Bearings SS2 Mathematics Lesson Note

ANGLE OF ELEVATION

This is the angle formed between the normal eye level and the line through which the observer views an object above.

In a ∆ABC, Angle ACB = Ó¨= Angle of elevation.

ANGLE OF DEPRESSION

This is the angle formed between the eye level of the observer and the object below when the observer is above the object in view.                    

Angle ABC = Ó¨ = Angle of Depression.           

The angle of elevation alternates with the angle of depression and problems involving angles of elevation and depression could be solved by using the basic trigonometric ratios and in some cases, the sine and cosine rule could be applied.

Sine Rule for BÂC; a   =    b    =     c

i.e. Sin A   =   Sin B  =    Sin C

ii. Cosine Rule:  

1. a2  =  b2 + c2 – 2bc Cos A
2. b2  =  a2 + c2 – 2ac Cos B 

iii. c2  =  a2 + b2 – 2ab Cos C

EXAMPLES

1. A ladder 50 m long rests against a vertical wall. If the ladder makes an angle of 65° with the ground, find the distance between the foot of the ladder and the wall.

Solution:                                                                   

Ladder = QR,     

Wall = QP

Distance between the foot of the ladder and the wall 

= PR                                           

 Cos 65°   =  PR

        50

                                                                                                                             Cross multiplying                                                                                                                 

 PR = 50 x Cos 65° = 50 x 0.4226                                                                                                                   PR = 21.13m

2. The angle of depression of an object on the ground from the top of a tower 60m high is 55°. Find the distance between the foot of the tower and the object to the nearest whole number.

Solution:                                                                                                       

Tower = AC,   Object = A

Distance between the foot of the tower and the object = BC

Cross multiplying; BC x Tan 55° = 60                                                                                               BC   = 60/tan 55°

BC   = 60/1.428 = 42.02m

BC   = 42m (nearest whole number)

ASSIGNMENT 

From the top of a building 10m high, the angle of elevation of a stone lying on the horizontal ground is 70°. Calculate correctly to 1 decimal place, the distance of the stone from the foot of the building and the distance of the stone from the top of the building