Evaluate each of the following functions and see if it is...

Evaluate each of the following functions and see if it is periodic. If periodic, find its period. (a) $f(t)=\cos \pi t+2 \cos 3 \pi t+3 \cos 5 \pi t$ (b) $y(t)=\sin t+4 \cos 2 \pi t$ (c) $g(t)=\sin 3 t \cos 4 t$ (d) $h(t)=\cos ^{2} t$ (e) $z(t)=4.2 \sin \left(0.4 \pi t+10^{\circ}\right)$ $+0.8 \sin \left(0.6 \pi t+50^{\circ}\right)$ (f) $p(t)=10$ $(\mathrm{g}) q(t)=e^{-\pi t}$

Evaluate each of the following functions and see if it is periodic. If periodic, find its period.
(a) $f(t)=\cos \pi t+2 \cos 3 \pi t+3 \cos 5 \pi t$
(b) $y(t)=\sin t+4 \cos 2 \pi t$
(c) $g(t)=\sin 3 t \cos 4 t$
(d) $h(t)=\cos ^{2} t$
(e) $z(t)=4.2 \sin \left(0.4 \pi t+10^{\circ}\right)$
$+0.8 \sin \left(0.6 \pi t+50^{\circ}\right)$
(f) $p(t)=10$
$(\mathrm{g}) q(t)=e^{-\pi t}$

Key Concepts

Periodicity

Periodicity refers to the property of a function to repeat its values at regular intervals over its domain. If there exists a positive number T such that f(t + T) = f(t) for all t, then T is called a period of the function. This concept is fundamental in understanding the behavior of many functions, especially in trigonometry and signal processing.

Fundamental Period of Sine and Cosine

Sine and cosine functions are inherently periodic with a basic period of 2?. When these functions are composed with a multiplier inside the argument—such as sin(?t) or cos(?t)—the period is altered to 2?/|?|. Recognizing this adjustment is key when analyzing the periodicity of scaled trigonometric functions.

Frequency Scaling in Trigonometric Functions

Scaling the argument of sine or cosine by a frequency factor changes the rate at which the function completes its cycles. Specifically, the period of a function like cos(k t) or sin(k t) is determined by dividing the standard period by the absolute value of the coefficient, resulting in a period of 2?/|k|. This principle is essential when assessing the contribution of each term in a sum of trigonometric functions.

Phase Shifts in Trigonometric Functions

Phase shifts occur when a constant is added to the argument of a trigonometric function, such as in cos(?t + ?) or sin(?t + ?). While phase shifts translate the function along the time axis, they do not alter the function’s period. Understanding this helps in identifying that the period of a phase-shifted function remains the same as that of its standard counterpart.

Sum of Periodic Functions

When multiple periodic functions are added together, the resulting function is periodic only if the individual periods are commensurate, meaning their ratios are rational numbers. The fundamental period of the sum can then be determined as the least common multiple (LCM) of the individual periods. Recognizing how periods interact is critical when evaluating the periodicity of combined signals.

Product-to-Sum Identities

Product-to-sum identities are trigonometric formulas that express the product of sine and cosine functions as a sum of sine or cosine terms. These identities are useful in simplifying products into sums, which can then be analyzed more readily for periodicity by determining the periods of the individual summed terms.

Constant Functions

A constant function, which outputs the same value regardless of the input, is technically periodic with any period. Since there is no change over time, the definition of periodicity is trivially satisfied, making constant functions a special case in the study of periodic behavior.

Exponential Functions

Exponential functions of the form e^(at) are generally non-periodic when the exponent a is a nonzero real number. Unlike trigonometric functions, exponential functions do not repeat their values at regular intervals, which is crucial when determining the periodicity of functions that include exponential components.

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