What Is Inverse Trigonometric Functions Class 11: Definition & Examples

Definition of Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse operations of the basic trigonometric functions: sine, cosine, and tangent.

• If $y = \sin x$, then $x = \sin^{-1} y$ or $x = \arcsin y$.
• Similarly, $x = \cos^{-1} y$ for cosine and $x = \tan^{-1} y$ for tangent.

These functions allow you to find the angle $x$ when the value of the trigonometric ratio $y$ is known.

Key point: Since trigonometric functions are periodic and not one-to-one over their entire domain, their inverse functions are defined with restricted domains to ensure they are functions.

Domain and Range of Inverse Trigonometric Functions

To define inverse trigonometric functions properly, we restrict domains of sine, cosine, and tangent so that each is one-to-one.

Example:

• $\sin^{-1}(0.5) = \frac{\pi}{6}$ because $\sin(\frac{\pi}{6}) = 0.5$

Understanding these domains and ranges is essential for correctly using inverse trig functions in Class 11 NCERT problems.

Common Inverse Trigonometric Functions and Their Formulas

Here are the most important inverse trigonometric functions used in Class 11:

• $y = \sin^{-1} x$ means $\sin y = x$, where $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
• $y = \cos^{-1} x$ means $\cos y = x$, where $y \in [0, \pi]$
• $y = \tan^{-1} x$ means $\tan y = x$, where $y \in (-\frac{\pi}{2}, \frac{\pi}{2})$

Key formulas:

$$\sin(\sin^{-1} x) = x, \quad \cos(\cos^{-1} x) = x, \quad \tan(\tan^{-1} x) = x$$

$$\sin^{-1}(\sin x) = x, \quad \text{if } x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$$

These formulas help simplify expressions and solve equations involving inverse trigonometric functions.

Worked Example: Finding an Angle Using Inverse Sine

Example: Find the angle $\theta$ if $\sin \theta = \frac{1}{2}$ and $\theta$ lies in $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Solution:

Since $\sin \theta = \frac{1}{2}$,

$$\theta = \sin^{-1} \left( \frac{1}{2} \right)$$

From known values,

$$\sin^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{6}$$

Thus, $\theta = \frac{\pi}{6}$ radians or 30°.

This example illustrates how inverse trigonometric functions help find angles from ratios, a common task in Class 11 NCERT problems.

Applications of Inverse Trigonometric Functions in Class 11

Inverse trigonometric functions have many applications in Class 11 Mathematics, especially in the chapter on Trigonometric Functions:

• Solving Trigonometric Equations: They help find angle solutions when trigonometric ratios are known.
• Integration and Differentiation: In calculus, inverse trig functions appear in integration formulas.
• Geometry Problems: Used to calculate angles in triangles and circles.
• Real-life Problems: Applied in physics and engineering to find angles from measurements.

Mastering these functions is important for scoring well in CBSE exams and understanding advanced mathematics.

Comparison of Trigonometric and Inverse Trigonometric Functions

Understanding the difference between trigonometric and inverse trigonometric functions is crucial:

This table helps clarify when to use each function in Class 11 problems.