Inverse Trigonometric Functions

What are Inverse Trigonometric Functions?

Inverse trigonometric functions help us find angles when the value of a trigonometric ratio is known. They play an important role in solving problems related to geometry, calculus, and real-life applications.

Inverse trigonometric functions are used to determine an angle from a given trigonometric value. For example, if $\sin \theta = \frac{1}{2}$, then $\theta = \sin^{-1}(\frac{1}{2})$. In inverse trigonometric functions class 12, you will learn how to apply inverse trigonometric functions formulas, understand the domain and range of inverse trigonometric functions, and interpret the graph of inverse trigonometric functions.

Inverse trigonometric functins are the reverse process of the trigonometric functions. In other words, the domain of the inverse function is the range of the original function and vice versa.

For example:

1. $\sin(\frac{\pi}{6}) = \frac{1}{2}$ ⇒ $\frac{\pi}{6} = \sin^{-1}(\frac{1}{2})$

2. $\cos(\pi) = -1$ ⇒ $\pi = \cos^{-1}(-1)$

3. $\tan(\frac{\pi}{4}) = 1$ ⇒ $\frac{\pi}{4} = \tan^{-1}(1)$

Types of Inverse Trigonometric Functions

1. Arcsine ($\sin^{-1}$) – Inverse of the sine function

2. Arccosine ($\cos^{-1}$) – Inverse of the cosine function

3. Arctangent ($\tan^{-1}$) – Inverse of the tangent function

4. Arccotangent ($\cot^{-1}$) – Inverse of the cotangent function

5. Arcsecant ($\sec^{-1}$) – Inverse of the secant function

6. Arccosecant ($\csc^{-1}$) – Inverse of the cosecant function

These functions are covered in inverse trigonometric functions class 12 and include important formulas, domain and range explanations, and graphs.

Domain and Range of Inverse Trigonometric Functions

Understanding the domain and range of inverse trigonometric functions is essential for solving problems. Below are the domains and ranges for each inverse function.

To define inverse trigonometric functions properly, the original trigonometric functions must be one-one and onto in the selected domain and range. For example, the sine function is defined for all real numbers, but its inverse is only defined when the domain is restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and the range to $[-1, 1]$.

Thus,

• The domain of $y = \sin^{-1}(x)$ is $[-1, 1]$

• The range of $y = \sin^{-1}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$

The same process applies to other functions like cosine, tangent, secant, etc.

Principal Domain / Principal Value Branch

By convention, the domain of inverse trigonometric functions is chosen to ensure the function is one-one and onto. For example, the principal domain for $y = \sin(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, which makes $y = \sin^{-1}(x)$ well-defined.

Graph of Inverse Trigonometric Functions

The graphs of inverse trigonometric functions are given as below:

$y=\sin ^{-1}(x)$

$\sin ^{-1}(x)$ is the inverse of the trignometric function $\sin x$.

$\mathrm{y}=\sin \mathrm{x}, \mathrm{x} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\mathrm{y} \in[-1,1]$

$y=\cos ^{-1}(x)$

$\cos ^{-1}(x)$ is the inverse of $\cos x$

$y=\tan ^{-1}(x)$

$\tan ^{-1}(x)$ is the inverse of $\tan x$.

$y=\operatorname{cosec}^{-1}(x)$

$\operatorname{cosec}^{-1}(x)$ is the inverse of $\operatorname{cosec} x$.

$y=\sec ^{-1}(x)$

$\sec ^{-1}(x)$ is the inverse of $\sec x$.

$y=\cot ^{-1}(x)$

$\cot ^{-1}(x)$ is the inverse of $\cot x$.

Inverse Trigonometric Functions Formulas

This section covers important inverse trigonometric functions formulas, including formulas for negative functions, reciprocal functions, complementary functions, sum and difference of angles, multiple angles, as well as differentiation and integration formulas. These formulas are essential for inverse trigonometric functions class 12 and are helpful in solving calculus problems.

