Trigonometry and Pythagoras’ theorem：練習問題

In a right-angled triangle, the hypotenuse has a length of \(10 \text{ cm}\). One of the acute angles is \(30^{\circ}\).
Calculate the length of the side opposite to the \(30^{\circ}\) angle.

From the top of a vertical cliff that is \(50 \text{ m}\) high, the angle of depression to a boat on the sea is \(25^{\circ}\).
Calculate the horizontal distance from the boat to the base of the cliff. Give your answer to 1 decimal place.

A square-based pyramid has a base side length of \(10 \text{ cm}\) and a vertical height of \(12 \text{ cm}\). The vertex \(V\) is directly above the center of the base.
Calculate the angle between a slant edge and the base of the pyramid. Give your answer to 1 decimal place.

In a right-angled triangle \(ABC\), the hypotenuse \(AB = 13 \text{ cm}\) and the side \(BC = 12 \text{ cm}\).
Calculate the length of the side \(AC\).

A right-angled triangle has a hypotenuse of \(15\text{ cm}\) and one shorter side of length \(9\text{ cm}\). Calculate the length of the third side.

In triangle \(ABC\), angle \(B = 90^{\circ}\), side \(AB = 6 \text{ cm}\), and side \(BC = 8 \text{ cm}\).
Calculate the length of the side \(AC\).

A vertical flagpole stands on horizontal ground. From a point on the ground \(15 \text{ m}\) away from the base of the flagpole, the angle of elevation to the top of the pole is \(38^{\circ}\).
Calculate the height of the flagpole, giving your answer to 3 significant figures.

A rectangular box (cuboid) has dimensions \(8 \text{ cm}\) by \(6 \text{ cm}\) by \(5 \text{ cm}\).
Calculate the length of the space diagonal connecting two opposite corners of the box. Give your answer correct to 2 decimal places.

A ladder of length 5 meters leans against a vertical wall. The base of the ladder is 1.5 meters from the bottom of the wall on horizontal ground.

(a) Calculate the height the ladder reaches up the wall. Give your answer to 2 decimal places.

(b) Calculate the angle the ladder makes with the horizontal ground. Give your answer to 1 decimal place.

A ship sails from port \( A \) on a bearing of \( 060^{\circ} \) for 15 km to reach point \( B \). From point \( B \), the ship changes course and sails on a bearing of \( 160^{\circ} \) for 20 km to reach point \( C \).

(a) Calculate the distance \( AC \). Give your answer to 3 significant figures.

(b) Calculate the bearing of \( A \) from \( C \). Give your answer to the nearest degree.