College Algebra & Trigonometry, 2018a

10.2. DOUBLE-ANGLE IDENTITIES 455
In the diagram below, we can see this more clearly:
c
90 ◦ − θ
a
θ
b
In the diagram above note that:
sin θ = a c =cos(90◦ − θ)
So, if we want an identity for sin(θ + θ), we’ll start with sin(α + β) which is equivalent
to cos(90 ◦ − (α + β)). We’ll use a trick here and restate this as:
sin(α + β) =cos(90 ◦ − (α + β))
=cos(90 ◦ − α − β)
= cos((90 ◦ − α) − β)
=cos(90 ◦ − α)cosβ + sin(90 ◦ − α)sinβ
=sinα cos β +cosα sin β
Now, we can use this to find an expression for sin 2θ = sin(θ + θ):
sin 2θ = sin(θ + θ)
=sinθ cos θ +cosθ sin θ
=2sinθ cos θ