Inverse Trigonometric Functions

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The chapter on Inverse Trigonometric Functions in Class 12 Maths (NCERT) deals with the inverses of trigonometric functions, their domains, ranges, and properties. Here’s a summary:

1. Definitions and Basic Concepts:

Need for Inverses: Trigonometric functions are not one-to-one over their entire domains, so their inverses are not defined unless we restrict the domains.
Principal Values: The principal value branch of an inverse trigonometric function is a specific range within which the inverse is defined. This ensures the inverse is a function (i.e., each input has only one output).
Notation: The inverse trigonometric functions are denoted as sin⁻¹, cos⁻¹, tan⁻¹, cosec⁻¹, sec⁻¹, and cot⁻¹. (Sometimes arcsin, arccos, etc., are also used.)

2. Domains and Ranges (Principal Value Branches):

This is crucial. You must memorize these:

Inverse Function	Domain	Principal Value Branch (Range)
sin⁻¹ x	[-1, 1]	[-π/2, π/2]
cos⁻¹ x	[-1, 1]	[0, π]
tan⁻¹ x	R (all real numbers)	(-π/2, π/2)
cosec⁻¹ x	R – (-1, 1) (i.e.,	x
sec⁻¹ x	R – (-1, 1) (i.e.,	x
cot⁻¹ x	R (all real numbers)	(0, π)

3. Properties of Inverse Trigonometric Functions:

These are essential for solving problems:

sin⁻¹(sin x) = x (if x is in the principal value branch of sin⁻¹)
cos⁻¹(cos x) = x (if x is in the principal value branch of cos⁻¹)
tan⁻¹(tan x) = x (if x is in the principal value branch of tan⁻¹)
Similar identities exist for cosec⁻¹, sec⁻¹, and cot⁻¹. Be careful about the restricted domains.
sin⁻¹(1/x) = cosec⁻¹ x
cos⁻¹(1/x) = sec⁻¹ x
tan⁻¹(1/x) = cot⁻¹ x (with adjustments for sign if x < 0)
sin⁻¹(-x) = -sin⁻¹ x
tan⁻¹(-x) = -tan⁻¹ x
cos⁻¹(-x) = π – cos⁻¹ x
tan⁻¹ x + cot⁻¹ x = π/2
sin⁻¹ x + cos⁻¹ x = π/2
Formulas for tan⁻¹ x + tan⁻¹ y, sin⁻¹ x ± sin⁻¹ y, cos⁻¹ x ± cos⁻¹ y. These are important but can be derived when needed.

4. Applications:

The chapter also includes problems involving simplifying expressions containing inverse trigonometric functions, often using trigonometric identities.

Key Points to Remember:

The principal value branches are essential. Always check if your angle is within the principal value range.
Trigonometric identities are frequently used in conjunction with inverse trigonometric functions.
Pay close attention to the domains and ranges.

Exercise 2.1

Find the principal values of the following:

1. Sin-1 (-1/2)

Ans :