Trigonometry：練習問題

Simplify the trigonometric expression \(\frac{\sin \theta}{\tan \theta}\).

Solve the equation \(2 \sin^2 x - \cos x = 1\) for \(0 \le x < 2\pi\), giving your answers in radians.

Express \(4\sin x - 3\cos x\) in the form \(R\sin(x-\alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\).

Find the principal value of \(\theta\), in radians, for which \(\sin \theta = \frac{\sqrt{3}}{2}\).

A sector of a circle has an area of \(20 \text{ cm}^2\) and a radius of \(4\) cm. Find the angle of the sector in radians.

Find all solutions for \(\tan x = -1\) in the interval \(0 \leq x < 2\pi\).

(a) Show that the equation \(6\sin^2\theta + \cos\theta - 5 = 0\) can be written as \(6\cos^2\theta - \cos\theta - 1 = 0\).

(b) Hence, solve the equation \(6\sin^2\theta + \cos\theta - 5 = 0\) for \(0^\circ \le \theta < 360^\circ\). Give your answers to one decimal place where appropriate.