Fourier Series Coefficients - how to separate e^(jwt) from the rest to find the coefficient?

[aopellopomelo]

I don't understand how to separate e^(jwt) to find the Fourier series coefficients; I've attached a screenshot of the procedure I don't understand, which is from the first red to the second red box.
Could someone explain to me why and how to do this? Thank you!

 

[ISTER_REG]

Hi, this is not hard... The idea is using Eulers formula, which is [MATH]e^{\pm i\phi} = i\cdot\sin(\phi) \pm \cos(\phi)[/MATH]. If you do that for [MATH]e^{i \frac{\pi}{6}} = i\sin(\frac{\pi}{6}) + \cos(\frac{\pi}{6}) = \frac{1}{2}i + \frac{\sqrt{3}}{2}[/MATH]

 

[ISTER_REG]

Sry, there was a typo in the formula, correct is: [MATH]e^{\pm i\phi} = \cos(\phi) \pm i\cdot\sin(\phi)[/MATH]
PS (for the Mods): I'm not able to edit my comment anymore, unfortunately, so please correct the previous comment.