Applications of Trigonometry：練習問題

A straight road has an inclination of \( 12^\circ \) to the horizontal. If a car travels 500 m along the road, find the vertical distance it has climbed, correct to the nearest meter.

A vertical tower AB of height 50 m stands on horizontal ground, where B is the base. From the top A, the angles of depression of two cars C and D on the ground are 30° and 45° respectively. It is given that B, C and D lie on the same straight line and the cars are on opposite sides of the tower. Find the distance between the two cars, correct to 3 significant figures.

A regular tetrahedron \(PABC\) has a base \(ABC\) which is an equilateral triangle of side length \(12\) cm. The vertex \(P\) is vertically above the center \(G\) of the base. If the length of each slant edge (e.g., \(PA\)) is \(10\) cm, find the angle between the lateral face \(PBC\) and the base plane \(ABC\), correct to one decimal place.

A hiker walks along a straight path that has a gradient of \(3 : 20\). If the hiker covers a horizontal distance of \(400\) m, find the vertical height gained by the hiker.

A plank \(4\) m long is used as a ramp. If the foot of the plank is \(2.5\) m from the base of the platform, what is the angle of elevation to the nearest degree?

A ship sails from port \(P\) for \(80\) km on a bearing of \(072^\circ\) to reach island \(Q\). Find the distance that the ship has moved due east from port \(P\), correct to 3 significant figures.

A hiker walks 5 km from point P on a bearing of \(045^\circ\), arriving at point Q. How far East is Q from P, correct to two decimal places?

Town \(B\) is \(60\) km from Town \(A\) on a bearing of \(N 15^\circ E\). Town \(C\) is \(80\) km from Town \(A\) on a bearing of \(S 75^\circ E\). Calculate the shortest distance between Town \(B\) and Town \(C\).

A point P is located at a distance of 150 m from point O on a bearing of \( 055^\circ \).
(a) Find the distance that point P is due North of point O, correct to 1 decimal place.
(b) Find the distance that point P is due East of point O, correct to 1 decimal place.

A rescue helicopter at point H is located \(60\) km from a control base B on a bearing of \(040^\circ\). At the same time, a distressed ship at point S is located \(80\) km from the helicopter H on a bearing of \(130^\circ\).

(a) Show that \(\angle BHS = 90^\circ\).
(b) Calculate the distance between the control base B and the ship S.
(c) Find the 3-digit bearing of the ship S from the base B, correct to the nearest degree.