# Basic Concepts of inverse Trigonometric Functions

1. Introduction:

• Inverse trigonometric functions are functions that "reverse" the effect of trigonometric functions.
• Denoted by ${\mathrm{sin}}^{-1}\left(x\right)$, ${\mathrm{cos}}^{-1}\left(x\right)$, ${\mathrm{tan}}^{-1}\left(x\right)$, ${\mathrm{cot}}^{-1}\left(x\right)$, ${\mathrm{sec}}^{-1}\left(x\right)$, and ${\mathrm{csc}}^{-1}\left(x\right)$.
2. Domain and Range:

• The domain of inverse trigonometric functions corresponds to the range of the corresponding trigonometric functions.
• The range of inverse trigonometric functions depends on the principal values and is often restricted to ensure single-valuedness.
3. Principal Values:

• Each inverse trigonometric function has principal values that restrict its range to ensure a unique value for a given input.
• For ${\mathrm{sin}}^{-1}\left(x\right)$ and ${\mathrm{cos}}^{-1}\left(x\right)$: [-π/2, π/2].
• For ${\mathrm{tan}}^{-1}\left(x\right)$: $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.
• For ${\mathrm{cot}}^{-1}\left(x\right)$: $\left(0,\pi \right)$.
• For ${\mathrm{sec}}^{-1}\left(x\right)$ and ${\mathrm{csc}}^{-1}\left(x\right)$: $\left[0,\pi \right]\cup \left[\pi ,2\pi \right]$.
4. Graphs of Inverse Trigonometric Functions:

• The graphs of inverse trigonometric functions exhibit specific characteristics due to their restricted domains and principal values.
• ${\mathrm{sin}}^{-1}\left(x\right)$ and ${\mathrm{cos}}^{-1}\left(x\right)$ have domain $\left[-1,1\right]$ and range $\left[0,\pi \right]$.
• ${\mathrm{tan}}^{-1}\left(x\right)$ has an asymptote at $±\frac{\pi }{2}$.
• ${\mathrm{cot}}^{-1}\left(x\right)$ has an asymptote at $x=0$ and $\pi$.
• ${\mathrm{sec}}^{-1}\left(x\right)$ and ${\mathrm{csc}}^{-1}\left(x\right)$ have asymptotes at $\left[-1,-1\right]$ and $\left[1,1\right]$ respectively.
5. Trigonometric Identities:

• Inverse trigonometric functions satisfy certain identities, such as:
• $\mathrm{sin}\left({\mathrm{sin}}^{-1}\left(x\right)\right)=x$.
• $\mathrm{cos}\left({\mathrm{cos}}^{-1}\left(x\right)\right)=x$.
• $\mathrm{tan}\left({\mathrm{tan}}^{-1}\left(x\right)\right)=x$.
• $\mathrm{cot}\left({\mathrm{cot}}^{-1}\left(x\right)\right)=x$.
• $\mathrm{sec}\left({\mathrm{sec}}^{-1}\left(x\right)\right)=x$.
• $\mathrm{csc}\left({\mathrm{csc}}^{-1}\left(x\right)\right)=x$.
6. Inverse Trigonometric Formulas:

• The inverse trigonometric functions have various identities and formulas, e.g.:
• ${\mathrm{sin}}^{-1}\left(x\right)+{\mathrm{cos}}^{-1}\left(x\right)=\frac{\pi }{2}$.
• ${\mathrm{tan}}^{-1}\left(x\right)+{\mathrm{cot}}^{-1}\left(x\right)=\frac{\pi }{2}$.
• ${\mathrm{sec}}^{-1}\left(x\right)+{\mathrm{csc}}^{-1}\left(x\right)=\frac{\pi }{2}$.
7. Applications:

• Inverse trigonometric functions are widely used in solving trigonometric equations, analyzing periodic phenomena, and in various scientific and engineering applications.
8. Special Values:

• Memorize the special values of inverse trigonometric functions, e.g., ${\mathrm{sin}}^{-1}\left(0\right)=0$, ${\mathrm{cos}}^{-1}\left(1\right)=0$, ${\mathrm{tan}}^{-1}\left(0\right)=0$, etc.