• Euler's Formula
• Euler's formaula with powers
• Some solutions

Euler’s formula

is reached by combining the results of the taylor series for e to a complex number, sin(x) and cos(y).

The Taylor series at x=a is defined below and is a polynomial estimation of the given function. When a=0 the series is called a Maclaurin series. A graphical illustration of the taylor series of sin can be found here: http://math.furman.edu/~dcs/java/taylor.html (requires java) :

This means that e to a complex number equals the following for the Maclaurin series. See how the components group into a real and imaginary component.

The Maclaurin series of cos at results in:

Use the following fact and that cos(0)=1 and sin(0)=0

The Maclaurin series results in:

Substituting the Maclaurin series of cos() and sin() into the series for e gives the result we were looking for:

Dealing with powers and Euler’s formula given above, the following is true.

 

Solution

Evaluating the folowing is done with referencing a trigonometry page like http://www.ping.be/~ping1339/gonio.htm#Sum-formulas