Everything Maths Grade 12

10.2 CHAPTER 10. TRIGONOMETRY

is undefined for some angles θ. Thus the identity is also undefined for these θ,

and hence is not valid for these angles. Also, forsome θ, we might have division

by zero in the LHS, which is not allowed. Thusthe identity won’t holdfor these

angles also.

Step 2 : Execute the strategy

LHS =

sinθ + 2 sinθ cosθ

1 + cosθ + (2 cos^2 θ− 1)

=

sinθ(1 + 2 cosθ)

cosθ(1 + 2 cosθ)

=

sinθ

cosθ

= tanθ

= RHS

We know that tanθ is undefined when θ = 90◦+ 180◦n, where n is an integer.

The LHS is undefined when 1 + cosθ + cos 2θ = 0. Thus we need to solvethis

equation.

1 + cosθ + cos 2θ = 0

=⇒ cosθ(1 + 2 cosθ) = 0

The above has solutionswhen cosθ = 0, which occurs when θ = 90◦+ 180◦n,

where n is an integer. These are the same values when tanθ is undefined. It

also has solutions when 1 + 2 cosθ = 0. This is true when cosθ =−^12 , and thus

θ = ...− 240 ◦,− 120 ◦, 120 ◦, 240 ◦,.... To summarise, the identity is not valid

when θ = ...− 270 ◦;− 240 ◦;− 120 ◦;− 90 ◦; 90◦; 120◦; 240◦; 270◦;...

Example 4: Trigonometric Equations

QUESTION

Solve the following equation for y without using a calculator:

1 − siny− cos 2y

sin 2y− cosy

=− 1

SOLUTION

Step 1 : Identify a strategy

Before we are able to solve the equation, we first need to simplify the left-hand

side. We do this by using the double-angle formulae.

Step 2 : Execute the strategy