The period of the signal

x(t) = 10 sin 12πt + 4 cos 18πt is

Concept:

To check whether a combination of signals is periodic or not, we have to take the ratio of individual time periods, if the ratio is rational then the combined signal is periodic else it is not periodic, and the period will be equal to the LCM of individual periods. 

Analysis:

x(t) = 10 sin 12πt + 4cos 18πt

x(t) = x₁(t) + x₂(t) 

x₁(t) = 10 sin 12πt

ω₁ = 12π 

T₁ = 2π/ω₁ = 1/6

x₂(t) = 4 cos 18πt 

ω₂ = 18π 

T₂ = 2π/ω₂ = 1/9

To check whether combined signal is periodic or not:

\(\frac{T_1}{T_2}=\frac{1/6}{1/9}= \frac{3}{2}\)

As the ratio is rational, therefore the combined signal is periodic. 

Time period = LCM {1/6, 1/9}

= \(\frac{LCM ~of ~(1,1)}{HCF ~of~(6,9)}\) = 1/3