Trigonometry：练习题

Simplify the trigonometric expression \(\sin \theta (\csc \theta - \sin \theta)\).

Find the values of \(x\) in the interval \(0 \le x \le \pi\) that satisfy the equation \(2 \sin(2x) = 1\).

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

Solve the equation \(\csc \theta = 2\) for \(0^\circ \le \theta \le 180^\circ\).

Find the values of \(\theta\) in the interval \(0 \le \theta \le \pi\) for which \(\sin(2\theta) = 0.5\).

State the amplitude and period of the function \( y = 3\sin(2x) \), where \( x \) is in degrees.

Show that \( \sin x \tan x + \cos x = \sec x \).

(a) Prove the identity \( \frac{\tan A}{1 + \sec A} + \frac{1 + \sec A}{\tan A} = 2 \text{cosec } A \).
(b) Hence, find all the values of \( \theta \) in the interval \( 0^\circ \le \theta \le 360^\circ \) such that \( \frac{\tan \theta}{1 + \sec \theta} + \frac{1 + \sec \theta}{\tan \theta} = 4 \sec \theta \).

A function is defined by \( f(x) = 5\sin(3x) - 2 \) for \( 0 \le x \le \pi \) radians.
(a) Write down the amplitude and the period of \( f \).
(b) Find the maximum and minimum values of \( f(x) \).

(a) Show that the equation \( 3\sec^2 \theta + 5\tan \theta - 7 = 0 \) can be written as \( 3\tan^2 \theta + 5\tan \theta - 4 = 0 \).
(b) Hence, solve the equation \( 3\sec^2 \theta + 5\tan \theta - 7 = 0 \) for \( 0^\circ \le \theta \le 180^\circ \), giving your answers correct to 1 decimal place.