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Sagot :
The period of the given function is T = 2π
What is the period of the function?
The period T of a function f(x) is such that:
f(x + T) = f(x).
In this case, our function is:
f(θ) = e^{iθ}
Remember that this can be written as:
f(θ) = cos(θ) + i*sin(θ)
So yes, this is in did a periodic function.
Then the period of the function f(θ) is the same as the period of the cosine and sine functions, which we know is T = 2π.
If you want to learn more about periodic functions, you can read:
https://brainly.com/question/26449711
The period of the considered function f(θ) = e^{iθ} is found to be P = 2π (assuming 'i' refers to 'iota' and 'e' refers to the base of the natural logarithm)
What is euler's formula?
For any real value θ, we have:
[tex]e^{i\theta} = \cos(\theta) + i\sin(\theta)[/tex]
where 'e' is the base of the natural logarithm, and 'i' is iota, the imaginary unit.
What are periodic functions?
Functions which repeats their values after a fixed interval, are called periodic function.
For a function [tex]y = f(x)[/tex], it is called periodic with period 'T' if we have:
[tex]y = f(x) = f(x + T) \: \forall x \in D(f)[/tex]
where D(F) is the domain of the function f.
Suppose that, the period of the function [tex]f(\theta) = e^{i \theta}[/tex] be P, then we get:
[tex]f(\theta + P) = f(\theta)\\\\e^{i(\theta)} = e^{i(\theta + P)}\\\\\cos(\theta) + i\sin(\theta) = \cos(\theta + P) + i\sin(\theta + P)[/tex]
When two complex numbers are equal, then their real parts are equal and their imaginary parts are equal.
That means,
[tex]\cos(\theta) + i\sin(\theta) = \cos(\theta + P) + i\sin(\theta + P)[/tex] implies that:
[tex]\cos(\theta) = \cos(\theta + P)\\\sin(\theta) = \sin(\theta + P)[/tex]
Also, we know that the period of sine and cosine function is [tex]2\pi[/tex]
Thus, we get:
[tex]P = 2\pi[/tex]
Thus, the period of the function [tex]f(\theta) = e^{i \theta}[/tex] is P = 2π
Learn more about periodic functions here:
brainly.com/question/12529476
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