Everything Maths Grade 12

10.3 CHAPTER 10. TRIGONOMETRY

�

�

�

�

�

A

C B

D

O

θ

x

x

SOLUTION

Step 1 : Identify a strategy

We want CB, and we have CD and BD. If we could get the angle BDCˆ ,

then we could use the cosine rule to determine BC. This is possible, as�ABD

is a right-angled triangle. We know this from circle geometry, that anytriangle

circumscribed by a circle with one side going through the origin, is right-angled.

As we have two anglesof�ABD, we know ADBˆ and hence BDCˆ. Using the

cosine rule, we can get BC^2.

Step 2 : Execute the strategy

ADBˆ = 180◦−θ− 90 ◦= 90◦−θ

Thus

BDCˆ = 180◦−ADBˆ

= 180◦− (90◦−θ)

= 90◦+θ

Now the cosine rule gives

BC^2 = CD^2 +BD^2 − 2 .CD.BD. cos(BDCˆ )

= x^2 +x^2 − 2 .x^2. cos(90◦+θ)

= 2x^2 + 2x^2 [ sin(90◦) cos(θ) + sin(θ) cos(90◦)]

= 2x^2 + 2x^2 [ 1. cos(θ) + sin(θ). 0]

= 2x^2 (1− sinθ)

Exercise 10 - 1

1. For the diagram on the right,