Inverse Trigonometric Functions — Study Notes

1. Introduction

Inverse trigonometric functions answer the question: given a value of a trigonometric ratio, what is the corresponding angle? Since trigonometric functions are periodic and hence many-to-one, they are not one-to-one on their natural domains, so their inverses must be defined by first restricting the domains so that the functions become bijective. The principal value branches of the six inverse trigonometric functions provide a standard choice of range, allowing each inverse to be single-valued.

This chapter establishes the domains and principal ranges of arcsin, arccos, arctan, arccosec, arcsec, and arccot, and develops the algebraic identities relating these functions, including the properties for positive and negative arguments and the sums and differences of inverse trigonometric functions. These identities are heavily tested in the board examinations and provide the simplification tools needed in calculus.

The chapter matters beyond the examination because inverse trigonometric functions appear throughout integration. Many standard integrals, such as those yielding $\arcsin$ and $\arctan$, and the technique of integration by substitution used to transform integrands into these forms, depend directly on a working knowledge of these functions. A student who understands the principal value conventions will find Chapter 5 and Chapter 7 substantially easier.

2. Domain and Principal Value Branches

For each trigonometric function, we restrict the domain to an interval on which the function is one-one and onto its range. The chosen range is called the principal value branch.

Sine Inverse

$y = \sin^{-1} x$ is defined for $x \in [-1, 1]$ with principal range $y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Here $\sin y = x$.

Cosine Inverse

$y = \cos^{-1} x$ is defined for $x \in [-1, 1]$ with principal range $y \in [0, \pi]$. Here $\cos y = x$.

Tangent Inverse

$y = \tan^{-1} x$ is defined for $x \in \mathbb{R}$ with principal range $y \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. Here $\tan y = x$.

Cosecant Inverse

$y = \csc^{-1} x$ is defined for $x \in (-\infty, -1] \cup [1, \infty)$ with principal range $y \in \left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]$.

Secant Inverse

$y = \sec^{-1} x$ is defined for $x \in (-\infty, -1] \cup [1, \infty)$ with principal range $y \in \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$.

Cotangent Inverse

$y = \cot^{-1} x$ is defined for $x \in \mathbb{R}$ with principal range $y \in (0, \pi)$. Here $\cot y = x$.

3. Graphs of Inverse Trigonometric Functions

The graph of $y = \sin^{-1} x$ is the reflection of the graph of $y = \sin x$, restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, across the line $y = x$. The graph is strictly increasing, passing through the origin, rising to $\frac{\pi}{2}$ at $x = 1$ and falling to $-\frac{\pi}{2}$ at $x = -1$.

The graph of $y = \cos^{-1} x$ is strictly decreasing, taking value $\pi$ at $x = -1$ and value $0$ at $x = 1$, with $\frac{\pi}{2}$ at $x = 0$.

The graph of $y = \tan^{-1} x$ is strictly increasing with two horizontal asymptotes $y = \pm \frac{\pi}{2}$; it passes through the origin, approaches $\frac{\pi}{2}$ as $x \to \infty$, and approaches $-\frac{\pi}{2}$ as $x \to -\infty$.

4. Principal Values of Standard Angles

Some standard principal values to memorise:

• $\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$, $\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$
• $\cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3}$, $\cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3}$
• $\tan^{-1}(1) = \frac{\pi}{4}$, $\tan^{-1}(-1) = -\frac{\pi}{4}$
• $\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6}$
• $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$
• $\tan^{-1}\left(-\sqrt{3}\right) = -\frac{\pi}{3}$, $\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

Note carefully that $\sin^{-1}(\sin x)$ equals x only when x lies in the principal range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$; otherwise the answer must be adjusted to land inside the principal branch.

5. Properties of Inverse Trigonometric Functions

Property 1: Negative Arguments

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

$$\cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \sec^{-1}(-x) = \pi - \sec^{-1}x, \quad \cot^{-1}(-x) = \pi - \cot^{-1}x$$

Property 2: Complementary Angles

$$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}, \quad \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}, \quad \sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$$

Property 3: Reciprocal Relations

$$\sec^{-1}\left(\frac{1}{x}\right) = \cos^{-1}x, \quad \csc^{-1}\left(\frac{1}{x}\right) = \sin^{-1}x, \quad \cot^{-1}\left(\frac{1}{x}\right) = \tan^{-1}x$$

6. Sum and Difference Identities

For appropriate values of x and y:

$$\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x + y}{1 - xy}\right), \quad xy < 1$$

$$\tan^{-1}x - \tan^{-1}y = \tan^{-1}\left(\frac{x - y}{1 + xy}\right), \quad xy > -1$$

$$\sin^{-1}x + \sin^{-1}y = \sin^{-1}\left(x\sqrt{1 - y^2} + y\sqrt{1 - x^2}\right), \quad x^2 + y^2 \leq 1$$

$$\cos^{-1}x + \cos^{-1}y = \cos^{-1}\left(xy - \sqrt{1 - x^2}\sqrt{1 - y^2}\right)$$

