Why is trigonometry important in calculus?

Question

I need to write short note why trigonometry is important is calculus and engineering mostly for presentation. I am not focusing on on what topic it specifically it appears (because I am guessing the list would be mile long) but by due to which of it's property. My guess is that it's solely due to it's connection with very simple geometric structures triangles and circles and complex numbers. I hope I am not being redundant and besides what other property can be attributed to trigonometric functions that it makes essential in calculus?

Answer 1

Trigonometry is important in calculus for a multitude of reasons. If you ever taken a course in Calculus with Analytic Geometry, the importance should be thoroughly emphasized. One key example of the usefulness of trigonometry in calculus is in geometric proofs. When transforming coordinates from Cartesian to polar, we find that $ x = r\cos{\theta} $, $y = r\sin{\theta} $, and $ \frac{y}{x} = \tan{\theta} $. If you recall that the limit of a small change in $y$ over a small change in $x$ as that small change in $x$ becomes zero is the derivative, which is TANgent to a curve.

Another example can be found in the methods of integration. Trigonometric substitution often provides vastly simpler integrands as compared to their more complicated partners. In these cases, the simplification and rationale can be derived from the trigonometric identities. If you have a term like $a^{2} - x^{2} $ you can set $ x = a\sin{\theta} $ and make use of the fact that the square of cosine plus the square of sine is unity.

Those are just a few examples I can think of off the top of my head.