The functions e x
,sinx, and cosx are related by the formula e ix
=cosx+isinx. This is Euler's formula, named for the Swiss mathematician Leonhard Euler. To derive this formula, we use the Taylor series for each function. Use the fact that the pattern i 1
=i,i 2
=1,i 3
=−i, and i 4
=1 repeats for higher powers of i to derive Euler's formula through the following steps. a) Find and simplify the Taylor series of e ix
b) Find and simplify the Taylor series of sin(ix). c) Use the results from parts (a) and (b) to show that e ix
=cosx+isinx.