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)`
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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|} }`.
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