Trigonometry：练习题

If \(\tan \theta = \frac{3}{4}\) and \(\theta\) is an acute angle, what is the value of \(\cos \theta\)?

The diagram shows the graph of \(y = a\cos(bx) + c\). The maximum point is \((0, 5)\) and the minimum point is \((\pi, -1)\). Find the values of \(a, b, c\).

Solve the equation \(\tan\theta + \cot\theta = 2.5\) for \(0^\circ \le \theta \le 180^\circ\). Give your answers to one decimal place.

Evaluate the exact value of \(\sin 30^\circ \cos 60^\circ + \tan 45^\circ\).

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

Find the exact value of \(\sin 150^\circ + \tan 315^\circ\).

Solve the equation \(2\cos^2\theta - \sin\theta - 1 = 0\) for \(0^\circ \le \theta \le 360^\circ\).

Express \( 3 \sin \theta - 4 \cos \theta \) in the form \( R \sin(\theta - \alpha) \), where \( R > 0 \) and \( 0^\circ < \alpha < 90^\circ \), giving the exact value of \( R \) and \( \alpha \) correct to one decimal place.

Solve the equation \( 2 \cos^2 x + 5 \sin x = 4 \) for \( 0^\circ \le x \le 360^\circ \).

The function \( \text{f}(\theta) \) is defined by \( \text{f}(\theta) = 3\sin\theta - 4\cos\theta \) for \( 0^\circ \le \theta \le 360^\circ \).

(a) Express \( \text{f}(\theta) \) in the form \( R\sin(\theta - \alpha) \), where \( R > 0 \) and \( 0^\circ < \alpha < 90^\circ \).

(b) Hence, state the maximum and minimum values of \( \text{f}(\theta) \).