Question

Determine which of the following signals are periodic. If a signal is periodic, determine its fundamental period.

1) cos(5t)

Answer 1

The signal cos(5t) is periodic with a fundamental period of 2π/5.

Step-by-step explanation:

A periodic signal repeats itself after a certain interval, known as the fundamental period. In this case, we examine the signal cos(5t). To determine if it is periodic, we need to check if there exists a positive constant T such that cos(5(t+T)) = cos(5t) for all t. The signal is periodic if such a T exists.

Let's analyze cos(5t):

cos(5(t+T)) = cos(5t + 5T)

To find T, we set this equal to cos(5t):

cos(5t + 5T) = cos(5t)

Applying the cosine angle sum identity, we get:

cos(5t)cos(5T) - sin(5t)sin(5T) = cos(5t)

For this equation to hold for all t, cos(5T) must be 1, and sin(5T) must be 0. This happens when 5T is a multiple of 2π. Therefore, T = 2π/5 is the fundamental period.

In summary, the signal cos(5t) is periodic with a fundamental period of 2π/5, as it repeats every
`\(2\pi/5\)` seconds.