Double angle formulas
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Statement
The "double angle formulas" refer to the following two formulas
- [ilmath]\forall \varphi,\psi\in\mathbb{R}[\sin(\varphi\pm\psi)\eq\sin(\varphi)\cos(\psi)\pm\cos(\varphi)\sin(\psi)][/ilmath]
- [ilmath]\forall\varphi,\psi\in\mathbb{R}[\cos(\varphi\pm\psi)\eq\cos(\varphi)\cos(\psi)\mp\sin(\varphi)\sin(\psi)][/ilmath]
However sometimes it is taken to mean the following two special cases:
- [ilmath]\forall\varphi\in\mathbb{R}[\sin(2\varphi)\eq2\sin(\varphi)\cos(\varphi)][/ilmath] and
- [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq(\cos(\varphi))^2-(\sin(\varphi))^2\big][/ilmath]
- Noting that [ilmath](\sin(\theta))^2+(\cos(\theta))^2\eq 1[/ilmath] we see that [ilmath](\cos(\theta))^2\eq 1-(\sin(\theta))^2[/ilmath] and [ilmath](\sin(\theta))^2\eq 1-(\cos(\theta))^2[/ilmath], yielding:
- [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq 1-2(\sin(\varphi))^2\big][/ilmath] and
- [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq 2(\cos(\varphi))^2-1\big][/ilmath]
- Either form is commonplace.
- Noting that [ilmath](\sin(\theta))^2+(\cos(\theta))^2\eq 1[/ilmath] we see that [ilmath](\cos(\theta))^2\eq 1-(\sin(\theta))^2[/ilmath] and [ilmath](\sin(\theta))^2\eq 1-(\cos(\theta))^2[/ilmath], yielding:
