 Inverse trigonometric functions

The inverse trigonometric functions invert trigonometric functions on branches where they are one-to-one. Standard real ranges are $[-\pi /2,\pi /2]$ for $\arcsin$, $[0,\pi ]$ for $\arccos$, and $(-\pi /2,\pi /2)$ for $\arctan$. Thus $\arcsin (1/2)=\pi /6$ but $\arcsin (\sin (5\pi /6))=\pi /6$: the inverse returns the selected branch. The notation ${\sin }^{-1}x$ usually means $\arcsin x$, not $1/\sin x$. Implicit differentiation gives $(\arcsin x{)}^{\prime }=(1-x^2{)}^{-1/2}$ for $|x|<1$.

Inverse trigonometric functions are the inverse operations of the standard trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). These functions are used to determine the angle that corresponds to a given value of a trigonometric function. The main inverse trigonometric functions include: 1. **Arc sine (arcsin or sin⁻¹)**: The inverse of the sine function. It returns the angle whose sine is a given value.