Everything Maths Grade 11

17.3 CHAPTER 17. TRIGONOMETRY

Activity: Reduction Formulae forFunction Values of 360 ◦±θ

1. Function values of (360◦− θ)

(a) In the figure P and P�lie on the cir-

cle with radius 2. OP makes an angle

θ = 30◦with the x-axis. P thus has coor-

dinates (

√

3;1). P�is the reflection of P

about the x-axis or the line y = 0. Using

symmetry, write down the coordinates of

P�.

(b) Using the coordinates for P’ deter-

mine sin(360◦− θ), cos(360◦− θ) and

tan(360◦− θ).

(c) From your results tryand determine a re-

lationship between the function values of

(360◦− θ) and θ.

�P

0 x

y

θ

�

θ

360 ◦− θ

P�

2

2

It is possible to have anangle which is larger than 360 ◦. The angle completes one revolution to give
360 ◦and then continues to give the required angle. We get the following results:

sin(360◦+ θ) = sin θ

cos(360◦+ θ) = cos θ

tan(360◦+ θ) = tan θ

Note also, that if k is any integer, then

sin(k 360 ◦+ θ) = sin θ

cos(k 360 ◦+ θ) = cos θ

tan(k 360 ◦+ θ) = tan θ

Example 3: Basic Use of a Reduction Formula

QUESTION

Write sin293◦as the function of an acute angle.

SOLUTION

We note that 293 ◦= 360◦− 67 ◦thus

sin293◦ = sin(360◦− 67 ◦)

=−sin67◦

where we used the factthat sin(360◦− θ) =−sin θ. Check, using your calculator, that these