Solve the equation \(\sec ^{2} 3 x+\tan 3 x-3=0\) for \(0^{\circ} \leqslant x \leqslant 120^{\circ}\).

(a) Solve the equation \(\tan^2(2x)-4\tan(2x)=0\) for \(0^\circ\leqslant x\leqslant180^\circ\).

(b) Solve the equation \(\operatorname{cosec}(y+1.2)=4\), where \(y\) is in radians and \(-5\lt y\lt 2\).

(a) Show that \(\dfrac{1-\sin x}{\cos x}+\dfrac{\cos x}{1-\sin x}=2\sec x\).

(b) Hence solve the equation \(\dfrac{1-\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}}+\dfrac{\cos \frac{\theta}{2}}{1-\sin \frac{\theta}{2}}=3\) for \(0^\circ\leqslant \theta\leqslant720^\circ\).

Solve the equation \(2\sin^3\theta=3\sin\theta\cos\theta\) for \(-\dfrac{\pi}{2}\leqslant\theta\leqslant\dfrac{\pi}{2}\).

(a) Solve the equation \((2-3 \cot x) \cos x=0\) for \(0\lt x \leqslant \frac{\pi}{2}\). (b) Solve the equation \(2 \operatorname{cosec}(2 \theta+1)-12 \sin (2 \theta+1)=5\), where \(\theta\) is in radians and \(-1 \leqslant \theta \leqslant 2\).

(a) Solve the equation \(3 \sec 3 x=\sqrt{3} \operatorname{cosec} 3 x\) for \(-120^{\circ} \leqslant x \leqslant 120^{\circ}\). (b) Solve the equation \(2 \cos \left(y+\frac{\pi}{3}\right) \sin \left(y+\frac{\pi}{3}\right)=\sin \left(y+\frac{\pi}{3}\right)\) for \(0 \leqslant y\lt 2 \pi\).

The function f is defined by \(\mathrm{f}(x)=15 \cos ^{2}(3 x+1.5)+7 \sin (3 x+1.5)-13\) for \(-0.3 \leqslant x \leqslant 0.5\), where \(x\) is in radians. (a) Solve the equation \(\mathrm{f}(x)=0\).

(b) Find the \(x\)-coordinates of the two stationary points on the curve \(y=\mathrm{f}(x)\).

(a) Show that \(\frac{\sin\theta\tan^2\theta}{1+\tan^2\theta}\) can be written as \(\sin^3\theta\).

(b) Hence solve the equation \(\frac{\sin3x\tan^23x}{1+\tan^23x}=\frac18\) for \(-180^\circ\leqslant x\leqslant180^\circ\).

Solve the equation \(\cot(y+1.5)=3\), where \(y\) is in radians and \(0\lt y\lt 6\).

(a) Solve the equation \(2 \operatorname{cosec}^{2} \theta-5=5 \cot \theta\) for \(-180^{\circ} \leqslant \theta \leqslant 180^{\circ}\). (b) Solve the equation \(3 \sin (2 \phi+1.5)=2\) for \(0\lt \phi\lt 5\), where \(\phi\) is in radians.

(a) Solve the equation \(7 \tan ^{2} \theta+5 \tan \theta-2=0\), for \(-180^{\circ} \leqslant \theta \leqslant 180^{\circ}\). (b) Solve the equation \(3 \sin (3 \phi-1.5)-2=0\), for \(0\lt \phi\lt 3\), where \(\phi\) is in radians.

Solve the equation \(\cot ^{2} 2 \theta+3 \operatorname{cosec} 2 \theta=9\) for \(-90^{\circ} \leqslant \theta \leqslant 90^{\circ}\).

Solve the equation \(4 \sin ^{2}\left(2 \alpha-\frac{\pi}{3}\right)=1\) for \(-\frac{\pi}{2} \leqslant \alpha \leqslant \frac{\pi}{2}\). Give your answers in terms of \(\pi\).

Solve the equation \(\sec \left(3 \theta-\frac{\pi}{2}\right)=2\) for \(-\frac{\pi}{2} \leqslant \theta \leqslant \frac{\pi}{2}\). Give your answers in exact form.

(a) Show that \(\sin ^{3} x\left(\frac{\operatorname{cosec} x}{\cot x}\right)\) can be written as \(\sin ^{2} x \tan x\). (b) Solve the equation \(\cos ^{2} x \tan x-\frac{1}{2} \tan x=0\) for \(-\pi\lt x\lt \pi\).

