Trigonometric Identities¶
Recall (doc/postdoc). Not a first course.
Job: polynomial relations in \(\sin,\cos\) from \(S^1\) / complex mult.; addition = representation of \(\mathbb{R}/2\pi\mathbb{Z}\).
Core¶
| Family | Identities |
|---|---|
| Pythagorean | \(\sin^2+\cos^2=1\); \(1+\tan^2=\sec^2\); \(1+\cot^2=\csc^2\) |
| Even/odd | \(\sin(-\theta)=-\sin\theta\); \(\cos(-\theta)=\cos\theta\) |
| Cofunction | \(\sin(\frac\pi2-\theta)=\cos\theta\) |
| Sum | \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\) |
| \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\) | |
| Double | \(\sin2A=2\sin A\cos A\); \(\cos2A=\cos^2A-\sin^2A\) |
| Power-reduce | \(\sin^2A=\frac{1-\cos2A}{2}\); \(\cos^2A=\frac{1+\cos2A}{2}\) |
Euler: \(e^{iA}e^{iB}=e^{i(A+B)}\) implies sum formulas via \(\operatorname{Re}/\operatorname{Im}\).
Domain: identities hold where both sides defined (avoid \(\cos=0\) in \(\tan\) forms).
Used later for¶
- Fourier analysis: product-to-sum, harmonics
- Complex analysis: analytic trig
- Integrals: Weierstrass \(t=\tan(x/2)\)
- Representation theory: characters of \(SO(2)\)
Check¶
Does \(\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\) hold for all real \(A,B\)?
No — undefined when \(\cos A=0\), \(\cos B=0\), or \(\cos(A+B)=0\) (denom \(1-\tan A\tan B=0\)). Holds on open set where all tans exist and denom \(\neq0\).