How to Solve Equations with Unknown Variables Using Trigonometric Identities

[cdummie]

Homework Statement

I have to solve the following equation:

Homework Equations

The Attempt at a Solution

I know that since the right side is 1 and on the left side i have i (imaginary number) it means that i could rewrite right side as cos0 + isin0 since it's the same, but what can i do with left side, it obviously can't stay like this because of the product, i have to get rid of it, i just don't have an idea.

 

[Samy_A]

What do you get when you replace the factors in the product using Euler's formula?

 

[cdummie]

What do you get when you replace the factors in the product using Euler's formula?

As far as i know, we have Пe^kx (П is from 1 to n) which is equal to:

e^x*e^2x*...*e^nx or

e^(1+2+...+n)x. Now, i am not 100% sure, but i believe is should represent 1+2+...+n as n(n+1)/2 and go back to the trigonometric form of number, and then simply find x since i know exact value of angle for the right side. Am i correct?

 

[Samy_A]

As far as i know, we have Пe^kx (П is from 1 to n) which is equal to:

e^x*e^2x*...*e^nx or

e^(1+2+...+n)x. Now, i am not 100% sure, but i believe is should represent 1+2+...+n as n(n+1)/2 and go back to the trigonometric form of number, and then simply find x since i know exact value of angle for the right side. Am i correct?

You forgot the $i$ in your exponent, as Euler's formula is $cos(y)+isin(y)=e^{iy}$.
You are correct about $\sum_{k=1}^n k =n(n+1)/2$.
As the product is equal to $1$, you can now indeed solve for x by applying Euler's formula again: the cosine of your exponent (not including the $i$) must be $1$.

 

[RUber]

How many solutions do you need? Just one, all of them for a fixed n, or all of them for any n?
$e^{iy} = 1$ is true for an infinite number of periodic terms.
For a large w, $e^{iwy}$ will have a much shorter period. Your w might get quite large as n grows.