# Inverse Trigonometric Functions Notes | SwaVid Inverse Trigonometric Functions, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/inverse-trigonometric-functions Source: https://www.swavid.com/cbse/class-12/maths/chapter/inverse-trigonometric-functions # Inverse Trigonometric Functions ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 12 Mathematics on SwaVid ### Need for domain restriction ### Principal value branches ### Sine and cosine inverses ### Tangent and cotangent inverses ### Secant and cosecant inverses ### Next chapter ### Related chapter ### Related chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 12 › Mathematics › Inverse Trigonometric Functions Notes written by SwaVid from the NCERT textbook. Class 12 Mathematics, Chapter 2. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter defines inverse trigonometric functions by restricting the domains of basic trigonometric functions to make them bijective. Students learn the domain, principal value branch, and basic evaluation rules for each inverse function. Source: NCERT Mathematics Part I (Reprint 2026-27) , Chapter 2, Inverse Trigonometric Functions Trigonometric functions are periodic and not one-to-one on their natural domains. To define their inverses, their domains are restricted to intervals where they are both one-to-one and onto. For each inverse trigonometric function, one specific range interval is designated as the principal value branch. The value of an inverse function lying in this standard branch is called its principal value. The function sin⁻¹ x has domain [-1, 1] and principal value range [-π/2, π/2]. The function cos⁻¹ x has domain [-1, 1] and principal value range [0, π]. The function tan⁻¹ x has domain R and principal value range (-π/2, π/2). The function cot⁻¹ x has domain R and principal value range (0, π). The function sec⁻¹ x has domain R - (-1, 1) and principal value range [0, π] - {π/2}. The function cosec⁻¹ x has domain R - (-1, 1) and principal value range [-π/2, π/2] - {0}. Try each question first. Then tap it to see the answer. Let y = sin⁻¹(-1/2), which means sin y = -1/2 We know that sin(π/6) = 1/2, so sin y = -sin(π/6) = sin(-π/6) The principal value branch of sin⁻¹ is [-π/2, π/2] and -π/6 lies in this interval Therefore, the principal value is -π/6 -π/6 Let y = cos⁻¹(-1/√2), which means cos y = -1/√2 We know that cos(π/4) = 1/√2, so cos y = -cos(π/4) = cos(π - π/4) = cos(3π/4) The principal value branch of cos⁻¹ is [0, π] and 3π/4 lies in this interval Therefore, the principal value is 3π/4 3π/4 Check if 2π/3 lies in [-π/2, π/2]; it does not Rewrite using trigonometric identity: sin(2π/3) = sin(π - π/3) = sin(π/3) Check if π/3 lies in [-π/2, π/2]; it does sin⁻¹(sin(π/3)) = π/3 π/3 Matrices . Relations and Functions . Continuity and Differentiability . Integrals . The 13 chapters of Class 12 Mathematics . CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Matrices. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 2 NCERT book - 80 marks, theory paper - 3 quick checks - Treating sin⁻¹(x) as 1 / sin(x) . Remember that sin⁻¹ x denotes the inverse function, not an algebraic exponent. The reciprocal of sin x is written as (sin x)⁻¹ or cosec x. - Writing cos⁻¹(-x) = -cos⁻¹(x) . The range of cos⁻¹ x is [0, π], which has no negative values. Instead, let y = cos⁻¹(-x), which means cos y = -x, and find the angle y in [0, π]. - Assuming sin⁻¹(sin θ) = θ for any real number θ . The identity holds only when θ lies inside [-π/2, π/2]. For angles outside this interval, rewrite the angle using periodicity into the principal value branch first. ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 12](https://www.swavid.com/cbse/class-12) - [Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 13 chapters](https://www.swavid.com/cbse/class-12/maths/chapters) - [NCERT Mathematics Part I (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/matrices) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/relations-and-functions) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/continuity-and-differentiability) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/integrals) - [Class 12 Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup)