Expressing the addition of two sinusoidal waves this form.

[Armin]

Homework Statement

Express the following in the form x = Re{Ae^iαe^iωt}

(a) x= cos(ωt) + sin(wt)
(b) x= sin(ωt +π/4) + cos(ωt)
(c) x= 2cos(ωt+π/3) + (√3)sin(ωt)-cos(ωt)

Homework Equations

cos x = 1/2 e^ix + 1/2 e^-ix
sin x = − i/ 2e^ix + i/2 e^−ix

The Attempt at a Solution

To be honest, I have no clue where to start. I do not know what the Re (Short for Real) asks for. There is no imaginary number in the functions but there are imaginary numbers in the form it needs to be expressed. Any help to point me in the right direction will be appreciated.

-A

 

[Buzz Bloom]

Hi Armin:

First, you need to correct some typos. You have "w" is some places where you meant "ω".

Second, you need to know that sin(ωt) also has a form involving 1/2 e^ix and 1/2 e^-ix. This should be another relevant equation under (2).

Third, you need to do a bit of manipulation to get e^ix and e^-ix as expressions in terms of the sin and cos.

Then you should be able to see the use of Re{...}. This function with a complex argument gives the real part of the argument.

Hope this helps.

Regards,
Buzz

 

[Armin]

Hi Armin:

First, you need to correct some typos. You have "w" is some places where you meant "ω".

Second, you need to know that sin(ωt) also has a form involving 1/2 e^ix and 1/2 e^-ix. This should be another relevant equation under (2).

Third, you need to do a bit of manipulation to get e^ix and e^-ix as expressions in terms of the sin and cos.

Then you should be able to see the use of Re{...}. This function with a complex argument gives the real part of the argument.

Hope this helps.

Regards,
Buzz

I did what you told me to and I got (1/2+i/2)e^-iωt+(1/2-i/2)e^iωt

and I do know that e^iωt=cosωt+isinωt

But I do not know what the Re[..] does.

 

[haruspex]

I do not know what the Re[..] does.

The Re() function extracts the real part of its complex argument. Re(x+iy)=x.

 

[haruspex]

I would start from the other end of the problem. Expand e^iαe^iωt using cos and sin, then apply Re() to it.