Regarding the trigonometric series,

Question

Why does cos start with a 1 and why are the powers of x even numbers

Also for sin, why does sin start with an x and why are the powers odd?

Answer 1

You must be referring to the expansion so sinx and cosx

sinx = x - x^3/3! + x^5/5! - x^7/7! + ... and
cosx = 1 - x^2/2! + x^4/4! - x^6/6! ...

I don't know if you have learned about the Taylor Series expansion.
Here is a page that develops the sinx expansion

http://www.intmath.com/Series-expansion/2_Maclaurin-series.php

Answer 2

The above reference should help you to answer the questions:

Why does cos start with a 1 and why are the powers of x even numbers
- think of what cos(0) equals, and
- is cos(x) an even or odd function?

Also for sin, why does sin start with an x and why are the powers odd?
- Evaluate lim sin(x)/x as x->0
- Is sin(x) an even or odd function?