TITLE: Is there a more efficient method of trig mastery than rote memorization?
QUESTION [10 upvotes]: I would like to get alot better at trig than I am. What is the best/most efficient method?
Thanks much in advance
Joe

REPLY [6 votes]: The complex-valued function $h(\theta)=\cos(\theta)+i \sin (\theta)$ is uniquely determined by the following three properties: 

$h$ is continuous;
$h(\theta+\phi)=h(\theta)h(\phi)$;
$\lvert h(\theta)\rvert=1$, $h(2\pi)=1$.

This means that any other property of $\cos$ and $\sin$, including all trigonometric formulas, can be derived from the above. 
Those properties are especially easy to remember if you write $e^{i \theta}=\cos(\theta)+i\sin(\theta)$ (this great idea is due to Euler), and this notation provides also a convenient algebraic way of deriving trig identities. 
Example. 
We want to expand $\cos^2(\theta)$. Write $\left( e^{i\theta}\right)^2= e^{i 2 \theta}$. The left hand side is 
$$(\cos \theta + i \sin \theta)(\cos \theta + i \sin \theta)=\cos^2\theta-\sin^2\theta+i(2\sin\theta \cos \theta),$$
while the right hand side is 
$$\cos 2\theta + i \sin 2 \theta.$$
Equating real parts we get 
$$\cos^2\theta=\sin^2\theta+\cos 2\theta,$$
and since $\lvert e^{i \theta}\rvert^2=1$, that is $\cos^2\theta+\sin^2\theta=1$, we conclude 
$$\cos^2\theta=\frac{1+\cos 2 \theta}{2}.$$