More about trigonometric functions：練習問題

Convert the angle \(315^\circ\) into radians, expressing the answer in terms of \(\pi\).

In \( \triangle ABC \), if \( \tan A = 2 \) and \( \tan B = 3 \), find the value of \( \tan C \).

Simplify the trigonometric expression:

$$E = \cos^2 x + \cos^2 \left(x + \frac{2\pi}{3}\right) + \cos^2 \left(x - \frac{2\pi}{3}\right)$$

Simplify the expression \( \sec^2 \theta - \frac{1}{\csc^2 \theta} - \tan^2 \theta \).

Simplify the expression \( \frac{1 - \cos 4\theta}{\sin^2 2\theta} \), assuming \( \sin 2\theta \ne 0 \).

Find the exact value of \( \cos \frac{5\pi}{12} + \cos \frac{\pi}{12} \).

Let \(0 < x < \frac{\pi}{2}\).

(a) Find a pair of constants \(h\) and \(k\) such that
\(\frac{\sec x + \tan x}{\csc x + \cot x} + \frac{\sec x - \tan x}{\csc x - \cot x} \equiv h \sec x \csc x + k\).

(b) Solve the equation \(\frac{\sec x + \tan x}{\csc x + \cot x} + \frac{\sec x - \tan x}{\csc x - \cot x} = 2\).

Solve the trigonometric equation \(sin(5\theta) + sin(3\theta) = cos(\theta) \) for \(0 \le \theta < \pi \).

(a) Convert \( 150^\circ \) to radians, expressing the answer in terms of \( \pi \).
(b) A sector of a circle has radius 6 cm and a central angle of \( \frac{2\pi}{3} \) radians. Find the exact value of its arc length and area.

(a) Prove the trigonometric identity:

$$ \frac{\cos 2x - \cos 4x}{\sin 4x + \sin 2x} = \tan x $$

(b) Hence, solve the equation \( \frac{\cos 2x - \cos 4x}{\sin 4x + \sin 2x} = 1 \) for \( 0 \le x < 2\pi \), ensuring the denominator is non-zero.