Everything Maths Grade 12

CHAPTER 10. TRIGONOMETRY 10.2

Example 2: Trigonometric Identities 2

QUESTION

Deduce a formula for tan(α +β) in terms of tanα and tanβ.

Hint: Use the formulaefor sin(α +β) and cos(α +β)

SOLUTION

Step 1 : Identify a strategy

We can express tan(α + β) in terms of cosines and sines, and then use the

double-angle formulae for these. We then manipulate the resulting expression in

order to get it in terms of tanα and tanβ.

Step 2 : Execute strategy

tan(α +β) =

sin(α +β)

cos(α +β)

=

sinα. cosβ + cosα. sinβ

cosα. cosβ− sinα. sinβ

=

sin α .cos β

cos α .cos β+

cos α .sin β

cos α .cos β

cos α .cos β

cos α .cos β−

sin α .sin β

cos α .cos β

=

tanα + tanβ

1 − tanα. tanβ

Example 3: Trigonometric Identities 3

QUESTION

Prove that

sinθ + sin 2θ

1 + cosθ + cos 2θ

= tanθ

In fact, this identity is not valid for all values of θ. Which values are those?

SOLUTION

Step 1 : Identify a strategy

The right-hand side (RHS) of the identity cannotbe simplified. Thus weshould

try simplify the left-handside (LHS). We can alsonotice that the trig function on

the RHS does not havea 2 θ dependence. Thus wewill need to use the double-

angle formulae to simplify the sin 2θ and cos 2θ on the LHS. We knowthat tanθ