Trigonometry：练习题

Given that \(\tan \theta = 2\), find the value of \(\sin 2\theta\).

Prove the identity \(\frac{\sin 3\theta}{\sin \theta} - \frac{\cos 3\theta}{\cos \theta} = 2\).

Which of the following is an intermediate step in the proof?

Solve the equation \(2 \cot^2 \theta - 5 \operatorname{cosec} \theta + 1 = 0\) for \(0^\circ \le \theta \le 360^\circ\).

Solve the equation \(\sin^2 \theta + 3 \sin \theta \cos \theta + 2 \cos^2 \theta = 0\) for the interval \(0^\circ \le \theta \le 180^\circ\).

Simplify the expression \(\sin(x+45^\circ) + \cos(x+45^\circ)\).

Solve the equation $$4\cos x = 2$$ for $$0^\circ \le x \le 360^\circ$$.

Express \( 8 \sin x + 15 \cos x \) in the form \( R \sin(x + \alpha) \), where \( R > 0 \) and \( 0^\circ < \alpha < 90^\circ \). Give the value of \( \alpha \) correct to one decimal place.

Solve the equation \( 2\sec^2 x - \tan x = 5 \) for \( 0^\circ \le x \le 360^\circ \).

(a) By expanding $$ \sin(x+30^\circ) $$ and $$ \cos(x-30^\circ) $$, show that the equation $$ \sin(x+30^\circ) = 2 \cos(x-30^\circ) $$ can be written in the form $$ \tan x = k $$, where $$ k $$ is an exact constant.
(b) Hence, solve the equation $$ \sin(x+30^\circ) = 2 \cos(x-30^\circ) $$ for $$ 0^\circ \le x \le 360^\circ $$, giving your answers correct to one decimal place.

(a) Show that the equation \(2 \sin x \tan x = 3 \cos x\) can be written as \(5 \cos^2 x - 2 = 0\) .

(b) Hence solve the equation \(2 \sin x \tan x = 3 \cos x\) for \(0^\circ \le x \le 360^\circ\) .