Tricks to learning trigonometry, math in general

Question

I don't know if it is appropriate to place this question here. Even though everyone says there is no tricks to learning math. The only thing there is is hardwork.

As you might have known, trig involves a lot of memorizing. Something I really have problem memorizing is half and double formulas. But generally, I would like to find out the derivations of the formula to help me in memorizing those formulas.

Are there better ways? In case even the derivation is hard to understand.

Do you know any sources of trig practice problem? Something I have discovered is that practicing is only part of what really help you with math. So if you have any recommendation of books to read, please put it here.

How much time a day do you guys spend on practice? This is my final question.

Answer 1

The amount of memorization in trig is minimal. The sum formulas $$\sin(x+y)=\sin x \cos y +\sin y \cos x$$ $$\cos(x+y)=\cos x \cos y -\sin x \sin y$$ are all you need to memorize since they are too hard to derive (although you could use de Moivre's formula as a mnemonic). Everything else can be reduced to simple derivations which you can preform on the spot. Including the unit circle. Memorization should result from practice. The trick to learning math is not to leave your apartment.

Answer 2

I found that after becoming familiar with complex numbers, remembering trigonometric formulas was much easier. This is because the theory of complex numbers provides the tools that make deriving such formulas particularly simple.

Answer 3

There definitely are tricks to memorizing formulas. For example, $\sin(x+y)$ has $\sin$ and $\cos$ paired up and summed, while $\cos(a+b)$ has the same trig functions together, but $\cos$ minus $\sin$. For $x-y$ you just flip the signs.

With a little inspection you'll notice the half angle formulas come from the identity $\cos(2A) = 1 - 2\sin^2(A) = 2 \cos^2(A) - 1$. This is just the $\cos(x+y)$ formula and the Pythagorean identity.

I hope these helped, everything's connected and related.