Note that the cosine function cos(x) is a trigonometric function. Here, we will learn about the domain, range, and period of cos(x).

## Domain of cos x

One knows that for any value of x, the cosine function cos(x) is defined. Thus, by the definition of the domain of a function, we can say that the domain of cos(x) is the set of all real numbers. In other words,

The domain of cos(x) = $(-\infty, \infty)$

For any real number $a$, the function cos(ax) is defined for all real numbers. So the domain of cos(ax) is the set of all real numbers. For example,

- Domain of cos(2x) = $(-\infty, \infty)$
- Domain of cos(3x) = $(-\infty, \infty)$
- Domain of cos(4x) = $(-\infty, \infty)$

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## Range of cos x

From the graph y=cos(x), one can see that the value of cos(x) oscillates between -1 and 1. That is, we have $-1 \leq \cos x \leq 1$ for all real numbers x. Thus, the range of cos(x) is the closed interval [-1, 1].

Range of cos(x) = [-1, 1]

The range of cos(ax) is the closed interval [-1, 1]. For example,

- Range of cos(2x) = [-1, 1]
- Range of cos(3x) = [-1, 1]
- Range of cos(4x) = [-1, 1]

## Period of cos x

For any natural number n, it is known that

$\cos(2n\pi+x)=\cos x$.

From this, we see that $2\pi$ is the smallest number (corresponding to n=1) such that

$\cos(2\pi+x)=\cos x$.

Thus, $2\pi$ is the period of cos(x).

**Question 1:**Find the period of cos(ax).

*Solution:*

The period of cos(ax) is $\dfrac{2\pi}{|a|}$. For example, the period of cos(3x) is equal to $\dfrac{2\pi}{3}$.

**Question 2:**Find the period of cos(8x)-cos(2x).

*Solution:*

The period of cos(8x) is $\dfrac{2\pi}{8}$ $=\frac{\pi}{4}$ and the period of cos(2x) is $\dfrac{2\pi}{2}=\pi$.

As the period of the difference of two periodic functions is the least common multiple (LCM) of their periods, we conclude that the period of cos(8x)-cos(2x) is

= LCM $\{\dfrac{\pi}{4}, \pi \}$

= $\pi$.

Therefore, the period of cos(8x)-cos(2x) is $\pi$.

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