(a) Given that \(\operatorname{cot}^2\theta=\frac{1}{y+2}\) and \(\operatorname{sec}\theta=x-4\), find \(y\) in terms of \(x\).

(b) Solve the equation

\(\sqrt3\,\operatorname{cosec}\left(2\phi+\frac{3\pi}{4}\right)=2,\)

for \(-\pi\lt\phi\lt\pi\), giving your answers in terms of \(\pi\).

Solve the equation

\(3\operatorname{sec}^2\left(2\theta+\frac{\pi}{6}\right)=4\)

for \(-\frac{\pi}{2}\lt \theta\lt \frac{\pi}{2}\), giving your answers in terms of \(\pi\).

Solve the equation

\(\displaystyle 3\operatorname{cosec}^2\left(\frac{2x}{3}-\frac{\pi}{3}\right)=4\)

for \(0\lt x\leq 3\pi\). Give your answers in terms of \(\pi\).

In this question, all angles are in radians.

(a) Solve the equation

\(\operatorname{sec}^2\theta=\tan\theta+3\)

for \(-\pi\lt \theta\lt \pi\).

(b) Show that, for \(0\lt \phi\lt \frac{\pi}{2}\),

\(\frac{\tan\phi}{\sqrt{1-\cos^2\phi}}=\operatorname{sec}\phi.\)

(c) Given that \(\operatorname{cosec}x=-\frac{17}{8}\) and that \(\frac{3\pi}{2}\lt x\lt 2\pi\), find the exact value of \(\operatorname{cot}x\).

(a) Solve the equation

\(3\operatorname{cosec}^{2}\left(2\phi-\frac{\pi}{3}\right)=4,\)

for \(0\lt \phi\lt \pi\). Give your solutions in terms of \(\pi\).

(b) Given that \(2x-1=\operatorname{cosec}^{2}\theta\) and \(y+1=\tan^{2}\theta\), find \(y\) in terms of \(x\).

(a)

(i) Show that

\(\sin x\tan x+\cos x=\operatorname{sec}x.\)

(ii) Hence solve the equation

\(\sin\frac{\theta}{2}\tan\frac{\theta}{2}+\cos\frac{\theta}{2}=4\)

for \(0\leqslant\theta\leqslant4\pi\), where \(\theta\) is in radians.

(b) Solve the equation

\(\operatorname{cot}(y+38^\circ)=\sqrt3\)

for \(0^\circ\leqslant y\leqslant360^\circ\).

(a)

(i) Write

\(6xy+3y+4x+2\)

in the form \((ax+b)(cy+d)\), where \(a,b,c\) and \(d\) are positive integers.

(ii) Hence solve the equation

\(6\sin\theta\cos\theta+3\cos\theta+4\sin\theta+2=0\)

for \(0^\circ\lt\theta\lt360^\circ\).

(b) Solve the equation

\(\frac12\operatorname{sec}\left(2\phi+\frac{\pi}{4}\right)=\frac1{\sqrt3}\)

for \(-\pi\lt\phi\lt\pi\), where \(\phi\) is in radians. Give your answers in terms of \(\pi\).

Solve the equation

\(\operatorname{cosec}^2\theta+2\operatorname{cot}^2\theta=2\operatorname{cot}\theta+9,\)

where \(\theta\) is in radians and

\(-\frac{\pi}{2}\lt \theta\lt \frac{\pi}{2}.\)

(a) Solve the equation \(3\operatorname{cosec}^2\theta-5=5\operatorname{cot}\theta\) for \(0^\circ\leq\theta\leq180^\circ\).

(b) Solve the equation \(\sin\left(\phi+\frac{\pi}{3}\right)=-\frac12\), where \(\phi\) is in radians and \(-\pi\leq\phi\leq\pi\). Give your answers in terms of \(\pi\).

(a) Solve

\(\tan(\alpha+45^\circ)=-\frac1{\sqrt2}\)

for \(0^\circ\leq\alpha\leq360^\circ\).

(b)(i) Show that

\(\frac1{\sin\theta-1}-\frac1{\sin\theta+1}=a\operatorname{sec}^2\theta,\)

where \(a\) is a constant to be found.

