Question

For each one of the following functions of t, indicate whether they are periodic or not. If periodic, indicate its period, and if not periodic, indicate why. Assume n is a positive integer.

(a) sin(2√2πt) + cos(√2πt)

Answer 1

The function sin(2√2πt) + cos(√2πt) is not periodic because the periods of the sine and cosine components, approximately 0.3536 seconds and 1.4142 seconds respectively, are not multiples of each other.

Step-by-step explanation:

To determine if the function sin(2√2πt) + cos(√2πt) is periodic and, if so, to find its period, we need to consider the individual periods of the sine and cosine components.

For sin(2√2πt), the period is calculated using the general formula for the period of a sine function, T = π/k, where k is the coefficient of t inside the trigonometric function. Here, k = 2√2π, hence the period of the sine component is T_s = 1/(2√2) or approximately 0.3536 seconds. similarly, for cos(√2πt), we again use the formula of the period of a cosine function, which gives us a period of T_c = 2π/√2π = √2 or approximately 1.4142 seconds. since the periods of the sine and cosine components are not multiples of each other, the function sin(2√2πt) + cos(√2πt) does not have a period that aligns to result in a periodic function. Thus, the function is not periodic because it does not repeat at regular intervals.