Is f(x) = cos^2(x) + sin^2(x) a periodic function?

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Homework Help Overview

The discussion revolves around the periodicity of the function f(x) = cos^2(x) + sin^2(x), with participants exploring whether this function can be classified as periodic based on its mathematical properties.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions the periodicity of the function, noting that both cos^2(x) and sin^2(x) are periodic, and wonders if their sum retains that property. Other participants assert that the sum equals 1, which is constant, and thus not periodic. There is also a discussion about the definition of periodic functions and the implications of a constant function being considered periodic.

Discussion Status

The discussion appears to have reached a point where participants agree that the function is not periodic, as it simplifies to a constant value. However, there is some exploration of the nuances of defining periodicity in the context of constant functions.

Contextual Notes

Participants reference the fundamental definition of periodic functions and the implications of a function being constant, indicating a need for clarity on these concepts.

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Homework Statement


Is f(x) = cos^2(x) + sin^2(x) a periodic function?


Homework Equations


sin^2(x) + cos^2(x) = 1


The Attempt at a Solution


This question is just something that randomly came to my mind (not a homework problem). I know cos^2(x) and sin^2(x) are both periodic functions, but is sin^2(x) + cos^2(x) a periodic function too? If so, what would be its fundamental frequency?
 
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No, sin^2(x) + cos^2(x) = 1 which we know is not periodic, but constant.
 
Zondrina said:
No, sin^2(x) + cos^2(x) = 1 which we know is not periodic, but constant.

Indeed, as Zondrina mentioned, it is not periodic. Typically, we say a function f(x) is periodic if there is a smallest positive integer P≠0 for which f(x+P)=f(x). The number P is then the period. Since sin^2(x)+cos^2(x)=1:=1(x), it is not periodic, because 1(x+P)=1=1(x) for any P (so in particular, there isn't a smallest one).
 
Last edited:
Technically a constant function is periodic but has no fundamental period, but it is kind of silly to refer to it in such a way.
 

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