Sinusoid to exponential w/ complex frequencies

[Number2Pencil]

Homework Statement

Express the given sinusoid in terms of exponentials and determine the complex frequency(s).

4(e^-2)cos(wt + pi/4)

Homework Equations

Euler's identity:

A(e^i*theta) = A[cos(theta) + i*sin(theta)]

The Attempt at a Solution

I'm really drawing a blank here. I tried writing in a "+ i*sin(wt + pi/4)" after the cosine but that really didn't help my any. I'm not even sure what type of math this is considered... please any starting point would be appreciated

 

[Mathdope]

Isn't there an expression for cos (z) in terms of complex exponentials? Check your text and try to use it. That seems to be what they are asking for.

 

[Number2Pencil]

I found this one online:

cos(theta) = (1/2)[e^(+i*theta) + e^(-i*theta)]

applying this to mine I get:

(4e^-2){(1/2)[e^(+i*(wt + 45)) + e^(-i*(wt + 45))]

rewrite as:

((4e^-2)/2)[(e^iwt)(e^i*45) + (e^-iwt)(e^-i*45)]

I guess that's my answer...but what about complex frequencies??

 

[Mathdope]

First off, change those 45s back to pi/4s. Too much of this stuff is invalid in degrees - always using radians when working in trig/exponential functions is a good habit to get into, unless you're specifically told not to use them (which will likely never happen).

If I'm remembering my Complex Analysis course correctly, the complex exponential function is periodic with period 2(Pi)i (that last i is i^2=-1, that i). Frequency generally refers to the number of cycles per standard period, but if you could give us the definition that you have it would be helpful.

It's likely that you could start from the original form and use the fact that cos(wt + pi/4)
will undergo one cycle as (wt + pi/4) ranges from 0 to 2Pi, but someone may certainly come in and correct me.