Trigonometry：練習問題

Evaluate \(\sin^2 30^\circ + \sin^2 60^\circ\).

Evaluate the expression \(\frac{\sin 30^\circ \cos 60^\circ}{\tan 45^\circ} + \sin^2 45^\circ\) without using a calculator.

Simplify the expression \(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta}\) for an acute angle \(\theta\).

In a right-angled triangle, if an acute angle \(\theta\) increases, which of the following statements about its trigonometric ratios is correct based on their properties?

A ship travels from point A to point B on a bearing of \( 120^\circ \) for a distance of \( 10 \) km. Calculate the vertical distance south that the ship has traveled from point A.

In a right-angled triangle ABC, with the right angle at B, if \(AB = 7\) cm and \(AC = 14\) cm, what is the value of \(\sin C\)?

If \( \sin(x + 15^\circ) = \cos(2x) \) and \( x \) is an acute angle, find the value of \( x \).

If \(\tan \theta = \frac{k}{3}\) and \(\cos \theta = \frac{3}{\sqrt{k^2+9}}\) for an acute angle \(\theta\), find the value of \(\sin^2 \theta + 2\cos^2 \theta\) in terms of \(k\).

In a right-angled triangle \(PQR\), \(\angle Q = 90^\circ\) and \(\angle P = 45^\circ\). The length of the hypotenuse \(PR\) is \(8\sqrt{2}\) cm.
(a) Find the measure of \(\angle R\).
(b) Find the length of the side \(PQ\).
(c) Find the length of the side \(QR\).
(d) Calculate the area of triangle \(PQR\).

A rectangular plot of land \(PQRS\) has a diagonal \(PR\) of length 26 m. The angle between the diagonal \(PR\) and the side \(PQ\) is \(22.6^\circ\).

(a) Find the lengths of the sides \(PQ\) and \(QR\), correct to 1 decimal place.
(b) Calculate the perimeter of the plot of land.
(c) Find the angle between the diagonal \(PR\) and the side \(PS\).