More about Trigonometry：練習問題

Consider the trigonometric function \(y = 3 - 2 \sin(4x - 60^{\circ})\). Which of the following is the maximum value of the function?

The figure shows a right pyramid with a square base \(ABCD\) of side length \(10 \text{ cm}\). The vertex \(V\) is located such that the foot of the perpendicular from \(V\) to the base is the center of the square. If the height of the pyramid is \(12 \text{ cm}\), find the angle between the slant edge \(VA\) and the base \(ABCD\) correct to the nearest degree.

Find the maximum value of the function \(f(\theta) = 4\sin^2 \theta + 6\cos \theta + 3\) for \(0^\circ \le \theta < 360^\circ\).

Consider the function \(y = 4 - 3\sin \theta\) for \(0^{\circ} \le \theta \le 360^{\circ}\). Find the range of the function.

Solve the trigonometric equation \(2 \tan \theta = \sin \theta\) for \(0^\circ \le \theta < 360^\circ\).

In \(\triangle ABC\), the lengths of sides \(AB\), \(BC\) and \(AC\) are \(13\text{ cm}\), \(14\text{ cm}\) and \(15\text{ cm}\) respectively. Use Heron's formula to find the area of \(\triangle ABC\).

In a cube \(ABCD-EFGH\), where \(ABCD\) is the horizontal base and \(E, F, G, H\) are vertices directly above \(A, B, C, D\) respectively, find the angle between the body diagonal \(HB\) and the base \(ABCD\), correct to 1 decimal place.

In a right triangular prism \( ABC-DEF \), the base \( ABC \) is an equilateral triangle with side length \( 4 \) cm and the height of the prism is \( 6 \) cm. If \( M \) is the midpoint of the edge \( EF \), find the angle between the line \( AM \) and the base \( ABC \) correct to the nearest \( 0.1^\circ \).

In \(\triangle PQR\), \(PQ = 15\text{ cm}\), \(\angle QPR = 35^\circ\) and \(\angle PQR = 75^\circ\).

(a) Find \(\angle PRQ\).

(b) Find the length of \(QR\) correct to 3 significant figures.

(c) Find the area of \(\triangle PQR\) correct to 3 significant figures.

A vertical pole \(OP\) stands at the corner \(O\) of a horizontal rectangular field \(OABC\). It is given that \(OA = 5\text{ m}\) and \(OC = 12\text{ m}\). The angle of elevation of the top of the pole \(P\) from the vertex \(B\) is \(30^\circ\).

(a) Find the length of the diagonal \(OB\).

(b) Find the height of the pole \(OP\) in metres, leaving your answer in surd form.

(c) Find the angle of elevation of \(P\) from vertex \(A\), correct to 1 decimal place.