Everything Maths Grade 11

17.3 CHAPTER 17. TRIGONOMETRY

SOLUTION

1. cos50◦= sin(90◦− 50 ◦) = sin40◦


2. sin320◦= sin(360◦− 40 ◦) =−sin40◦


3. cos230◦= cos(180◦+ 50◦) =−cos50◦
   =−sin(90◦− 50 ◦) =−sin40◦

Function Values of (θ− 90 ◦)

sin(θ− 90 ◦) =−cos θ and cos(θ− 90 ◦) = sin θ.

These results may be proved as follows:

sin(θ− 90 ◦) = sin[−(90◦− θ)]

=−sin(90◦− θ)

=−cos θ

similarly, cos(θ− 90 ◦) = sin θ

Summary

The following summarymay be made

second quadrant (180◦− θ) or (90◦+ θ) first quadrant (θ) or (90◦− θ)

sin(180◦− θ) = +sin θ all trig functions are positive

cos(180◦− θ) =−cos θ sin(360◦+ θ) = sin θ

tan(180◦− θ) =−tan θ cos(360◦+ θ) = cos θ

sin(90◦+ θ) = +cos θ tan(360◦+ θ) = tan θ

cos(90◦+ θ) =−sin θ sin(90◦− θ) = sin θ

cos(90◦− θ) = cos θ

third quadrant (180◦+ θ) fourth quadrant (360◦− θ)

sin(180◦+ θ) =−sin θ sin(360◦− θ) =−sin θ

cos(180◦+ θ) =−cos θ cos(360◦− θ) = +cos θ

tan(180◦+ θ) = +tan θ tan(360◦− θ) =−tan θ

Tip

1. These reduction
   formulae hold
   for any angleθ.
   For convenience,
   we usually work
   withθ as if it
   is acute, i.e.
   0 ◦<θ< 90 ◦.


2. When determin-
   ing function val-
   ues of 180 ◦±
   θ, 360 ◦±θ and
   −θ the functions
   never change.


3. When determin-
   ing function val-
   ues of 90 ◦±θ
   andθ− 90 ◦the
   functions changes
   to its co-function
   (co-co rule).

Extension: Function Values of(270

◦±θ)

Angles in the third andfourth quadrants may be written as 270 ◦± θ with θ an acute angle.

Similar rules to the above apply. We get

third quadrant (270◦− θ) fourth quadrant (270◦+ θ)

sin(270◦− θ) =−cos θ sin(270◦+ θ) =−cos θ

cos(270◦− θ) =−sin θ cos(270◦+ θ) = +sin θ