Resonance - from Feynman lectures

[rpthomps]

Good afternoon,

I am working my way through the Feynman lectures and I am stumped at Chapter 23, Resonance. Specifically, the derivation of equation 23.12. I have followed up to that point but the appearance of tan (theta) baffles me. The equation is below:

Any help would be greatly appreciated.

Ryan

 

[DrClaude]

You have to use Euler's formula along with a separation of the real and imaginary parts:
$$
\begin{align}
e^{-i \theta} &= \omega_0^2 - \omega^2 + i \gamma \omega \\
\cos \theta - i \sin \theta &= \omega_0^2 - \omega^2 + i \gamma \omega
\end{align}
$$
from which you get that $\cos \theta = \omega_0^2 - \omega^2$ and $\sin \theta = -\gamma \omega$

 

[rpthomps]

Thanks DrClaude. That's very helpful.