How do you find the amplitude and period of a function y = sin(2x) + cos(4x)?

Jun 4, 2017

The period of the function is $= \pi$
The amplitude is $= 1.56$

Explanation:

The period of the sum of $2$ periodic functions is the LCM of their periods.

The period of $\sin \left(2 x\right)$ is ${T}_{1} = \frac{2}{2} \pi = \pi$

The period of $\cos \left(4 x\right)$ is ${T}_{2} = \frac{2}{4} \pi = \frac{1}{2} \pi$

The LCM of ${T}_{1}$ and ${T}_{2}$ is $T = \pi$

To calculate the amplitude, we need the maximum and the minimum of the funtion $y$

$y = \sin 2 x + \cos 4 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 \cos 2 x - 4 \sin 4 x$

$= 2 \cos 2 x - 4 \cdot 2 \sin 2 x \cos 2 x$

$= 2 \cos 2 x \left(1 - 4 \sin 2 x\right)$

The max. and min. when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

That is,

$2 \cos 2 x \left(1 - 4 \sin 2 x\right) = 0$

$\implies$

$\cos 2 x = 0$, $\implies$, $2 x = \frac{\pi}{2}$ or $2 x = \frac{3}{2} \pi$

$\implies$, $x = \frac{\pi}{4}$ or $x = \frac{3}{4} \pi$

and

$1 - 4 \sin 2 x = 0$, $\implies$, $\sin 2 x = \frac{1}{4}$

$\implies$, $2 x = 0.253$, $\implies$, $x = 0.126$

So,

$y \left(\frac{\pi}{4}\right) = \sin \left(2 \cdot \frac{\pi}{4}\right) + \cos \left(4 \cdot \frac{\pi}{4}\right) = 1 - 1 = 0$

$y \left(\frac{3}{4} \pi\right) = \sin \left(2 \cdot \frac{3}{4} \pi\right) + \cos \left(4 \cdot \frac{3}{4} \pi\right) = - 1 - 1 = - 2$

$y \left(0.126\right) = \sin \left(2 \cdot 0.126\right) + \cos \left(4 \cdot 0.126\right) = 1.125$

Therefore,

The amplitude is $= \frac{M a x - \min}{2} = \frac{1.125 - \left(- 2\right)}{2} = 1.56$
graph{(y-sin(2x)-cos(4x))=0 [-3.523, 5.245, -2.154, 2.23]}