Period of sum of three trigonometric functions

Question

am trying to compute the period of the following: $$\cos(\pi t) + 2\cos(3\pi t) + 3\cos(5\pi t)$$

I know that given two sinusoids, the period is found from the ratio of the two sinusoids. but here:

$$\text{(period of the first term) }T_1 = 2$$ $$\text{(period of the second term) }T_2= 2/3$$ $$\text{(period of the third term) }T_3= 2/5$$

but where should I go from here. Can somebody please show me a general formula whenever I encounter a question that asks for the period of the product or sum of multiple sinusoids. Thanks in advance.

Answer 1

For the first function , the periods are

$$\color {red}{2}, 4, 6, 8, 10...$$ for the second $$2/3, 4/3, \color {red}{2}, 8/3,... $$ and for the third

$$2/5, 4/5, 6/5, 8/5,\color {red}{ 2},... $$

the smallest common period is $$T=2$$

Answer 2

$$2,\frac{2}{3},\frac{2}{5}\to\\\dfrac{30}{15},\dfrac{10}{15},\dfrac{6}{15}\\$$now factor $\dfrac{2}{15}$so $$T_{total}=\dfrac{2}{15}\underbrace{[15,5,3]}_{lcm}=\\\dfrac{2}{15}\times3\times5=2$$