{3\pi }{2}$.

Find

(i) $\sin 2\alpha$

(ii) $\cos 2\alpha$

Hence show that

$\sin \alpha =\sqrt{\frac{k+5\sqrt{k}}{2k}}\mathrm{and}\cos \alpha =-\sqrt{\frac{k-5\sqrt{k}}{2k}}$

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

The value of $\sin B=x$,  for  $\frac{\pi }{2}<B<\pi$.

Find the following in terms of $x$:

(i) $\sin 2B$

(ii) $\cos 2B$

(iii) $\tan 2B$.

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 the range of $x$ values for which

(i) the ‘plus’ version of the part (a) result should be used

(ii) the ‘minus’ version of the part (a) result should be used

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

Solve the equation

$\sin 2\theta -\sin \theta +\sqrt{3}\cos \theta =\frac{\sqrt{3}}{2}$

in the interval $0\le \theta \le 360°$.

Solve the equation

$\sin 2x\tan 2x=\frac{1}{2\cos 2x}$

in the interval $-90°\le x\le 90°$.

Solve the equation 

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

in the interval $-\pi \le x\le 2\pi$.

Use the fact that

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

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

Two functions, f  and g, are defined by

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

for  $-\frac{\pi }{6}\le x\le \frac{\pi }{3}$.

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.

Let OAB be an isosceles triangle with  $\mathrm{OA}=\mathrm{OB}$ and  $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\mathrm{\theta }$.

If the length of line segment AB is denoted by $p$, and the area of triangle OAB is denoted by $q$, show that

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

where

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

The diagram below shows circle sector OAB with centre O and angle at the centre  $\mathrm{A}\hat{\mathrm{O}}\mathrm{B}=\theta$.

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