What is Inverse Trigonometric Functions Class 11: Definition & Examples
By ConceptScroll Team · Published on 19 June 2026 · 4 min read
Inverse trigonometric functions in Class 11 are the functions that reverse the effect of trigonometric functions like sine, cosine, and tangent. They help find angles when the value of the trigonometric ratio is known, forming an essential part of the NCERT Mathematics syllabus.
Definition of Inverse Trigonometric Functions for Class 11
Inverse trigonometric functions are the inverse operations of the basic trigonometric functions: sine, cosine, and tangent. If $y = \sin x$, then the inverse function is written as $x = \sin^{-1} y$ or $x = \arcsin y$. This means that the inverse trigonometric function returns the angle $x$ when the value of the trigonometric ratio $y$ is known.
The main inverse trigonometric functions studied in Class 11 are:
- $\sin^{-1} x$ (arcsin)
- $\cos^{-1} x$ (arccos)
- $\tan^{-1} x$ (arctan)
- $\cot^{-1} x$ (arccot)
- $\sec^{-1} x$ (arcsec)
- $\csc^{-1} x$ (arccosec)
These functions are defined with restricted domains to ensure each inverse function is a proper function (one-to-one and onto).
Domain and Range of Inverse Trigonometric Functions
To make inverse trigonometric functions well-defined, their domains and ranges are restricted as follows:
| Function | Domain | Range (Principal Value) |
|---|---|---|
| $\sin^{-1} x$ | $[-1, 1]$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$ |
| $\cos^{-1} x$ | $[-1, 1]$ | $[0, \pi]$ |
| $\tan^{-1} x$ | $(-\infty, \infty)$ | $(-\frac{\pi}{2}, \frac{\pi}{2})$ |
| $\cot^{-1} x$ | $(-\infty, \infty)$ | $(0, \pi)$ |
| $\sec^{-1} x$ | $(-\infty, -1] \cup [1, \infty)$ | $[0, \pi]$, $\neq \frac{\pi}{2}$ |
| $\csc^{-1} x$ | $(-\infty, -1] \cup [1, \infty)$ | $[-\frac{\pi}{2}, \frac{\pi}{2}]$, $\neq 0$ |
Understanding these domains and ranges is crucial for solving problems involving inverse trigonometric functions.
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Properties and Formulas of Inverse Trigonometric Functions
Inverse trigonometric functions have several important properties and identities that simplify calculations and proofs:
- $\sin(\sin^{-1} x) = x$ for $x \in [-1,1]$
- $\cos(\cos^{-1} x) = x$ for $x \in [-1,1]$
- $\tan(\tan^{-1} x) = x$ for all real $x$
- $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$ for $x \in [-1,1]$
- $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$ for all real $x$
These formulas help in simplifying expressions and solving trigonometric equations in Class 11 NCERT exercises.
Worked Example: Finding an Angle Using Inverse Trigonometric Functions
Example: Find the angle $\theta$ if $\sin \theta = \frac{1}{2}$.
Solution:
We use the inverse sine function:
$$ \theta = \sin^{-1} \left(\frac{1}{2}\right) $$
From the unit circle or standard values:
$$ \sin^{-1} \left(\frac{1}{2}\right) = \frac{\pi}{6} \text{ or } 30^\circ $$
So, the angle $\theta$ is $30^\circ$ or $\frac{\pi}{6}$ radians.
This example illustrates how inverse trigonometric functions help find angles when trigonometric ratios are given.
Difference Between Trigonometric and Inverse Trigonometric Functions
Understanding the difference between trigonometric and inverse trigonometric functions is essential:
| Aspect | Trigonometric Functions | Inverse Trigonometric Functions |
|---|---|---|
| Purpose | Find ratio from angle | Find angle from ratio |
| Example | $\sin 30^\circ = \frac{1}{2}$ | $\sin^{-1} \left(\frac{1}{2}\right) = 30^\circ$ |
| Domain | All real numbers (angles) | Restricted to ensure uniqueness |
| Range | $[-1,1]$ for sine and cosine | Angles in principal value ranges |
This comparison helps Class 11 students grasp the concept clearly.
Applications of Inverse Trigonometric Functions in Class 11 Mathematics
Inverse trigonometric functions are widely used in Class 11 Mathematics, especially in the chapter on Trigonometric Functions. Their applications include:
- Solving trigonometric equations where the angle is unknown
- Calculating angles in geometry and coordinate problems
- Deriving formulas for integration and differentiation in calculus
- Understanding periodic functions and their inverses
By mastering inverse trigonometric functions, students can solve complex problems efficiently and score well in CBSE exams.
Frequently asked questions
What is the principal value of inverse sine function?
The principal value of $\sin^{-1} x$ lies between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.
Can inverse trigonometric functions take any real number as input?
No, their domains are restricted. For example, $\sin^{-1} x$ only accepts values from $-1$ to $1$.
How do inverse trigonometric functions help in solving equations?
They allow finding the angle when the trigonometric ratio is known, making equation solving easier.
Are inverse trigonometric functions part of the Class 11 NCERT syllabus?
Yes, they are a key topic in the Trigonometric Functions chapter of Class 11 NCERT Mathematics.
What is the difference between $\sin^{-1} x$ and $\frac{1}{\sin x}$?
$\sin^{-1} x$ is the inverse sine function, while $\frac{1}{\sin x}$ is the cosecant function, which is different.
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