# Complex exponentials (simplify the expression)

1. Simplify ei6x(1+e-i10x)/(1+ei2x)

2. i have no idea how to simplify this its supposed to be in terms of cosines

3. i dont how i can simplify this such that i can use the 1/2(e^x +e^-x) = cosx formula

fzero
Homework Helper
Gold Member
You need the Euler formula $$e^{ix} = \cos x + i \sin x$$.

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

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, ive 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 cant get the e^2 out of there so i really have no idea how, can someone show me the steps of simplifying this

fzero
Homework Helper
Gold Member
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 in to be graded.)

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?!?!

Exponent rules :D...

fzero
Homework Helper
Gold Member
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

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

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