A special and heavily used case:

$$\tan^{-1}x + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{2} \text{ for } x > 0$$

7. Converting Trigonometric Expressions

A standard examination technique is to express a trigonometric expression in terms of a single inverse function by substitution. For example, to evaluate $\cos(2\cos^{-1}x)$, use the double angle formula:

$$\cos(2\cos^{-1}x) = 2\cos^2(\cos^{-1}x) - 1 = 2x^2 - 1$$

Similarly, for expressions like $\tan\left(\frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right)$, set $\cos^{-1}\left(\frac{2}{3}\right) = \theta$ so that $\cos\theta = \frac{2}{3}$, then use the half-angle formula $\tan\frac{\theta}{2} = \sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$.

Another common simplification rewrites products like $\sin^{-1}x$ with a trigonometric substitution. For example, $\sin^{-1}\left(2x\sqrt{1-x^2}\right) = 2\sin^{-1}x$ when $|x| \leq \frac{1}{\sqrt{2}}$, using the identity $\sin 2\theta = 2\sin\theta\cos\theta$.

Quick Revision Tables

Table 1: Principal Value Branches

Table 2: Standard Principal Values

Table 3: Useful Identities

Mind Map

Important Diagrams (SVG)

Diagram 1: Principal Branches of sin^(-1) and cos^(-1)

Common Mistakes

1. Confusing $\sin^{-1} x$ with $(\sin x)^{-1} = \csc x$. The notation $\sin^{-1} x$ denotes the inverse sine function, never the reciprocal.
2. Writing $\sin^{-1}(\sin x) = x$ for all x. This identity holds only when $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$; otherwise the value must be adjusted to the principal branch.
3. Using $\cos^{-1}(-x) = -\cos^{-1}x$. The correct identity is $\cos^{-1}(-x) = \pi - \cos^{-1}x$, since the principal range of cos inverse is $[0, \pi]$.
4. Forgetting the condition $xy < 1$ when applying the $\tan^{-1}x + \tan^{-1}y$ formula; without the condition the result may be off by $\pi$.
5. Taking the principal value of $\sin^{-1}\left(-\frac{1}{2}\right)$ as $\frac{11\pi}{6}$ or $-\frac{11\pi}{6}$. The principal value is $-\frac{\pi}{6}$, which lies in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
6. Using the domain of $\sin^{-1}x$ for $\sec^{-1}x$. The domains differ; secant inverse excludes $(-1, 1)$.
7. Forgetting that $\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$ holds on the common domain $(-\infty, -1] \cup [1, \infty)$.
8. In simplification problems, applying $\tan^{-1}x + \tan^{-1}y$ without verifying that x and y give the principal branch, which changes signs in the answer.
9. Writing $\cos^{-1}(\cos x) = x$ for x outside $[0, \pi]$. Adjust the answer so that it falls within $[0, \pi]$.

Exam Tips

1. Always state the principal value branch used before evaluating an inverse function; examiners award marks for stating the range.
2. Memorise the principal values table for arguments 0, 1/2, 1/sqrt(2), sqrt(3)/2, and 1, for sine, cosine, and tangent inverses.
3. When evaluating $\sin^{-1}(\sin x)$ for large angles, first reduce x by subtracting multiples of $2\pi$, then adjust into $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
4. Use the identity $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ to replace a stubborn cos inverse with a sine inverse before applying sum formulas.
5. For prove-type questions on $\tan^{-1}x + \tan^{-1}y$, check the sign condition $xy < 1$ and add or subtract $\pi$ accordingly; write the condition explicitly.
6. In integration chapters, learn to recognise integrands that produce $\sin^{-1}\left(\frac{x}{a}\right)$, $\tan^{-1}\left(\frac{x}{a}\right)$, and $\sec^{-1}\left(\frac{x}{a}\right)$; this chapter's ranges decide the constants.
7. For MCQ problems on $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$, the answer depends on the range of x: it is $2\sin^{-1}x$ for $|x| \leq \frac{1}{\sqrt{2}}$ and $\pi - 2\sin^{-1}x$ for $x > \frac{1}{\sqrt{2}}$. Test a sample value to confirm.
8. Practise converting between inverse functions using right-triangle substitutions, e.g., if $\sin^{-1}x = \theta$, then $\cos\theta = \sqrt{1 - x^2}$.

Conclusion

Inverse trigonometric functions extend the idea of solving equations to angle determination and form a compact and elegant part of the syllabus. The careful restriction of domains to obtain principal value branches is the key structural idea, and the standard values, negative-argument rules, complementary identities, and sum-difference formulas are the working tools. Because these functions appear as answers to a large family of integrals and as the foundation of many substitutions in calculus, this short chapter has an outsized importance. Mastery of the principal ranges and the sign rules will prevent the most frequent errors in both objective and long-answer questions, and it prepares the student for the continuity, differentiability, and integration chapters that follow.