What are inverse trigonometric functions?

Inverse trigonometric functions return the principal value of an angle corresponding to a given trigonometric ratio. The six inverse functions are \(\sin^{-1}x\), \(\cos^{-1}x\), \(\tan^{-1}x\), \(\cot^{-1}x\), \(\sec^{-1}x\), and \(\csc^{-1}x\).

Why are domains of trigonometric functions restricted before defining their inverses?

Trigonometric functions are periodic and not one-one over their entire domains. Restricting the domain to a principal interval makes them one-one, allowing inverse functions to exist.

What is the principal value range of \(\sin^{-1}x\)?

The principal value range of \(\sin^{-1}x\) is \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\), while its domain is \([-1,1]\).

What are the principal value intervals of all inverse trigonometric functions?

\(\sin^{-1}x:\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\), \(\cos^{-1}x:[0,\pi]\), \(\tan^{-1}x:\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\), \(\cot^{-1}x:(0,\pi)\), \(\sec^{-1}x:[0,\pi]-\left\{\dfrac{\pi}{2}\right\}\), and \(\csc^{-1}x:\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]-\{0\}\).

What is the domain of the inverse tangent function?

The domain of \(\tan^{-1}x\) is the set of all real numbers \(\mathbb{R}\), and its principal value lies in \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\).

How do you evaluate expressions like \(\sin^{-1}(\sin x)\)?

First determine whether \(x\) lies in the principal interval \(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\). If not, reduce it to an equivalent angle within the principal interval before applying the inverse sine function.

What is the identity relating inverse sine and inverse cosine?

For every \(x\in[-1,1]\), the identity is \(\sin^{-1}x+\cos^{-1}x=\dfrac{\pi}{2}\). It is one of the most frequently used identities in Board and JEE examinations.

What are the most common mistakes in inverse trigonometric functions?

Common mistakes include confusing inverse functions with reciprocal functions, ignoring principal value intervals, using \(\sin^{-1}(\sin x)=x\) for all real \(x\), and forgetting domain restrictions.

How are inverse trigonometric functions useful in CBSE and JEE Mathematics?

They are extensively used in differentiation, integration, limits, trigonometric equations, coordinate geometry and calculus. Questions based on principal values and identities are regularly asked in CBSE Board and competitive examinations.

How can I score full marks in the Inverse Trigonometric Functions chapter?

Memorize the principal value intervals, learn standard inverse values, practice principal value problems from different quadrants, revise important identities, and always check the domain before simplifying inverse trigonometric expressions.

What is the principal value range of \(\sin^{-1}x\)?

The principal value range of \(\sin^{-1}x\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\).

What is the principal value range of \(\cos^{-1}x\)?

The principal value range of \(\cos^{-1}x\) is \([0,\pi]\).

What is the principal value range of \(\tan^{-1}x\)?

The principal value range of \(\tan^{-1}x\) is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).

What is the domain of \(\sin^{-1}x\) and \(\cos^{-1}x\)?

The domain of both \(\sin^{-1}x\) and \(\cos^{-1}x\) is \([-1,1]\).

What is the domain of \(\tan^{-1}x\)?

The domain of \(\tan^{-1}x\) is the set of all real numbers, \(\mathbb{R}\).

NCERT Class 12 Maths Chapter 2: Inverse Trigonometric Functions

6 CBSE Marks

★★★★★ Difficulty

6 Topics

Medium-High JEE Weight

Topics Covered

6 key topics in this chapter

Principal Value Branch

Domain & Range of sin⁻¹, cos⁻¹, tan⁻¹

Graphs of Inverse Trigonometric Functions

Complementary Function Identities

Sum & Difference Properties

Simplification Using Substitution

Study Resources

📖 NCERT Notes ✏️ MCQ Practice 🏆 JEE / CBSE PYQs 📝 Exercise Solutions ✅ True / False Quiz

𝑓 Key Formulae

Essential mathematical expressions for this chapter — understand derivations, not just results.

Principal Range sin⁻¹

\[\sin^{-1}x : [-1,1] \to \left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]\]

📌 Principal value branch

Complementary Pair

\[\sin^{-1}x + \cos^{-1}x = \tfrac{\pi}{2}\]

📌 Holds for all x ∈ [−1,1]

Complementary Pair II

\[\tan^{-1}x + \cot^{-1}x = \tfrac{\pi}{2}\]

📌 Same identity family

Reciprocal Identity

\[\sin^{-1}\left(\tfrac{1}{x}\right) = \csc^{-1}x,\; |x|\ge 1\]

📌 Careful with domain restriction

Addition Formula

\[\tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\tfrac{x+y}{1-xy}\right)\]

📌 Valid only when xy < 1

🎯 Exam-Ready Insights

Important points to remember — curated from CBSE Board question patterns.

Every inverse trig function has a fixed principal value branch — CBSE explicitly tests knowledge of these domain-range pairs.

The addition formula for tan⁻¹ has three cases depending on the sign of xy; always check which case applies before applying it blindly.

Simplification problems often require a substitution like x = tanθ or x = sinθ before the identity becomes visible.

Graphs are reflections of the corresponding trig function graphs restricted to the principal branch — sketch them to internalise range.

🏆 Competitive Exam Strategy

Targeted tips for JEE Main, JEE Advanced, NEET, BITSAT, and CBSE Boards.

JEE Main

Evaluate compound expressions like sin⁻¹(sin(7π/6)) by first reducing the angle into the principal range — do not assume it equals the inner angle.

JEE Advanced

Problems combine inverse trig identities with algebraic simplification of surds; substitution is usually the fastest path.

BITSAT

Rapid-fire questions test only the principal value ranges — memorise all six branches cold.

⚠️ Common Mistakes to Avoid

Assuming sin⁻¹(sin x) = x for all x — true only within the principal range [−π/2, π/2].

Ignoring the domain restriction |x| ≥ 1 for sec⁻¹ and cosec⁻¹.

Applying the tan⁻¹ addition formula when xy = 1, where the formula breaks down entirely.

💡 Key Takeaways

◆

Principal value branches exist so that inverse trig functions are well-defined (single-valued).

◆

sin⁻¹x, tan⁻¹x, and cosec⁻¹x are all odd functions; cos⁻¹x, sec⁻¹x, and cot⁻¹x are not.

◆

Most problems reduce to a right-triangle or substitution trick rather than rote formula recall.

Inverse Trigonometric Functions