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Inverse trigonometric functions are a short chapter with a high error rate. The formulas fit on one page, but each comes with a condition, and JEE questions are written to test exactly those conditions. A student who writes sin⁻¹(sin 2) = 2 has used a formula outside its range. This post gives you the principal ranges, a method for undoing "inverse of a function" expressions, and the identities with their conditions, followed by a practice routine.
What does the syllabus include?
The JEE (Main) 2026 syllabus lists "inverse trigonometrical functions their properties" under Trigonometry, and differentiation of inverse trigonometric functions under calculus. The NCERT Class 12 chapter on inverse trigonometric functions defines the principal value branches and their properties. Check the syllabus for your exam year.
Why do these functions need principal ranges?
Sine is not one-one on all real numbers: sin(π/6) and sin(5π/6) are both 1/2. To define an inverse, we restrict sine to an interval where it is one-one and takes every value from -1 to 1 once. That interval is the principal value branch. NCERT's summary table gives:
sin⁻¹x: domain [-1, 1], range [-π/2, π/2]
cos⁻¹x: domain [-1, 1], range [0, π]
tan⁻¹x: domain all reals, range (-π/2, π/2)
cot⁻¹x: domain all reals, range (0, π)
sec⁻¹x: domain outside (-1, 1), range [0, π] without π/2
cosec⁻¹x: domain outside (-1, 1), range [-π/2, π/2] without 0
Learn these with a unit circle in front of you, not as a list. The research backs that up. Keith Weber (2005) compared two college trigonometry courses. Students in a lecture-based course developed a very limited understanding of trigonometric functions, while students taught with an experimental approach that treated sine and cosine as processes, supported by unit-circle activities, developed a deep understanding.
Also note: sin⁻¹x is not (sin x)⁻¹. The second is 1/sin x, which is cosec x.
How do you simplify sin⁻¹(sin x) when x is outside the range?
The rule is sin⁻¹(sin x) = x only when x is in [-π/2, π/2]. Otherwise, find the angle in the principal range with the same sine.
Worked example 1: sin⁻¹(sin(2π/3)). 2π/3 is outside [-π/2, π/2]. Since sin(π - θ) = sin θ, sin(2π/3) = sin(π/3), and π/3 is in range. Answer: π/3.
Worked example 2: sin⁻¹(sin 2). Here 2 is in radians, about 2 × 57.3° ≈ 114.6°, which is more than π/2 ≈ 1.571. Use the same shift: π - 2 ≈ 1.142, which lies in range. Answer: π - 2.
Worked example 3: cos⁻¹(cos(7π/6)). 7π/6 is outside [0, π]. Since cos(2π - θ) = cos θ, cos(7π/6) = cos(5π/6), and 5π/6 is in [0, π]. Answer: 5π/6. Check: cos(5π/6) = -√3/2 = cos(7π/6).
Worked example 4: tan⁻¹(tan(3π/4)). Tangent has period π, so tan(3π/4) = tan(3π/4 - π) = tan(-π/4). Answer: -π/4.
Which identities matter, and when do they apply?
sin⁻¹x + cos⁻¹x = π/2, for x in [-1, 1]. Same for tan⁻¹x + cot⁻¹x = π/2, for all real x.
sin⁻¹(-x) = -sin⁻¹x and tan⁻¹(-x) = -tan⁻¹x, but cos⁻¹(-x) = π - cos⁻¹x. Students often give cos⁻¹ the odd-function rule by mistake.
tan⁻¹x + tan⁻¹y = tan⁻¹((x + y)/(1 - xy)), valid when xy < 1. If x > 0, y > 0 and xy > 1, add π.
Worked example 5: tan⁻¹(1/2) + tan⁻¹(1/3). Here xy = 1/6 < 1, so the formula applies directly: (1/2 + 1/3)/(1 - 1/6) = (5/6)/(5/6) = 1. Answer: tan⁻¹1 = π/4.
Worked example 6: tan⁻¹2 + tan⁻¹3. Now xy = 6 > 1 with both positive. The formula gives tan⁻¹((5)/(1 - 6)) = tan⁻¹(-1) = -π/4, which cannot be right, since both angles are positive. Add π: the answer is π - π/4 = 3π/4. A calculator check agrees: tan⁻¹2 ≈ 1.107 and tan⁻¹3 ≈ 1.249 add to about 2.356, which is 3π/4.
Common slips and how to avoid them
Writing sin⁻¹(sin x) = x without checking the range.
Treating cos⁻¹ as an odd function.
Using the tan⁻¹ addition formula when xy > 1.
Mixing degrees and radians, as in sin⁻¹(sin 2).
Forgetting the domain [-1, 1] for sin⁻¹ and cos⁻¹ in a "find the domain" question.
A practice routine
Draw the six principal ranges on unit circles from memory.
Do ten "inverse of function" simplifications, including two with plain radian values like 2 or 5.
Do five tan⁻¹ sums, at least two with xy > 1, and check each with a calculator.
Mix in domain questions such as the domain of sin⁻¹(2x - 1).
This chapter uses ideas from two others: trigonometric identities and domain and range of functions. When you are stuck on which branch a question needs, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Today, simplify sin⁻¹(sin 5) and cos⁻¹(cos 5) on paper, then check both with a calculator.