(b)(ii) Hence solve

\(\frac1{\sin3\phi-1}-\frac1{\sin3\phi+1}=-8\)

for \(-\dfrac{\pi}{3}\leq\phi\leq\dfrac{\pi}{3}\) radians.

(a) Solve

\(3\operatorname{cot}^2 x-14\operatorname{cosec}x-2=0\)

for \(0^\circ\lt x\lt 360^\circ\).

(b) Show that

\(\frac{\sin^4y-\cos^4y}{\operatorname{cot} y}=\tan y-2\cos y\sin y.\)

Solve the equations

(a) \(5\operatorname{sec}^2 A+14\tan A-8=0\), for \(0^\circ\leq A\leq180^\circ\),

(b) \(5\sin\left(4B-\frac{\pi}{8}\right)+2=0\), for \(-\frac{\pi}{4}\leq B\leq\frac{\pi}{4}\) radians.

(a) Given that \(2\cos x=3\tan x\), show that

\(2\sin^2x+3\sin x-2=0.\)

(b) Hence solve

\(2\cos\left(2\alpha+\frac{\pi}{4}\right) =3\tan\left(2\alpha+\frac{\pi}{4}\right)\)

for \(0\lt \alpha\lt \pi\) radians, giving your answers in terms of \(\pi\).

(a)(i) Show that \(\operatorname{sec}\theta-\dfrac{\tan\theta}{\operatorname{cosec}\theta}=\cos\theta\).

(a)(ii) Solve \(\operatorname{sec}2\theta-\dfrac{\tan2\theta}{\operatorname{cosec}2\theta}=\dfrac{\sqrt3}{2}\) for \(0^\circ\leq\theta\leq180^\circ\).

(b) Solve \(2\sin^2\left(\phi+\frac{\pi}{3}\right)=1\) for \(0\lt \phi\lt 2\pi\) radians.

(a)(i) Show that \(\dfrac{\operatorname{cosec}\theta-\operatorname{cot}\theta}{\sin\theta}=\dfrac{1}{1+\cos\theta}\).

(a)(ii) Hence solve \(\dfrac{\operatorname{cosec}\theta-\operatorname{cot}\theta}{\sin\theta}=\dfrac52\) for \(180^\circ\lt \theta\lt 360^\circ\).

(b) Solve \(\tan(3\phi-4)=-\dfrac12\) for \(0\lt \phi\lt \frac{\pi}{2}\) radians.

(a) Solve \(6\sin^2x-13\cos x=1\) for \(0^\circ\leq x\leq360^\circ\).

(b) (i) Show that, for \(-\dfrac{\pi}{2}\lt y\lt \dfrac{\pi}{2}\), \(\dfrac{4\tan y}{\sqrt{1+\tan^2y}}\) can be written in the form \(a\sin y\), where \(a\) is an integer.

(ii) Hence solve \(\dfrac{4\tan y}{\sqrt{1+\tan^2y}}+3=0\) for \(-\dfrac{\pi}{2}\lt y\lt \dfrac{\pi}{2}\) radians.

(a)(i) Use the factor theorem to show that \(2x-1\) is a factor of \(p(x)\), where \(p(x)=4x^3+9x-5\).

(ii) Write \(p(x)\) as a product of linear and quadratic factors.

(b)(i) Show that

\(13\tan x\operatorname{sec}x-4\sin x-5\operatorname{sec}^2x=0\)

can be written as

\(4\sin^3x+9\sin x-5=0.\)

(ii) Using your answers to part (a)(ii) and part (b)(i), solve

\(13\tan x\operatorname{sec}x-4\sin x-5\operatorname{sec}^2x=0\)

for \(0\lt x\lt 2\pi\) radians.

(i) Show that

\(\frac{\tan x}{1+\operatorname{sec}x}+\frac{1+\operatorname{sec}x}{\tan x}\equiv\frac{2}{\sin x}.\)

(ii) Hence solve

\(\frac{\tan x}{1+\operatorname{sec}x}+\frac{1+\operatorname{sec}x}{\tan x}=1+3\sin x\)

for \(0^\circ\lt x\lt180^\circ\).

(i) Show that

\(\frac{\operatorname{cosec}x-\operatorname{cot} x}{1-\cos x}\equiv\operatorname{cosec}x.\)

(ii) Hence solve

\(\frac{\operatorname{cosec}x-\operatorname{cot} x}{1-\cos x}=2\)

for \(0^\circ\lt x\lt180^\circ\).

