JEE Maths›Inverse Trigonometric Functions›Question Types

Inverse Trigonometric Functions - How It Appears in JEE

8 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Find the Principal Value

Pattern

Find the principal value of an inverse trig expression like sin inverse(sin x) or cos inverse(cos x)

How to recognize

Asks for the value of sin inverse(sin(7pi/6)), cos inverse(cos(-pi/3)), or similar compositions where the argument is outside the principal range.

→ First check if the argument is in the principal range. If not, use periodicity and symmetry to bring it into the principal range. sin inverse(sin x) = x only when x is in [-pi/2, pi/2].

Apply Complementary Pair Identities

Pattern

Use sin inverse(x) + cos inverse(x) = pi/2 or similar pair identities to simplify expressions

How to recognize

Expression contains both sin inverse and cos inverse of the same argument, or tan inverse and cot inverse of the same argument.

→ Replace one function using the pair identity. For example, cos inverse(x) = pi/2 - sin inverse(x). This simplifies the expression to a single inverse trig function.

Simplify Using Substitution

Pattern

Simplify expressions like sin inverse(2x*sqrt(1-x^2)) or tan inverse((3x-x^3)/(1-3x^2)) using trig substitution

How to recognize

The expression inside the inverse trig function matches a double-angle or triple-angle trig identity.

→ Substitute x = sin(theta), x = cos(theta), or x = tan(theta) as appropriate. The expression inside becomes a multiple-angle formula, and the inverse cancels with the forward function. Check domain restrictions carefully.

Evaluate Composition of Trig and Inverse Trig

Pattern

Find the value of sin(cos inverse x), tan(sin inverse x), or similar compositions

How to recognize

A trig function applied to an inverse trig function, like sin(tan inverse(3/4)) or cos(2*sin inverse(1/3)).

→ Draw a right triangle. If theta = sin inverse(x), then sin(theta) = x and cos(theta) = sqrt(1-x^2). Use this triangle to find any trig ratio of theta.

Solve Inverse Trig Equations

Pattern

Solve equations involving inverse trig functions like tan inverse(x) + tan inverse(2x) = pi/4

How to recognize

An equation with inverse trig functions set equal to a constant or another inverse trig function.

→ Use the sum/difference formulas for tan inverse. Apply tan to both sides after combining. Solve the resulting algebraic equation. Always verify solutions satisfy domain restrictions.

Sum of Inverse Trig Series

Pattern

Find the sum of a series of inverse trig terms, typically telescoping tan inverse series

How to recognize

A summation involving tan inverse terms with arguments like 1/(1+r+r^2), 1/(2r^2), or similar patterns.

→ Express each term as a difference using tan inverse(a) - tan inverse(b) = tan inverse((a-b)/(1+ab)). The series telescopes, leaving only the first and last terms.

Graph and Range Problems

Pattern

Determine the range, domain, or graph properties of inverse trig expressions

How to recognize

Asks for the range of f(x) = sin inverse(x^2 - 1) or the domain of cos inverse(2x - 1), or asks about graph properties.

→ For domain: solve the inequality so the argument lies in the valid domain (e.g., -1 <= 2x-1 <= 1). For range: the output is restricted to the principal value range, further narrowed by the argument's range.

Sum and Difference of Two Inverse Trig Values

Pattern

Find tan inverse(x) + tan inverse(y) or sin inverse(x) + sin inverse(y) with specific values

How to recognize

Asks to evaluate or simplify the sum or difference of two specific inverse trig values.

→ For tan inverse: use the sum formula with the xy < 1 or xy > 1 condition. For sin inverse: use sin inverse(x) + sin inverse(y) = sin inverse(x*sqrt(1-y^2) + y*sqrt(1-x^2)) when the conditions are met.

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