Some planar geom:
$$ \textrm{Sine rule: }a/\sin A=b/\sin B=c/ \sin C $$
$$ \textrm{Cosine rule: }a^2=b^2+c^2-2bc\cos A $$
R-formulae:
$$ a \sin \theta + b\cos \theta = R \sin (\theta + \alpha)=R\cos(\theta -\alpha) \quad \\ \textrm{where} \quad R=\sqrt{a^2+b^2}, \quad \alpha=\arctan \frac{b}{a} \textrm{ or arctan} \frac{a}{b} \textrm{for cos} $$
Sum to product:
The form looks quite similar to the addition formulae
$$ \sin u+\sin v=2\sin\frac{u+v}{2}\cos\frac{u-v}{2} \\ \sin u-\sin v=2\sin\frac{u-v}{2}\cos\frac{u+v}{2} \\ \cos u + \cos v=2\cos\frac{u+v}{2}\cos\frac{u-v}{2} \\ \cos u - \cos v=-2\sin\frac{u+v}{2}\sin\frac{u-v}{2} $$
Half angle formulae:
$$ \tan \frac{x}{2}=\frac{\sin x}{1+\cos x} $$
Very blatant approximation:
$$ \frac{dy}{dx} \approx \frac{\Delta y}{\Delta x} \approx \frac{y-y_0}{x-x_0} $$
Useful:
$$ f(x,y)=kx^ay^b \implies \frac{\dot{f}}{f}=a\frac{\dot{x}}{x}+b\frac{\dot{y}}{y} $$
Taylor series:
$$ f(x)=\sum^{\infty}_{n=0}\frac{f^{(n)}(a)}{n!}(x-a)^n=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+ \cdots $$
Can be applied by expanding to first or second order if $x-a \ll a$. Only works for analytic functions (see video by morphocular)
If $x \gg 1$, any addition or subtraction can be ignored: $x+1 \approx x$