What is the Fourier Transform of cos(theta) and sin(theta + pi/2)?

[frenzal_dude]

Homework Statement

Hi, I tried to work out the FT of cos(theta), and sin(theta + pi/2) which should both give the exact same FT since they are the same function.

However I get two different results as shown in the .jpg.

Homework Equations

I used the 'time shifting' property to get that exponential in the second part.

Thanks in advance

The Attempt at a Solution

 

[vela]

Multiply the exponential factor through and use the fact that $\delta(f-f_0) h(f) = \delta(f-f_0) h(f_0)$.

 

[frenzal_dude]

Multiply the exponential factor through and use the fact that $\delta(f-f_0) h(f) = \delta(f-f_0) h(f_0)$.

Thanks for your help! I never realized that.

I managed to work it out to almost the same as the cos function FT, however I am still left with the minus sign before the 2nd delta function, how do you get rid of that?

 

[vela]

Check your signs. One factor will be just j and the other one will be -j, which conveniently flips the sign of that term.

 

[frenzal_dude]

Check your signs. One factor will be just j and the other one will be -j, which conveniently flips the sign of that term.

Yes you are right. I forgot to sub in f=-f0 for the 2nd exponential.

It works out perfectly.

Thanks a lot for your help!