Chapter 4, Problem 14CT

To determine

To find:To find the period of the mentioned trigonometric function.

Answer to Problem 14CT

The period of the trigonometric function $\sin 2\pi x+2\cos \pi x$ is $2$ .

Explanation of Solution

Given information:

The mentioned trigonometric function in the problem is $\sin 2\pi x+2\cos \pi x$ .

Formula used:

Period of $\sin x=2\pi ,\cos x=2\pi$ then the period of $\sin ax=\frac{2\pi }{a},\cos ax=\frac{2\pi }{a}$ .

Graph: Below mentioned is the graph of the mentioned trigonometric function $\sin 2\pi x+2\cos \pi x$ from the interval $\left[0,2\pi \right]$ .

  

The graph mentioned above is in real number line on x axis and y axis.

From the above mentioned graph we observe that the graph is repeating on every $1$ length of $2\pi$ cycle. So the function $\sin 2\pi x+2\cos \pi x$ is periodic

Calculation:

Consider the statement that mentioned trigonometric function in the problem is $\sin 2\pi x+2\cos \pi x$ .

From the above mentioned graph the period of $\sin 2\pi x+2\cos \pi x$ is two.

Now, Recall that Period of $\sin x=2\pi ,\cos x=2\pi$ then the period of $\sin ax=\frac{2\pi }{a},\cos ax=\frac{2\pi }{a}$ .

Therefore the period of $\sin 2\pi x=\frac{2\pi }{2\pi }=1$ and $2\cos \pi x=\frac{2\pi }{2\pi }=1$ .

Therefore the period of $\sin 2\pi x+2\cos \pi x$ is $1+1=2$