The sine function sin x is a trigonometric function. In this post, we will learn about the domain, range, and period of sin x.

## Domain of sin x

We know that the sine function sin x is computable for any real number x. Thus, the domain of sin x is the set of all real numbers.

The domain of sin(x) = The set of real numbers = `(-infty, infty)`

For the same reason, we can compute sin ax for any real number. So the domain of sin ax is also the set of all real numbers. For example,

- Domain of sin(2x) = `(-infty, infty)`
- Domain of sin(3x) = `(-infty, infty)`
- Domain of sin(4x) = `(-infty, infty)`

**Also Read:**

## Range of sin x

From the graph y=sin(x), one can see that sin(x) oscillates between -1 and 1. In other words, the value of sin(x) lies between -1 and 1 for any real number x. Thus, the range of sin x is the closed interval [-1, 1].

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

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

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

## Period of sin x

We know that

`sin(2npi+x)=sin x` for any natural number `n`. From this, we see that `2pi` is the smallest number (corresponding to n=1) such that

`sin(2pi+x)=sin x`.

Thus, `2pi` is the period of sin x.

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

*Solution:*

The period of sin(ax) is `frac{2pi}{|a|}`.

**Question 2:**Find the period of sin(ax)+cos(bx).

*Solution:*

The period of sin(ax) is `frac{2pi}{|a|}` and the period of cos(bx) is `frac{2pi}{|b|}`.

We know that the period of the sum of two periodic functions is the least common multiple (LCM) of their periods, we conclude that the period of sin(ax)+cos(bx) is

= LCM `{frac{2pi}{|a|}, frac{2pi}{|b|} }`.

**Also Read:**