Cofunction identities,
and the fact that cosine is an even function
and sine is an odd function:
\(\sin(\theta) = \cos\bigl(90°\!-\theta\bigr)\)
\(\cos(\theta) = \sin\bigl(90°\!-\theta\bigr)\)
\(\cos(-\theta) = \cos(\theta)\)
\(\sin(-\theta) = -\sin(\theta)\)
Pythagorean identities:
\(\sin^2(\theta) + \cos^2(\theta) = 1\)
\(\tan^2(\theta) + 1 = \sec^2(\theta)\)
\(1 + \cot^2(\theta) = \csc^2(\theta)\)
Sum-of-angles formulas:
\(\displaystyle \cos(\theta+\phi) = \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)\)
\(\displaystyle \sin(\theta+\phi) = \sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi)\)
\(\displaystyle \tan(\theta+\phi) = \frac{\tan(\theta)+\tan(\phi)}{1 - \tan(\theta)\tan(\phi)}\)
Double-angle formulas and half-angle formulas:
\(\cos(2\theta) = 2\cos(\theta)\sin(\theta) \)
\(\cos\Bigl(\frac{\theta}{2}\Bigr) = \sqrt{\frac{1}{2}\bigl(1+\cos(\theta)\bigr)}\)
\(\sin(2\theta) = \cos^2(\theta)-\sin^2(\theta) \)
\(\sin\Bigl(\frac{\theta}{2}\Bigr) = \sqrt{\frac{1}{2}\bigl(1-\cos(\theta)\bigr)}\)
Extracurricular bonus fact — the trigonometric functions can be defined as complex exponential functions:
\(\displaystyle \mathrm{e}^{\mathfrak{i}\theta} = \cos(\theta) + \mathfrak{i}\sin(\theta) \)
\(\displaystyle \cos(\theta) = \frac{\mathrm{e}^{\mathfrak{i}\theta} + \mathrm{e}^{-\mathfrak{i}\theta}}{2} \)
\(\displaystyle \sin(\theta) = \frac{\mathrm{e}^{\mathfrak{i}\theta} - \mathrm{e}^{-\mathfrak{i}\theta}}{2\mathfrak{i}} \)