Everything Maths Grade 11

CHAPTER 17. TRIGONOMETRY 17.5

Therefore the area of�ABC is:

1

2

c. h

=

1

2

c. b. sinAˆ

=

1

2

c. a. sinBˆ

Similarly, by drawing the perpendicular between point B and line AC we can show that:

1

2

c. b. sinAˆ =

1

2

a. b. sinCˆ

Therefore the area of�ABC is:

1

2

c. b. sinAˆ =

1

2

c. a. sinBˆ =

1

2

a. b. sinCˆ

If we divide through by^12 a. b. c, we get:

sinAˆ

a

=

sinBˆ

b

=

sinCˆ

c

This is known as the sine rule and applies to any triangle, right-angled ornot.

Example 14: Lighthouses

QUESTION

A� �B

�

C

127 ◦ 255 ◦

There is a coastline withtwo lighthouses, one oneither side of a beach. The two lighthouses

are 0 , 67 km apart and one is exactly due east of the other. The lighthouses tellhow close a

boat is by taking bearings to the boat (remember – a bearing is an anglemeasured clockwise

from north). These bearings are shown. Use thesine rule to calculate how far the boat is from

each lighthouse.