What is Inverse Trigonometric Functions Class 12: Definition & Concepts

Definition of Inverse Trigonometric Functions in Class 12

Inverse trigonometric functions are the inverse relations of the six basic trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. For example, if $y = \sin x$, then $x = \sin^{-1} y$ or $x = \arcsin y$. These functions allow us to find an angle when the value of its trigonometric ratio is known.

Since trigonometric functions are periodic and not one-to-one over their entire domain, their inverses are defined by restricting domains to principal values. This ensures each inverse function is a proper function with a unique output for every input in its domain.

In Class 12 NCERT Mathematics, the focus is on the principal branches of inverse sine, cosine, and tangent functions, which are widely used in problems and applications.

Principal Branches and Domains of Inverse Trigonometric Functions

To define inverse trigonometric functions properly, we restrict the domain of the original functions to intervals where they are one-to-one. These intervals are called principal branches.

For example, $\sin^{-1} x$ (arcsin) is defined only for $x \in [-1,1]$ and its output is an angle in $[-\frac{\pi}{2}, \frac{\pi}{2}]$. This restriction ensures the inverse is a function and can be used reliably in calculations.

Understanding these domains and ranges is crucial for solving inverse trigonometric problems in exams.

Common Inverse Trigonometric Functions and Their Formulas

The most commonly used inverse trigonometric functions in Class 12 are:

• $\sin^{-1} x$ or $\arcsin x$
• $\cos^{-1} x$ or $\arccos x$
• $\tan^{-1} x$ or $\arctan x$

Each has specific domain and range:

• $\sin^{-1} x$: Domain $[-1,1]$, Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$
• $\cos^{-1} x$: Domain $[-1,1]$, Range $[0, \pi]$
• $\tan^{-1} x$: Domain $(-\infty, \infty)$, Range $(-\frac{\pi}{2}, \frac{\pi}{2})$

Key formulas:

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

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

(Only when $x$ lies in the principal domain of the inverse function.)

Worked example:

Find $\theta$ if $\sin \theta = \frac{1}{2}$ and $\theta$ is in the principal range.

Solution:

$$\theta = \sin^{-1} \frac{1}{2} = \frac{\pi}{6}$$

This example shows how inverse functions help find angles from ratios.

Properties and Graphs of Inverse Trigonometric Functions

Inverse trigonometric functions have unique properties that help in solving equations and simplifying expressions.

Properties include:

• Domain and Range: As discussed, each inverse function has a limited domain and range.
• Odd and Even Functions:
• $\sin^{-1}(-x) = -\sin^{-1} x$ (odd function)
• $\tan^{-1}(-x) = -\tan^{-1} x$ (odd function)
• $\cos^{-1}(-x) = \pi - \cos^{-1} x$
• Addition formulas:
• $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \left(\frac{x + y}{1 - xy}\right)$, when $xy < 1$

Graphs:

The graphs of inverse trig functions are reflections of their original functions about the line $y = x$ within the principal domain. For example, the graph of $y = \sin^{-1} x$ is the reflection of $y = \sin x$ restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Understanding these properties and graphs helps in visualising and solving problems efficiently.

Applications of Inverse Trigonometric Functions in Class 12 Mathematics

Inverse trigonometric functions are widely used in various Class 12 topics and real-world applications:

• Solving trigonometric equations: Finding angles when ratios are known.
• Calculus: Differentiation and integration involving inverse trig functions.
• Geometry: Calculating angles and lengths in triangles and circles.
• Physics and Engineering: Modelling wave functions, oscillations, and more.

Example: Differentiate $y = \sin^{-1} x$.

Using the formula:

$$\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1$$

This derivative is essential for many calculus problems in Class 12.

Mastering inverse trigonometric functions will boost your problem-solving skills for exams and competitive tests.