I'm going to write out the whole problem as it is given to me (bad grammar and all) even though some of the info may be irrelevant to finding a solution.
A charge $e$ moving along a straight line undergoes simple harmonic motion with frequency $\omega$ and amplitude $a$. Since the charge is moving with an acceleration, it emits electromagnetic radiation. The angular distribution of the instantaneous (time-dependent) intensity of the radiation is $$I(\theta,t)=\kappa \frac{\sin^2\theta \cos^2\omega t}{(1+\beta\cos\theta\sin\omega t)^5}$$ where $\theta$ is the angle between the direction of the oscillations and the direction of the emission, $\kappa$ is an irrelevant for us constant, $$\beta=\frac{\omega a}{c},0\leq\beta<1$$ is the ratio of the maximal speed of the charge to the speed of light. Find the angular distribution of the averaged intensity of the emission, $$I(\theta)=\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}I(\theta,t)dt$$ Use contour integration and use a computer algebra system to calculate the relevant residue(s).
I'm not even sure how to start this problem. If it only had $\cos$ or $\sin$ terms i would Use Euler's to rewrite in terms of the Real or Imaginary parts of $e^{i\theta}$ but with with these two different trig functions i'm not sure how i would do that. Another problem I'm having is understanding what the problem is even asking. It wants me to find $I(\theta)$ but it wants me to integrate over $t$. Does that mean my solution will be in terms of $\theta$? In which case i can factor out the $\sin^2\theta$ from the numerator? About the only thing i'm remotely sure about is that after i somehow convert this to a function $f(z)$ I'll have to find the singularity(s) enclosed within the integration contour and use Residues to determine a solution. But i'm not even sure how to begin converting this to a usable function $f(z)$.