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Periodicity of Functions

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Exercises

Find the period of the function $f(x) = \sin(3x) + \cos(2x)$ .

Reference Answer(3 个标签)

periodic functionperiod of sum functionsleast common multiple

Solution Approach: We need to find the periods of $\sin(3x)$ and $\cos(2x)$ respectively, then find their least common multiple.

Detailed Steps:

Period of $\sin(3x)$ : $T_1 = \frac{2\pi}{3}$
Period of $\cos(2x)$ : $T_2 = \frac{2\pi}{2} = \pi$
Find the least common multiple: $\frac{2\pi}{3} = \frac{2\pi}{3}$ , $\pi = \frac{3\pi}{3}$ Least common multiple is $2\pi$

Answer: The period of this function is $2\pi$ .

Determine whether the function $f(x) = \sin^2 x + \cos^2 x$ is a periodic function. If so, find its period.

Reference Answer(3 个标签)

periodic functiontrigonometric identitiesconstant function

Solution Approach: Simplify the function expression using trigonometric identities.

Detailed Steps:

Using the identity: $\sin^2 x + \cos^2 x = 1$
Therefore $f(x) = 1$ , this is a constant function
A constant function is periodic, any non-zero real number is a period
The fundamental period does not exist (because any arbitrarily small positive number is a period)

Answer: This function is periodic but has no fundamental period.