Difficulty understanding trigonometric functions

I have taken several calculus classes as part of highschool and general science degrees. Sin, cos, tan, and their inverse functions have always been this 'black box' to me that I never understood... instead I just memorized the operations and got through the classes that way.

My current understanding is that they are functions that 'map' an angle in a right-angle triangle to the ratio of two given sides (sin being opposite+hypoteneuse, and so forth).

So in this example that I have drawn, we have a triangle with dimensions [4, 7, 8.06]. If you want to find theta, you just take the inverse cosine of ADJACENT (4) and HYPOTENEUSE (8.06). Since cosine is relating the RATIO of these two sides to the FOCUS ANGLE, it maps 0.4962 (the ratio) to 60 degrees.

My question is... how does my calculator DO that math? Is sine/cos really a complex polynomial function that is 'black boxed' to sine, cos, tan, etc. for simplicity's sake? How does it turn 0.4962 into 60?

Thank you for any insight, this idea bothers me a lot!

• Also, your calculater finds the values using the Taylor Series Expansions of the functions, which is essentially that it just plugs values into a formula. See: en.m.wikipedia.org/wiki/Taylor_series Commented Feb 6, 2019 at 18:00