Question

What is the fundamental period of $2\cos (3x)$ ?

Answer 1

We say that a function f(x) is a periodic function in trigonometry if there exists a T>0 in such a way that $f(x+T)=f(x)$ for all values of x. If T is the smallest positive real number such that $f(x+T)=f(x)$for all x, then T is called the fundamental period of f(x).

Complete step-by-step answer:
As we know, the period of $\cos (p)=2\pi$ where p is representing the angle of cosine function.
And we can see from this function that $\cos (0)$ to $\cos (2\pi )$ represents the one full period. In the given expression $2\cos (3x)$ the coefficient 2 only modifies the amplitude. The angle 3x in place of x stretches the value of x by a factor of 3.
That is $\cos (0)$ to $\cos \left( 3.\left( \dfrac{2\pi }{3} \right) \right)$ represents the one full period.
So, the fundamental period of $\cos (3x)$ is $\dfrac{2\pi }{3}$ .
We know that the period of cos(x) is $2\pi$ , hence the period of cos(3x) would be $\dfrac{2\pi }{3}$ which means it would repeat itself three times between 0 and $2\pi$ .

Note: The distance between the repetition of any function is called the period of that function. For a trigonometric function we can say that the length of one complete cycle is called the period and x=0 is taken as the starting point for any trigonometric function. Mainly we consider sine, cosine, tangent functions and rest all can be obtained from them.