What is Inverse Trigonometric Functions Class 12: Definition & Concepts

Definition of Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse relations of the 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 the angle $x$ when the value of the trigonometric ratio is known.

Since trigonometric functions are periodic and not one-to-one over their entire domain, their inverses are defined by restricting the domain to principal values where the function is one-to-one. This ensures the inverse is a proper function.

Common inverse functions:

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

These are fundamental in Class 12 NCERT Mathematics and form the basis for solving various problems.

Principal Values and Domain Restrictions

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

These restrictions ensure that the inverse functions are well-defined and single-valued. For example, $\sin^{-1} x$ always returns an angle in $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Understanding these domains and ranges is essential for solving problems and graphing inverse trig functions.

Properties and Formulas of Inverse Trigonometric Functions

Inverse trigonometric functions follow several important properties and formulas useful for simplification and problem-solving:

• $\sin(\sin^{-1} x) = x$, for $x \in [-1,1]$
• $\cos(\cos^{-1} x) = x$, for $x \in [-1,1]$
• $\tan(\tan^{-1} x) = x$, for all real $x$

Key identities include:

$$\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$$ $$\tan^{-1} x + \tan^{-1} \frac{1}{x} = \frac{\pi}{2}, \quad x > 0$$

Example: Find $\sin^{-1}\frac{1}{2}$.

Since $\sin \frac{\pi}{6} = \frac{1}{2}$, we have:

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

These formulas are frequently tested in Class 12 exams and help simplify expressions involving inverse trig functions.

Graphical Representation of Inverse Trigonometric Functions

Graphs of inverse trigonometric functions are reflections of their original functions about the line $y = x$, restricted to principal domains.

• $y = \sin^{-1} x$ is defined on $[-1,1]$ with range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
• $y = \cos^{-1} x$ is defined on $[-1,1]$ with range $[0, \pi]$.
• $y = \tan^{-1} x$ is defined for all real $x$ with range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Understanding these graphs helps visualize the behaviour of these functions and solve inequalities or equations involving inverse trigonometric functions.

Tip: Sketching the graphs aids in remembering domain and range restrictions.

Applications of Inverse Trigonometric Functions in Class 12

Inverse trigonometric functions have many applications in Class 12 Mathematics, particularly in:

• Solving trigonometric equations
• Calculus: differentiation and integration involving inverse trig functions
• Geometry: finding angles in triangles when sides are known
• Physics: problems involving oscillations and waves

Example problem: Solve for $x$ if $\sin^{-1} x = \frac{\pi}{4}$.

Since $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$,

$$x = \frac{\sqrt{2}}{2}$$

Mastering these functions helps in understanding advanced topics and scoring well in exams.

Worked Example: Simplify $\tan^{-1} 1 + \tan^{-1} 2$

Use the formula:

$$\tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right), \quad \text{if } ab < 1$$

Here, $a = 1$, $b = 2$, so $ab = 2 > 1$, formula does not directly apply. Instead use:

$$\tan^{-1} a + \tan^{-1} b = \pi + \tan^{-1} \left( \frac{a + b}{1 - ab} \right), \quad \text{if } ab > 1$$

Calculate:

$$\frac{1 + 2}{1 - 2} = \frac{3}{-1} = -3$$

Therefore,

$$\tan^{-1} 1 + \tan^{-1} 2 = \pi + \tan^{-1} (-3) = \pi - \tan^{-1} 3$$

This example shows how to handle sums of inverse tangent functions, a common Class 12 problem.