# Domain of sin x | Range of sin x | Period of sin x

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)

## 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|} }.