Inverse Trigonometric Functions

part 1

Basic Inverse Trig Functions

• Need for inverse trigonometric functions
• Domain and range restrictions
• Inverse sine and cosine functions
• Graphs of sin⁻¹ and cos⁻¹

part 2

Other Inverse Trig Functions

• Inverse tangent and cotangent
• Inverse secant and cosecant
• Graphs of all six inverse trig functions

part 3

Properties and Identities

• Elementary properties of inverse trig functions
• Important identities and their proofs
• Solving equations using inverse trig functions

sin⁻¹(−x) = −sin⁻¹x

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

★ board favourite

cos⁻¹(−x) = π − cos⁻¹x

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

★ board favourite

sin⁻¹x + cos⁻¹x = π/2

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

★ board favourite

tan⁻¹x + cot⁻¹x = π/2

$\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$

★ board favourite

tan⁻¹x + tan⁻¹y (|xy| < 1)

$\tan^{-1}\frac{x+y}{1-xy}$

★ board favourite

tan⁻¹(−x) = −tan⁻¹x

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

★ frequently asked

sec⁻¹x + cosec⁻¹x = π/2

$\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$

★ frequently asked

tan⁻¹x − tan⁻¹y

$\tan^{-1}\frac{x-y}{1+xy}$

★ frequently asked

2tan⁻¹x (as sin⁻¹ and cos⁻¹)

$\sin^{-1}\frac{2x}{1+x^2} = \cos^{-1}\frac{1-x^2}{1+x^2} = 2\tan^{-1}x$

★ frequently asked

Simplify: tan⁻¹[ (√(1 + x²) − 1) / x ], where x ≠ 0.

Find the principal value of cos⁻¹(−1/2) + 2 sin⁻¹(1/2).

Find the principal value of sin⁻¹(−1/2).

Write the domain and the principal value branch (range) of the function y = sec⁻¹x.

Prove that tan⁻¹(1/2) + tan⁻¹(2/11) = tan⁻¹(3/4).

Find the value of tan⁻¹(√3) − cot⁻¹(−√3).

Solve for x: tan⁻¹(2x) + tan⁻¹(3x) = π/4.

Prove that 2 tan⁻¹(1/5) + sec⁻¹(5√2/7) + 2 tan⁻¹(1/8) = π/4.

Express sin⁻¹[ (sin x + cos x)/√2 ] in the simplest form, where −π/4 < x < π/4.

How many marks is Inverse Trigonometric Functions worth in the Class 12 board exam?

Around 3 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Principal value branch — master the standard identities.

Is Inverse Trigonometric Functions easy or hard?

It's rated medium — a moderate-difficulty chapter in Class 12 Maths. Most students need about 3 hours to cover it well.

What are the important topics in Inverse Trigonometric Functions?

The chapter covers 7 NCERT topics in 3 areas: Basic Inverse Trig Functions; Other Inverse Trig Functions; Properties and Identities.

What questions come from Inverse Trigonometric Functions in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Simplification, Principal Values, Domain and Range, Properties of Inverse Trig — 1- to 3-mark questions.