Double angle formulas
From Maths
Statement
The "double angle formulas" refer to the following two formulas
- ∀φ,ψ∈R[sin(φ±ψ)=sin(φ)cos(ψ)±cos(φ)sin(ψ)]
- ∀φ,ψ∈R[cos(φ±ψ)=cos(φ)cos(ψ)∓sin(φ)sin(ψ)]
However sometimes it is taken to mean the following two special cases:
- ∀φ∈R[sin(2φ)=2sin(φ)cos(φ)] and
- ∀φ∈R[cos(2φ)=(cos(φ))2−(sin(φ))2]
- Noting that (sin(θ))2+(cos(θ))2=1 we see that (cos(θ))2=1−(sin(θ))2 and (sin(θ))2=1−(cos(θ))2, yielding:
- ∀φ∈R[cos(2φ)=1−2(sin(φ))2] and
- ∀φ∈R[cos(2φ)=2(cos(φ))2−1]
- Either form is commonplace.
- Noting that (sin(θ))2+(cos(θ))2=1 we see that (cos(θ))2=1−(sin(θ))2 and (sin(θ))2=1−(cos(θ))2, yielding:
