Question

Write down the domain and range of $\sin x$.

Answer 1

Hint: $\sin x$ is defined for all real $x$ and its absolute value can never be greater than 1.

The given function is $\sin x$.
We know that $\sin x$ is defined for all real values of $x$ i.e. it gives some definite value for all real numbers. Thus, the domain of $\sin x$ is:
$ \Rightarrow x \in R$
And we also know that the absolute value of $\sin x$ can never be greater than 1. So we have:
$
   \Rightarrow \left| {\sin x} \right| \leqslant 1, \\
   \Rightarrow - 1 \leqslant \sin x \leqslant 1 \\
$
Thus, the value of $\sin x$ lies from -1 to 1.
Hence, the range of $\sin x$ is:
$ \Rightarrow \sin x \in \left[ {\begin{array}{*{20}{c}}
  { - 1,}&1
\end{array}} \right]$.

Note: $\sin x$ and $\cos x$ are periodic functions with period $2\pi $. Their value repeats itself after $x = 2\pi $. They also have the same domain and range.