I think I've reinvented trigonometry at least fifteen times in my life when I needed to animate something roughly circular, spherical or wave-like. I still have no freakin clue which thing is a sin, cos or tan. I failed precalc (and they still let me work on video games!) But I do remember that a hypotenuse is the square root of A^2 + B^2 and from that I can pretty much figure out the rest.
As someone who did trigonometry every week for over a decade, I can say that reinventing it often will really slow you down. There is merit to memorizing the identities (of course, you should understand them well enough to derive them if you need to).
I was also better at solving problems involving trigonometry than anyone I knew - including my professors.
You could, of course, argue that if you needed to use it every week, you would know all the identities merely by using them so much. Don't assume this is correct, though. One of the reasons I used them so much is because I had memorized them. My colleagues who used them as often as I did and who didn't memorize them did not, in fact, have them burnt into memory after so much usage.
I had trouble with that, always going back to the "SohCahToa" mnemonic. Nowadays, I always use the unit circle. Cos is abs[cissa] (horizontal), Sin is ordinate (vertical). Rotate the unit circle (or your drawing) to align as needed, scale with the radius: https://en.wikipedia.org/wiki/Unit_circle
You can derive that back from the Pythagorean theorem, but only if you have the sine and cosine definition in mind (SohCahToa).
> I still have no freakin clue which thing is a sin, cos or tan.
Atleast two methods I still recall:
1. Look up the SOH CAH TOA method.
2. Knowing that Sin(0) = 0 and Cos(0) = 1 can give a hint which one to use. You may know an equation should have a Sin or Cos but unsure which one so you pick the one that gives the expected results at 0 degrees. Works great for engineering type of problems.
