complex numbers - average of a function

Question

Express $\sin^4(x)$ in multiple angle form. Hence, deduce the average of this function over one cycle.

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I've expressed it in multiple angle form:

$sin^4(x) = \dfrac{1}{8} cos(4x) - \dfrac{1}{2} cos(2x) + \dfrac{3}{8}$

BUT how do i deduce the average of the function over one cycle?

Answer 1

Hint: $\cos x$ has period $2 \pi$ and average $0$ over one cycle $[0,2 \pi]$:

$$\int_0^{2 \pi} \cos x \,dx = -\sin x \,\Big|_0^{2\pi} = 0$$

Then $\cos n x$, whose period is $2\pi/n$, has average $0$ over any cycle $[a,a+2 k \pi/n], k \in\mathbb{Z}$. So:

$$\int_0^{2 \pi} sin^4(x) \,dx = \frac{1}{8} \int_0^{2 \pi} cos(4x) \,dx - \frac{1}{2} \int_0^{2 \pi} cos(2x) \,dx + \frac{3}{8} \int_0^{2 \pi} \,dx = 0+0+ \cdots $$