Trigonometry：练习题

Find the exact value of \(\sin 150^\circ + \cos 240^\circ\).

Solve the equation
\(4\sin^2\theta = 3\)
for \(0 \le \theta \le \pi\).
Give your answers in terms of \(\pi\).

Solve the equation \(2\cos^2 x + \sin x - 1 = 0\) for the interval \(0^\circ \le x < 360^\circ\).

Find all solutions to the equation
\(2\sin\theta - 1 = 0\)
for \(0^\circ \le \theta \le 360^\circ\).

Solve the equation
\(2\cos^2\theta - 3\cos\theta + 1 = 0\)
for \(0 \le \theta \le 2\pi\).
Give your answers in terms of \(\pi\).

Using the addition formula for \(\sin(A - B)\), find the exact value of \(\sin 15^\circ\).
Give your answer in the form \(\frac{\sqrt{6} - \sqrt{2}}{k}\), where \(k\) is an integer.

A sector of a circle with radius \( r \) and center angle \( \theta \) radians has an area of \( 24\text{ cm}^2 \) and an arc length of \( 8\text{ cm} \). Find the value of \( r \) and the value of \( \theta \).

Find the exact solutions of the equation \( 2\cos^2 \theta + 5\sin \theta = 4 \) for the interval \( 0 \le \theta \le 2\pi \).

A sector of a circle of radius \(8\text{ cm}\) has an angle of \(1.25\) radians at the centre.
(a) Find the length of the arc of the sector.
(b) Find the area of the sector.

(a) Use the identity \(\sin(A - B) = \sin A\cos B - \cos A\sin B\) with \(A = 45^\circ\) and \(B = 30^\circ\) to show that the exact value of \(\sin 15^\circ\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\).
(b) Solve the equation
\(4\sin\left(\theta - 15^\circ\right) = \sqrt{6} - \sqrt{2}\)
for \(0^\circ \le \theta \le 360^\circ\).