Double angle formulas
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Revision as of 18:42, 9 June 2017 by Alec (Talk | contribs) (Created page with "==Statement== The "''double angle formulas''" refer to the following two formulas * {{M|\forall \varphi,\psi\in\mathbb{R}[\sin(\varphi\pm\psi)\eq\sin(\varphi)\cos(\psi)\pm\cos...")
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:
