Mathematics > trigonometry > Inverse Trigonometric Functions

Case Study - 3

If a function $f:X \to Y$ defined as $f(x)=y$ is one-one and onto, then we can define a unique function $g:Y \to X$ such that $g(y)=x$, where $x \in X$ and $y=f(x)$, $y \in Y$. Function $g$ is called the inverse of function $f$.

The domain of sine function is $R$ and function sine: $R \to R$ is neither one-one nor onto. The following graph shows the sine function.

Let sine function be defined from set $A$ to $[-1, 1]$ such that inverse of sine function exists, i.e., $\sin^{-1}x$ is defined from $[-1, 1]$ to $A$.

(ii) If $\sin^{-1}(x)$ is defined from $[-1, 1]$ to its principal value branch, find the value of $\sin^{-1}\left(-\frac{1}{2}\right) - \sin^{-1}(1)$.