# Expressing Sine Signal as Cosine?

1. May 19, 2013

Apologies for my other threads. My browser had a weird hiccup and now somehow there are many that are empty. Please delete the others! Sorry!

1. The problem statement, all variables and given/known data

I need to express the real part of e^(-t)*sin(3t + ∏) in the form e^(αt)*A*cos(ωt + θ) where A, α, ω and θ are all real numbers.

2. Relevant equations

See above. Also, Euler's Formula: e^jθ = cosθ + jsinθ

3. The attempt at a solution

My efforts so far have given me the following:

Use Euler's Formula to convert: (cos(-t) + jsin(-t))*(sin(3t + ∏)

(cos(t) - jsin(t))*(sin(3t + ∏))

cos(t)sin(3t + ∏) + jsin(t)*sin(3t + ∏)

Since the later part of the sum is imaginary...

cos(t)sin(3t + ∏)

Now, I have two questions: 1) Have I erred in using Euler's formula with no complex number j in the exponent initially? 2) What more is there to be done? I thought about initially trying to re-express in a single step knowing that sinx = cos(x - pi/2), but graphing the two did not seem to yield identical results.

Any help would be greatly appreciated!

2. May 19, 2013

### Dick

Sure you have erred by using Euler's formula with no j. Since everything else in the problem is real, you should let t be complex. Write t=b+cj. Use sin(z)=(e^(jz)-e^(-jz))/(2j).