Complex exponentials (simplify the expression)

[Luongo]

1. Simplify e^i6x(1+e^-i10x)/(1+e^i2x)
2. i have no idea how to simplify this its supposed to be in terms of cosines
3. i don't how i can simplify this such that i can use the 1/2(e^x +e^-x) = cosx formula

 

[fzero]

You need the Euler formula e^{ix} = \cos x + i \sin x.

 

[Luongo]

You need the Euler formula e^{ix} = \cos x + i \sin x.

i tried that, i got garbage it MUST be in terms of cosines

 

[fzero]

You might want to show some work to see where your problem is. (Also you might find e^{6ix} = e^{5ix} e^{ix} useful.)

 

[Luongo]

You might want to show some work to see where your problem is. (Also you might find e^{6ix} = e^{5ix} e^{ix} useful.)

well, I've multiplied it out i got e^6x+e^-4x and then i did e^2x * e^4x + e^-4x
in the numerator to try to get it in cosine form but i can't get the e^2 out of there so i really have no idea how, can someone show me the steps of simplifying this

 

[fzero]

You also want to simplify the denominator, so you don't want to multiply the numerator through by the whole e^{6ix} factor. (Also you're leaving out the factors of i in your exponentials, which is a bit confusing, but you shouldn't do it in anything you turn into be graded.)

 

[Luongo]

You might want to show some work to see where your problem is. (Also you might find e^{6ix} = e^{5ix} e^{ix} useful.)

i got it! you're a genius, how on Earth did you see that??

 

[Kevin_Axion]

Exponent rules :D...

 

[fzero]

i got it! you're a genius, how on Earth did you see that??

You know that

\frac{ e^{ia} + e^{-ia}}{2} = \cos a

so if you see

1 + e^{ib}

you want to rewrite that as

e^{ib/2} ( e^{-ib/2} + e^{ib/2} ).

 

[Luongo]

You know that

\frac{ e^{ia} + e^{-ia}}{2} = \cos a

so if you see

1 + e^{ib}

you want to rewrite that as

e^{ib/2} ( e^{-ib/2} + e^{ib/2} ).

thanks a lot i appreciate it i was hung up on this question for a while

 

[Luongo]

thanks a lot i appreciate it i was hung up on this question for a while

for simplifying in terms of sines... can i use the same formula except the negative sign is between the 2 expos?

 

[fzero]

for simplifying in terms of sines... can i use the same formula except the negative sign is between the 2 expos?

Yes.