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!
Best Answer
Most calculators will use an approximation method - for example, a truncated Taylor series - which has error small enough that it will be well below what will appear on the display. I am not sure what specific implementation any calculator uses, though I assume it is not as simple as a Taylor polynomial (since those grow without bounds towards infinity).