Trigonometry：練習問題

Find all values of \(x\) in the interval \(0^\circ \le x \le 360^\circ\) which satisfy the equation \(4 \sin^2 x - 1 = 0\).

Find the maximum value of the function \(f(x) = 3 \cos x + 4 \sin x + 10\).

A sector of a circle has radius \(r\) cm and angle \(\theta\) radians. Given that the perimeter of the sector is 20 cm and its area is 24 cm\(^2\), find the possible values of \(r\) and \(\theta\).

In triangle \(ABC\), \(AB = x\) cm, \(BC = (x + 2)\) cm, \(AC = 5\) cm and \(\angle ABC = 60^\circ\). Calculate the value of \(x\).

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

Solve the equation \(2 \cos^2 x - 5 \cos x + 2 = 0\) for the interval \(0^\circ \le x < 360^\circ\).

Figure 1 shows a sector of a circle with radius \(r\) cm and angle \(\theta\) radians. Given that the sector has a fixed area of \(50 \text{ cm}^2\), express the perimeter \(P\) in terms of \(r\) and find the value of \(r\) for which \(P\) is stationary.