Fundamental period of $5 \sin(5 \pi t) + 5 \sin(3 \pi t)$

Question

f(t) = $5 \sin(5 \pi t) + 5 \sin(3 \pi t)$

$ w_1 = 5 \pi$

$w_2 = 3 \pi $

Formula to find combined period is:

$ T = \frac{2 \pi}{gcd} = \frac{2 \pi}{\pi} = 2 $

gcd = greatest common divisor of angular frequencies $5 \pi$ and $3 \pi $, which is $ \pi$

Wolfram agrees that period is T $= 2$

Is this formula legit?

Answer 1

We can say, $$5(\sin (5πt)+\sin (3πt))$$$$$$ $$\frac{5}{2}(\sin (\frac{8πt}{2})\cos (\frac{2πt}{2}))$$$$$$ $$\frac{5}{2}(\sin (4πt)\cos (πt))$$$$$$ $$5(\sin(πt)\cos(2πt) \cos^2(πt))$$$$$$ Here $\sin(πt)$ has period $2$ $$$$ $\cos(2πt)$ has period $1$ $$$$ $\cos^2(πt)$ has period $1$ $$$$ Therefore total period is $2$