(a) Solve

\(3\operatorname{cot}^2\left(y-\frac{\pi}{4}\right)=1\)

for \(0\lt y\lt\pi\) radians.

(b) Solve

\(7\operatorname{cot} z+\tan z=7\operatorname{cosec}z\)

for \(0^\circ\leq z\leq360^\circ\).

(a) (i) Show that \(\dfrac{(1-\sin A)(1+\sin A)}{\sin A\cos A}=\operatorname{cot}A\).

(ii) Hence solve \(\dfrac{(1-\sin3x)(1+\sin3x)}{\sin3x\cos3x}=\dfrac12\) for \(0^\circ\leq x\leq180^\circ\).

(b) Solve \(10\tan^2y-\operatorname{sec}y-1=0\) for \(0\leq y\leq2\pi\) radians.

(a) Solve \(3\cos^2\theta+4\sin\theta=4\) for \(0^\circ\leqslant\theta\leqslant180^\circ\).

(b) Solve \(\sin2\phi=\sqrt3\cos2\phi\) for \(-\dfrac{\pi}{2}\leqslant\phi\leqslant\dfrac{\pi}{2}\) radians.

(a) Solve

\(10\cos^2x+3\sin x=9\)

for \(0^\circ\lt x\lt360^\circ\).

(b) Solve

\(3\tan2y=4\sin2y\)

for \(0\lt y\lt\pi\) radians.

(a) Solve

\(10\cos^2x+3\sin x=9\)

for \(0^\circ\lt x\lt360^\circ\).

(b) Solve

\(3\tan2y=4\sin2y\)

for \(0\lt y\lt\pi\) radians.

Solve

\(\operatorname{sec} x=\operatorname{cot} x-5\tan x\)

for \(0^\circ\lt x\lt 360^\circ\).

(a) Solve

\(2\sin\left(x+\frac{\pi}{4}\right)=\sqrt3\)

for \(0\lt x\lt \pi\) radians.

(b) Solve

\(3\operatorname{sec} y=4\operatorname{cosec}y\)

for \(0^\circ\lt y\lt 360^\circ\).

(c) Solve

\(7\operatorname{cot} z-\tan z=2\operatorname{cosec}z\)

for \(0^\circ\lt z\lt 360^\circ\).

Solve the equation

(i) \(4\sin\left(3x-\dfrac{\pi}{4}\right)=3\) for \(0\leqslant x\leqslant \dfrac{\pi}{2}\) radians,

(ii) \(2\tan^2 y+\operatorname{sec}^2 y=14\operatorname{sec} y+3\) for \(0^\circ\leqslant y\leqslant 360^\circ\).

Solve the equation

(a) \(2\lvert \sin x\rvert=1\) for \(-\pi\le x\le \pi\) radians,

(b) \(3\tan(2y+15^\circ)=1\) for \(0^\circ\le y\le 180^\circ\),

(c) \(3\operatorname{cot}^2 z=\operatorname{cosec}^2 z-7\operatorname{cosec} z+1\) for \(0^\circ\le z\le 360^\circ\).

(a) Solve \(3\operatorname{cosec}2x-4\sin2x=0\) for \(0^\circ\leq x\leq180^\circ\).

(b) Solve \(3\tan\left(y-\dfrac{\pi}{4}\right)=\sqrt3\) for \(0\leq y\leq2\pi\) radians, giving your answers in terms of \(\pi\).

(a) Solve \(2\operatorname{cot}(z+35^\circ)=5\) for \(0^\circ\leq z\leq360^\circ\).

(b) (i) Show that \(\dfrac{\operatorname{sec}\theta}{\operatorname{cot}\theta+\tan\theta}=\sin\theta\).

(ii) Hence solve \(\dfrac{\operatorname{sec}3\theta}{\operatorname{cot}3\theta+\tan3\theta}=-\dfrac{\sqrt3}{2}\) for \(-\dfrac{\pi}{2}\leq\theta\leq\dfrac{\pi}{2}\), giving your answers in terms of \(\pi\).

(a) Show that

\(\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=2\operatorname{cosec}x.\)

(b) Solve the following equations.

(i) \(\operatorname{cot}^2y+\operatorname{cosec}y-5=0\), for \(0^\circ\le y\le360^\circ\).

(ii) \(\cos\left(2z+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt3}{2}\), for \(0\le z\le\pi\) radians.