Inverse Trigonometric Functions Class 12 Chapter Notes

Must remember

• Inverse trigonometric functions are defined as single-valued functions only after restricting the range of original trig functions to their principal value branches.
• sin⁻¹x, cos⁻¹x, sec⁻¹x and cosec⁻¹x have restricted domains; tan⁻¹x and cot⁻¹x are defined for all real x.
• sin⁻¹x means the angle y in [-π/2, π/2] such that sin y = x; it is not 1/sin x.
• Core complementary identities: sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2, with valid domain conditions.
• Odd functions: sin⁻¹(-x), tan⁻¹(-x), cosec⁻¹(-x) equal the negative of the original value.
• Non-odd transformations: cos⁻¹(-x) = π - cos⁻¹x, cot⁻¹(-x) = π - cot⁻¹x, sec⁻¹(-x) = π - sec⁻¹x.
• For f⁻¹(f(x)), first check whether x lies in the principal range; otherwise reduce the angle to an equivalent principal value.
• In tan⁻¹ addition formulas, branch correction may be needed when xy > 1; do not blindly apply tan⁻¹((x + y)/(1 - xy)).
• Board answers should mention range restrictions when defining inverse functions or proving identities.

Key definitions

Inverse trigonometric function

A function that gives the principal angle corresponding to a trigonometric ratio, such as sin⁻¹x giving y when sin y = x.

Principal value

The unique angle chosen from a fixed interval so that an inverse trigonometric function becomes single-valued.

Domain

The set of all permissible input values of a function; for sin⁻¹x and cos⁻¹x it is [-1, 1].

Range

The set of output values of a function; for tan⁻¹x it is (-π/2, π/2).

One-one function

A function in which distinct inputs have distinct outputs; trigonometric functions are restricted to one-one intervals before taking inverse.

Branch

A selected interval of the original trigonometric function on which it is one-one and hence invertible.

Composite inverse expression

An expression like sin⁻¹(sin x) or cos(cos⁻¹x), where domain and principal range decide the simplification.

Branch correction

The adjustment by π or -π in inverse tangent formulas to keep the final value in the correct principal range.

Concept: why principal ranges are needed

Trigonometric functions such as sin x, cos x and tan x are many-one over R because the same value repeats for infinitely many angles. For example, sin x = 1/2 at π/6, 5π/6, 13π/6 and many more angles. So a direct inverse would not be a function unless we choose exactly one output angle. CBSE uses principal value branches to make inverse trigonometric functions single-valued. Thus sin⁻¹(1/2) is π/6, not 5π/6, because π/6 lies in the principal range [-π/2, π/2]. Similarly cos⁻¹(-1/2) is 2π/3 because cos inverse has range [0, π]. Most Chapter 2 questions are not difficult algebraically; they test whether you remember the allowed output interval. Always ask: what is the domain of the inverse function, and what range must the final angle belong to?

Domain and principal range table

Formula sheet: identities used in CBSE questions

How to simplify inverse trig expressions

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Short worked examples

• Evaluate sin⁻¹(1/2) + cos⁻¹(-1/2). sin⁻¹(1/2) = π/6 because π/6 ∈ [-π/2,π/2]. cos⁻¹(-1/2) = 2π/3 because 2π/3 ∈ [0,π]. Sum = π/6 + 2π/3 = 5π/6.
• Evaluate tan⁻¹1 + tan⁻¹2 + tan⁻¹3. tan⁻¹1 + tan⁻¹2 has xy = 2 > 1 and both are positive, so it equals π + tan⁻¹((1+2)/(1-2)) = π + tan⁻¹(-3) = π - tan⁻¹3. Adding tan⁻¹3 gives π.
• Simplify cos⁻¹(cos 5π/3). cos⁻¹ always returns a value in [0,π]. Since cos 5π/3 = 1/2 and the principal angle with cosine 1/2 in [0,π] is π/3, the answer is π/3.

Board exam traps and marking cues

Do not write inverse trigonometric values without checking principal range. For example, sin⁻¹(sin 5π/6) is not 5π/6 because 5π/6 is outside [-π/2,π/2]; the correct value is π/6. Similarly, cos⁻¹(cos θ) is equal to θ only when θ ∈ [0,π]. In proof questions, start with a substitution such as y = sin⁻¹x, mention the allowed interval for y, and then use ordinary trigonometric identities. In tan⁻¹ formulas, show branch correction when xy > 1; this often carries method marks. Avoid writing cosec⁻¹x as 1/sin⁻¹x or sec⁻¹x as 1/cos⁻¹x.

Quick revision checks

• Q: What is the principal range of cos⁻¹x? A: [0, π].
• Q: Evaluate sin⁻¹(-√3/2). A: -π/3, because it lies in [-π/2, π/2].
• Q: Is tan⁻¹x defined for every real x? A: Yes, domain of tan⁻¹x is R and range is (-π/2, π/2).
• Q: Simplify cos⁻¹(cos 7π/6). A: 5π/6, because cos 7π/6 = -√3/2 and cos⁻¹(-√3/2) = 5π/6.

Frequently Asked Questions