At the end of this lesson, students should be able to:

- Solve trigonometric problems in two dimensions involving angle of depression.

**Angle of depression**

The angle of depression is the angle below the horizontal through which a line of view is lowered. When you are standing on a high point and look down at a boat, the angle through which your line of sight turns from looking straight ahead (the horizontal) is called the angle of depression.

__Example 1__

From the top of a vertical cliff, 19m high, Cynthia sees a boat out at sea. The angle of depression from Cynthia to the boat is 73^{o}. How far from the base of the cliff is the boat?

__Solution __

From figure 1, the angle formed when Cynthia looks down the cliff to the boat is the angle of depression. Figure ii shows a right – angled triangle that is gotten from figure 1.

__Example 2__

A person standing on top of a cliff 50 m high is in line with two buoys whose angles of depression are 18^{0} and 20^{0}. Calculate the distance between the buoys.

__Solution__

The problem is illustrated in the diagram below where the two buoys are C and D and the observer is A.

__Unit 10 Lesson 15: Exercise __

1) An aero plane receives a signal from a point X on the ground. If the angle of depression of point X from the aero plane is 30^{o}, calculate the height at which the plane is flying if the horizontal distance between the plane and the point X is 43 m.

2) A and B are two villages. If the horizontal distance between them is 12km and the vertical distance between them is 2km, calculate

(i) The shortest distance between the two villages.

(ii) The angle of depression of A from B.

3) An aircraft is flying at an altitude of 4000m and is 10km from the airport. If a passenger can see the airport, what is the angle of depression?

4) A fishing boat detects a school of fish at a depth of 12 m, and at a distance of 0.4 km away.

a) At what angle of depression was the detecting sonar beam, to the nearest degree?

b) At what depth was a school of fish when detected 500 m away at an angle of depression of 2.1^{0}?

5) A bird flies from the top of a 12m tall tree at an angle of depression of 34^{o}, to catch a worm on the ground.

a) How far does the bird actually fly?

b) How far was the worm from the base of the tree?

6) Stephen wants to work out the height of a building. He stands about 50m away from the building. The angle of depression from the top of the building to Stephen is about 15^{o}. How tall is the building?

7) A man standing 200m from the base of a television transmitter looks at a bird on top of it and notices that the angle of depression from the bird to where he stands is 65^{o}. How high is the television transmitter?

8) Man standing on top of a cliff 80 m high is in between two buoys whose angles of depression are 17^{0} and 21^{0}. Calculate the distance between the buoys.

9) Two hot air balloons are 1 km apart in the air. If the angle of elevation of the higher from the lower balloon is 20^{0}. Calculate giving your answer to the nearest meter;

(i) The vertical height between the two balloons.

(ii) The horizontal distance between the two balloons.

10) A man on top of a 100 m tower sees a dog standing at the base of the tower. The distance from the base of the tower to the dog is 300m. What is the angle of depression of the dog from the man?

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