If $f(x)= \sin^4(6x) + \cos^4(6x)$ is a sine wave, find amplitude, phase, and period.

Question

If $f(x)= \sin^4(6x) + \cos^4(6x)$ is a sine wave, find amplitude, phase, and period.

I don't know how to work with $\sin^4$ and $\cos^4$, i tried to use the formula but it doesn't work because of the $4$th degree. Any hints?

Answer 1

Hint: Since $\sin{\theta}\cos{\theta}=\frac{\sin 2\theta}{2}$ and $\sin^2\theta=\frac{1-\cos 2\theta}{2}$ $$\sin^4a+\cos^4a=(\sin^2a+\cos^2a)^2-2\sin^2a\cos^2a=1-\frac{\sin^22a}{2}$$ $$=1-\frac{1-\cos4a}{4}=\frac34+\frac{\cos 4a}{4}$$ Now put $a=6x$

Answer 2

Hint: use the Pythagorean identity and then the double angle formulae (see the same link, if necessary).

Answer 3

$16f=(2i\sin y)^4+(2\cos y)^4$

Using How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?,

$$16f=(e^{iy}-e^{-iy})^4+(e^{iy}+e^{-iy})^4$$

$$=2(2\cos4y)+2\binom42$$ where $y=6x$