Inverse Trigonometry JEE Main: Complete Guide 2026

Inverse trigonometry is a moderate-weightage chapter in JEE Main Mathematics, contributing one to two questions per session. The chapter tests domain and range of inverse functions, principal values, key identities, and compositions of trigonometric and inverse trigonometric functions. The chapter rewards careful memorization of the restricted domain definitions — which are counterintuitive and error-prone without systematic learning — combined with fluency in applying the key identities. Once mastered, inverse trigonometry becomes a reliable mark-earner.

Domains, Ranges, and Principal Values

The six trigonometric functions are not one-to-one over their natural domains, so their inverses are defined on restricted domains where they are monotone and one-to-one. The principal value branches and their domains and ranges must be memorized precisely. Arcsin: domain [-1, 1], range [-π/2, π/2]. Arccos: domain [-1, 1], range [0, π]. Arctan: domain (-∞, ∞), range (-π/2, π/2). Arccot: domain (-∞, ∞), range (0, π). Arcsec: domain (-∞,-1]∪[1,∞), range [0,π] excluding π/2. Arccsc: domain (-∞,-1]∪[1,∞), range [-π/2,π/2] excluding 0.

JEE Main tests the evaluation of specific inverse trigonometric expressions — like arcsin(sin(5π/6)) — where the input is outside the restricted range and the answer is not simply the argument but must be converted to the principal value. These evaluations require knowing the domain and range precisely and applying the periodic properties of the trigonometric functions correctly. For the direct trigonometry context, connect with our trigonometry guide.

Key Identities and Composition Formulas

The fundamental identities relate arcsin and arccos (they sum to π/2), arctan and arccot (they sum to π/2), and arcsec and arccsc (they sum to π/2). These complementary identities are tested directly and as components of simplification problems. The relationship between arctan x and arctan(1/x) depends on whether x is positive or negative, and this conditional form is a reliable source of JEE Main questions testing whether students know the precise statement.

The addition formulas for arctan — expressing arctan x + arctan y in terms of arctan((x+y)/(1-xy)) — are powerful and appear in many JEE Main problems about simplifying sums of inverse tangents. The analogous formulas for arcsin and arccos are also tested. The double-angle and half-angle formulas for inverse trigonometric functions are tested less frequently but appear in more complex problems. Take a free mock test on inverse trigonometry to practice these identity applications.

Graphs of Inverse Trigonometric Functions

The graph of arcsin x is a portion of the curve y = sin⁻¹x with domain [-1,1] and range [-π/2, π/2] — increasing and S-shaped. The graph of arccos x has domain [-1,1] and range [0,π] — decreasing. The graph of arctan x has domain all reals and range (-π/2, π/2) — increasing with horizontal asymptotes at ±π/2. JEE Main tests these graphs through questions about whether a function is increasing or decreasing, about the behavior at boundary points, and about the sketch of the graph of a composition like arcsin(2x-1) or arctan(1/x).

Sketching the graph of simple compositions and transformations of inverse trigonometric functions is a skill tested in JEE Main through questions about the range of a composite expression, the maximum and minimum values of a function involving inverse trigonometry, and the number of solutions of an equation involving these functions.

Equations and Inequalities Involving Inverse Trigonometry

Solving equations and inequalities involving inverse trigonometric functions requires careful attention to the domains of the expressions involved. Common JEE Main problem types: finding the number of real solutions of an equation like arcsin x = arccos x or 2 arcsin x = π/6, and solving inequalities like arctan x > π/4. These problems test both the key relationships between inverse functions and the ability to reason correctly about the restricted domains.

For a complete math preparation, this chapter connects directly to our trigonometry guide and our sets, relations, and functions guide (for the domain and inverse function concepts). Follow our 30-day math plan and sign up free to access our inverse trigonometry question bank with 100+ solved problems.