Question

How do you find the domain and range of arcsin (sinx)?

Answer 1

Hint: In this question, we need to find the domain and range of a trigonometric function arcsin (sinx). For this we will first evaluate this function and after that we will calculate domain and range as per the domain and range of solution of the function. Domain is the interval which tells us the value of x that we can put in the function such that function remains well defined. Range is the interval in which the solution lies when we put the value of x in the function.

Complete step by step solution:
We are given the function as arcsin (sinx). We need to find the domain and range of the function.
For this let us first understand what domain and range are. For a function f(x), all the values of x which make the function well defined lies in the interval called domain. Range is the interval in which the solution f(x) lies when values of x are put from the domain.
For finding the domain and range of arcsin (sinx), let us simplify it. We know that arcsin means that we are dealing with the inverse of sin. So let us take sinx = u.
The inverse of sinx = u will give us arcsin = x.
Now putting the value of u we get $\arcsin \left( \sin x \right)=x$.
So the function arcsin (sinx) will have the same domain and range as that of x.
The function f(x) thus can be defined as f(x) = x.
We know that for any value of x, the function is well defined. Therefore the domain of the function will be all the real numbers i.e. domain $\left( -\infty ,\infty \right)$.
Since the function gives us the same values all that of x and x can be between $\left( -\infty ,\infty \right)$. So range will also be the same. Therefore, the range of the function $\left( -\infty ,\infty \right)$.
Hence domain of arcsin (sinx) $\Rightarrow \left( -\infty ,\infty \right)$.
Range of arcsin (sinx) $\Rightarrow \left( -\infty ,\infty \right)$.

Note: Students should note that arcsin can also be written as ${{\sin }^{-1}}$. Here we have calculated one of the values which is sufficient to find domain and range. Students can get confused between domain and range. Note that values of x are domain and values of f(x) are range.