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

In summary, the function f(x) = cos^2(x) + sin^2(x) is not a periodic function because it is equal to a constant, 1, and therefore does not have a fundamental period.
  • #1
jkface
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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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  • #2
No, sin^2(x) + cos^2(x) = 1 which we know is not periodic, but constant.
 
  • #3
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).
 
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  • #4
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.
 

1. What is a periodic function?

A periodic function is a mathematical function that repeats itself at regular intervals. This means that the output values of the function repeat in a predictable pattern as the input values increase.

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

Yes, f(x) = cos^2(x) + sin^2(x) is a periodic function. This is because both cos^2(x) and sin^2(x) are periodic functions with a period of 2π. Therefore, their sum will also have a period of 2π.

3. What is the period of f(x) = cos^2(x) + sin^2(x)?

The period of f(x) = cos^2(x) + sin^2(x) is 2π. This can be seen by observing that both cos^2(x) and sin^2(x) have a period of 2π, and when added together, their sum also has a period of 2π.

4. Can a periodic function have multiple periods?

No, a periodic function can only have one period. This is because the period is the smallest interval at which the function repeats itself, and there cannot be multiple smallest intervals.

5. Is every function with a period of 2π a periodic function?

No, not every function with a period of 2π is a periodic function. A function must also have a repeating pattern at regular intervals in order to be considered periodic. For example, the function f(x) = x^2 has a period of 2π, but it does not repeat itself in a predictable pattern and therefore is not considered a periodic function.

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