Everything Maths Grade 12

CHAPTER 10. TRIGONOMETRY 10.2

we have that

cos(α−β) = sin(90− (α−β))

= sin((90−α) +β))

= sin(90−α) cosβ + cos(90−α) sinβ

= cosα cosβ + sinα sinβ

Derivation ofsin2α EMCCK

We know that
sin(α +β) = sinα cosβ + cosα sinβ

When α = β, we have that

sin(2α) = sin(α +α) = sinα cosα + cosα sinα

= 2 sinα cosα

Derivation ofcos2α EMCCL

We know that
cos(α +β) = cosα cosβ− sinα sinβ

When α = β, we have that

cos(2α) = cos(α +α) = cosα cosα− sinα sinα

= cos^2 α− sin^2 α

However, we can also write
cos 2α = 2 cos^2 α− 1

and
cos 2α = 1− 2 sin^2 α

by using
sin^2 α + cos^2 α = 1.

Activity: Thecos2α Identity

Use

sin^2 α + cos^2 α = 1

to show that:

cos 2α = 2 cos^2 α− 1 = 1− 2 sin^2 α