Solve the equation
\(\sqrt2\cos(3x+1.2)=2\sin(3x+1.2),\)
where \(x\) is in radians, for \(-1.5\leq x\leq1.5\).
Solve the equation
\(\operatorname{cot}\left(2x+\frac{\pi}{3}\right)-\sqrt3=0,\)
where \(-\pi\lt x\lt \pi\) radians. Give your answers in terms of \(\pi\).
Solve the equation
\(\operatorname{cot}^2\left(2x-\frac{\pi}{3}\right)=\frac13,\)
where \(x\) is in radians and \(0\leq x\lt \pi\).
Solve \(1+\sqrt2\sin(x+50^\circ)=0\), for \(-180^\circ\leq x\leq180^\circ\).
(a) Find \(\int_0^\pi \sin\theta\,\mathrm{d}\theta\).
(b) Given that \(0\lt \alpha\lt \frac{\pi}{2}\), show that \(\frac{\sec\alpha}{\cot\alpha+\tan\alpha}\) can be written as \(\sin\alpha\).
Show that \(\tan \theta+\cot \theta\) can be written as \(\sec \theta \operatorname{cosec} \theta\).
(a) Show that
\(\frac{\sin x\tan x}{1-\cos x}=1+\operatorname{sec} x.\)
(b) Solve the equation
\(5\tan x-3\operatorname{cot} x=2\operatorname{sec} x\)
for \(0^\circ\leq x\leq360^\circ\).
(i) Show that
\(\frac{\operatorname{cosec}\theta}{\operatorname{cot}\theta+\tan\theta}=\cos\theta.\)
It is given that
\(\int_0^a \frac{\operatorname{cosec}2\theta}{\cot2\theta+\tan2\theta}\,d\theta=\frac{\sqrt3}{4}, \qquad 0\lt a\lt \frac{\pi}{4}.\)
(ii) Using your answer to part (i), find \(a\) in terms of \(\pi\).
(a) Show that
\(\frac{\tan^2\theta+\sin^2\theta}{\cos\theta+\operatorname{sec}\theta}=\tan\theta\sin\theta.\)
(b) Given that \(x=3\sin\phi\) and \(y=\dfrac3{\cos\phi}\), find the numerical value of \(9y^2-x^2y^2\).
(i) Prove that \(\sin x(\operatorname{cot} x+\tan x)=\operatorname{sec} x\).
(ii) Hence solve the equation \(|\sin x(\operatorname{cot} x+\tan x)|=2\) for \(0^\circ\le x\le360^\circ\).
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.