Expectation of product of cosine and sine

Question

$\theta\sim U(-\pi,\pi)$.
When $\theta$ follows uniform distribution, what is the expected value of the producot of cosine and sine, i.e. $$E[\sin\theta \cos\theta] = \ ?$$

Answer 1

Since the probability density function of $\mathcal{U}\big([-\pi,\pi]\big)$ is $\displaystyle x \, \mapsto \, \frac{1}{2\pi} \mathbf{1}_{[-\pi,\pi]}(x)$, by definition, if $\theta \sim \mathcal{U}\big( [-\pi,\pi] \big)$, we have :

$$ \begin{align*} \mathrm{E}\big[ \cos(\theta)\sin(\theta) \big] &= {} \int \cos(x)\sin(x) \frac{1}{2\pi} \mathbf{1}_{[-\pi,\pi]}(x) \, dx \\[2mm] &= \frac{1}{2\pi} \int_{-\pi}^{\pi} \cos(x)\sin(x) \, dx \\[2mm] &= \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{2} \sin(2x) \, dx \\[2mm] &= \frac{1}{4\pi} \Big[ -\frac{1}{2} \cos(2x) \Big]_{-\pi}^{\pi} \\[2mm] &= 0. \end{align*} $$

Answer 2

Hint $$E[\cos\theta \sin\theta] = \frac1{2\pi} \int_{-\pi}^\pi \sin\theta \cos\theta \mathrm d\theta$$

Note that $\sin\theta\cos\theta = \frac12 \sin(2\theta)$ is an odd function. What do you know about symmetric integrals of odd functions?

Answer 3

$$\begin{align} \mathsf E[\cos\theta\sin\theta] &= \frac 1{2\pi}\int_{-\pi}^\pi \cos \theta\sin \theta\operatorname d \theta \\ &= \frac 1{4\pi}\int_{-\pi}^\pi \sin (2\theta)\operatorname d \theta \end{align}$$

Answer 4

Hint:

$\sin(-\theta)\cos(-\theta)=-\sin\theta\cos\theta$ and $\theta$ and $-\theta$ have the same distribution.

So: $$\mathbb E\sin\theta\cos\theta=\mathbb E\sin(-\theta)\cos(-\theta)=-\mathbb E\sin\theta\cos\theta$$