Formulas for Negative Functions

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

• $\tan^{-1}(-x) = -\tan^{-1}(x)$, for all $x \in \mathbb{R}$

• $\csc^{-1}(-x) = -\csc^{-1}(x)$, for all $x \in \mathbb{R} - (-1,1)$

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

• $\sec^{-1}(-x) = \pi - \sec^{-1}(x)$, for all $x \in \mathbb{R} - (-1,1)$

• $\cot^{-1}(-x) = \pi - \cot^{-1}(x)$, for all $x \in \mathbb{R}$

Formulas for Reciprocal Functions

• $\sin^{-1}\left(\frac{1}{x}\right) = \csc^{-1}(x)$, for all $x \in (-\infty,-1] \cup [1,\infty)$

• $\cos^{-1}\left(\frac{1}{x}\right) = \sec^{-1}(x)$, for all $x \in (-\infty,-1] \cup [1,\infty)$

• $\tan^{-1}\left(\frac{1}{x}\right) = \begin{cases}\cot^{-1} x, & x>0\ -\pi + \cot^{-1} x, & x<0\end{cases}$

Formulas for Complementary Functions

The sum of complementary functions results in a right angle:

• $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$, for all $x \in [-1,1]$

• $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$, for all $x \in \mathbb{R}$

• $\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}$, for all $x \in (-\infty,-1] \cup [1,\infty)$

Sum and Difference Formulas of Inverse Trigonometric Functions

Below are the sum and difference formulas of inverse trigonometric functions like $\tan^{-1}$, $\sin^{-1}$, and $\cos^{-1}$. These formulas help in simplifying expressions and solving equations in inverse trigonometric functions class 12.

Sum of Angles in Terms of $\tan^{-1}$

• $\tan^{-1} x + \tan^{-1} y = \begin{cases} \tan^{-1}\left(\frac{x+y}{1-xy}\right), & x>0, y>0, xy<1\\ \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right), & x>0, y>0, xy>1\\ -\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right), & x<0, y<0, xy>1 \end{cases}$

Difference of Angles in Terms of $\tan^{-1}$

• $\tan^{-1} x - \tan^{-1} y = \begin{cases} \tan^{-1}\left(\frac{x-y}{1+xy}\right), & xy > -1\\ \pi + \tan^{-1}\left(\frac{x-y}{1+xy}\right), & x>0, y<0, xy<-1\\ -\pi + \tan^{-1}\left(\frac{x-y}{1+xy}\right), & x<0, y>0, xy<-1 \end{cases}$

Sum and Difference of Angles in Terms of $\sin^{-1}$

• $\sin^{-1} x + \sin^{-1} y = \begin{cases} \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right), & -1 \leq x, y \leq 1, x^2+y^2 \leq 1\\ \pi - \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right), & x>0, y>0, x^2+y^2>1\\ -\pi - \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right), & -1 \leq x, y <0, x^2+y^2>1\end{cases}$

• $\sin^{-1} x - \sin^{-1} y = \begin{cases} \sin^{-1}\left(x\sqrt{1-y^2} - y\sqrt{1-x^2}\right), & -1 \leq x, y \leq 1, x^2+y^2 \leq 1\\ \pi - \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right), & x>0, y<0, x^2+y^2>1\\ -\pi - \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right), & -1 \leq x<0, 0<y\leq1, x^2+y^2>1\end{cases}$

Sum and Difference of Angles in Terms of $\cos^{-1}$

• $\cos^{-1} x + \cos^{-1} y = \cos^{-1}\left(xy - \sqrt{1-x^2}\sqrt{1-y^2}\right)$, if $0 < x, y \leq 1$

• $\cos^{-1} x - \cos^{-1} y = \begin{cases} \cos^{-1}\left(xy + \sqrt{1-x^2}\sqrt{1-y^2}\right), & 0 \leq x, y \leq 1, x \leq y\\ -\cos^{-1}\left(xy + \sqrt{1-x^2}\sqrt{1-y^2}\right), & 0 < x, y \leq 1, x > y \end{cases}$

Multiple Angle Formulas of Inverse Trigonometric Functions

Below are the multiple angle formulas of inverse trigonometric functions such as $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$. These formulas are useful for solving advanced problems in inverse trigonometric functions class 12 and calculus.

Double and Triple Angle Formulas in Terms of $\sin^{-1}$

• $2\sin^{-1} x = \begin{cases} \sin^{-1}(2x\sqrt{1-x^2}), & -\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}\\ \pi - \sin^{-1}(2x\sqrt{1-x^2}), & x > \frac{1}{\sqrt{2}}\\ -\pi - \sin^{-1}(2x\sqrt{1-x^2}), & x < -\frac{1}{\sqrt{2}} \end{cases}$

• $3\sin^{-1} x = \begin{cases} \sin^{-1}(3x - 4x^3), & -\frac{1}{2} \leq x \leq \frac{1}{2}\\ \pi - \sin^{-1}(3x - 4x^3), & x > \frac{1}{2}\\ -\pi - \sin^{-1}(3x - 4x^3), & x < -\frac{1}{2}\end{cases}$

