Problems — 9.3 Right-angle Trigonometry

1

**Tree height.** A tree casts a 5 m shadow when the angle of elevation of the sun is 40°. (a) Find the tree height. (b) Find the slant distance from the tip of the shadow to the top of the tree.

Working space

2

**Building observation.** A pedestrian 1.7 m tall stands 40 m from a 31 m monument. (a) Find the angle of elevation of the top of the monument from the pedestrian's eyes. (b) The pedestrian moves to 20 m from the monument. Find the new angle.

Working space

3

**Zipline.** A zipline runs from a 154 m UN building to a 443 m Empire State Building, separated horizontally by 1488 m. (a) Find the zipline length. (b) Find the angle of elevation from the UN to the ESB.

Working space

4

**Bearings.** A boat sails 12 km on a bearing of 60°. (a) Find the easterly displacement. (b) Find the northerly displacement. (c) The boat then sails 10 km on bearing 150°. Find the resulting position.

Working space

5

**Pyramid trigonometry.** A pyramid has square base side 10 cm, apex 12 cm above the centre. (a) Find the half-diagonal of the base. (b) Find the slant edge length. (c) Find the angle the slant edge makes with the base.

Working space

6

**Cuboid diagonal trigonometry.** A cuboid has length 8, width 6, height 4. (a) Find the base diagonal. (b) Find the space diagonal. (c) Find the angle the space diagonal makes with the base.

Working space

7

**Exact-value triangles.** For a 30-60-90 triangle with shorter leg 1: (a) Find the hypotenuse. (b) Find the longer leg. (c) State the ratios sin, cos, tan for 30° and 60°.

Working space

8

**45-45-90 triangle.** A square has side 1. (a) Find the diagonal. (b) Find sin 45°, cos 45°, tan 45°.

Working space

9

**Angle of depression problem.** A coastguard 50 m above sea level sees a boat with angle of depression 8°. (a) Find the horizontal distance to the boat. (b) The boat moves so the angle is 12°. Find the new distance. (c) How fast does the boat's distance to the coastguard's vertical change?

Working space

10

**Find the side or angle.** A right triangle has hyp 20 cm and angle 25°. (a) Find both legs. (b) Find the area. (c) Find the perimeter.

Working space

11

**3D trig in a cuboid.** A box is 12 cm × 8 cm × 6 cm. A diagonal is drawn from one corner to the opposite corner. (a) Find the base diagonal. (b) Find the space diagonal. (c) Find the angle between the space diagonal and the longest face.

Working space

12

**Application: roof slope.** A house has a roof that rises 3 m over a horizontal span of 5 m. (a) Find the angle of the roof above horizontal. (b) Find the slant length of the roof. (c) Find the area of one side of the roof, if the house is 10 m long.

Working space