Double and Triple Angle Formulas in Terms of $\cos^{-1}$

• $2\cos^{-1} x = \begin{cases} \cos^{-1}(2x^2 - 1), & 0 \leq x \leq 1\\ 2\pi - \cos^{-1}(2x^2 - 1), & -1 \leq x \leq 0\end{cases}$

• $3\cos^{-1} x = \begin{cases} \cos^{-1}(4x^3 - 3x), & \frac{1}{2} \leq x \leq 1\\ 2\pi - \cos^{-1}(4x^3 - 3x), & -\frac{1}{2} \leq x \leq \frac{1}{2}\\ 2\pi + \cos^{-1}(4x^3 - 3x), & -1 \leq x \leq -\frac{1}{2}\end{cases}$

Multiple Angles in Terms of $\tan^{-1}$ and $\sin^{-1}$

• $2\tan^{-1} x = \begin{cases} \sin^{-1}\left(\frac{2x}{1+x^2}\right), & -1 \leq x \leq 1\\ \pi - \sin^{-1}\left(\frac{2x}{1+x^2}\right), & x>1\\ -\pi - \sin^{-1}\left(\frac{2x}{1+x^2}\right), & x<-1\end{cases}$

Multiple Angles in Terms of $\tan^{-1}$ and $\cos^{-1}$

• $2\tan^{-1} x = \begin{cases} \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), & 0 \leq x < \infty\\ -\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), & -\infty < x \leq 0\end{cases}$

Differentiation of Inverse Trigonometric Functions

The derivatives of inverse trigonometric functions are essential for calculus problems:

• $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, x \neq \pm 1$

• $\frac{d}{dx}(\cos^{-1} x) = \frac{-1}{\sqrt{1-x^2}}, x \neq \pm 1$

• $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$

• $\frac{d}{dx}(\cot^{-1} x) = \frac{-1}{1+x^2}$

• $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}, x \neq \pm 1$

• $\frac{d}{dx}(\csc^{-1} x) = \frac{-1}{|x|\sqrt{x^2-1}}, x \neq \pm 1$

Integration of Inverse Trigonometric Functions

The integrals of inverse trigonometric functions are widely used in calculus:

• $\int \sin^{-1} x ,dx = x\sin^{-1}x + \sqrt{1-x^2} + C$

• $\int \cos^{-1} x ,dx = x\cos^{-1}x - \sqrt{1-x^2} + C$

• $\int \tan^{-1} x ,dx = x\tan^{-1}x - \frac{1}{2}\ln|1+x^2| + C$

• $\int \csc^{-1} x ,dx = x\csc^{-1}x + \ln|x+\sqrt{x^2-1}| + C$

• $\int \sec^{-1} x ,dx = x\sec^{-1}x - \ln|x+\sqrt{x^2-1}| + C$

• $\int \cot^{-1} x ,dx = x\cot^{-1}x + \frac{1}{2}\ln|1+x^2| + C$

Inverse Trigonometric Functions Table

This table summarizes the domain and range of inverse trigonometric functions, which is an important topic for inverse trigonometric functions class 12 and solving problems.

Difference between Trigonometric Functions and Inverse Trigonometric Functions

Below is a table showing the key differences between trigonometric functions and inverse trigonometric functions, covering their definitions, domains, ranges, and applications in inverse trigonometric functions class 12.

List of Topics according to NCERT/JEE MAIN

Below is a list of important topics related to the NCERT syllabus and JEE Main, helping you focus on key areas for preparation.

Important Books for Inverse Trigonometric Functions

Below are the recommended books for studying inverse trigonometric functions, including theory, examples, and practice questions, useful for inverse trigonometric functions class 12 and competitive exams like JEE Main.

NCERT Resources

Below are helpful NCERT resources such as textbooks, solutions, and notes that explain inverse trigonometric functions clearly, making it easier to prepare for board exams and entrance tests.

NCERT Maths Notes for Class 12th Chapter 2 - Inverse Trigonometric Functions

NCERT Maths Solutions for Class 12th Chapter 2 - Inverse Trigonometric Functions

NCERT Maths Exemplar Solutions for Class 12th Chapter 2 - Inverse Trigonometric Functions

NCERT Subjectwise Resources

Below are subject-wise NCERT resources offering notes, solutions, exemplar problems, and solved examples for class 12 and competitive exam preparation.

Practice Questions based on Inverse Trigonometric Functions

Below are practice questions based on inverse trigonometric functions, including problems from NCERT exercises and JEE Main-level questions, to help you strengthen your understanding and problem-solving skills.