Class 12 Maths - CBSE
Inverse Trigonometric Functions
The chapter 'Inverse Trigonometric Functions' in Class 12 CBSE Mathematics extends the study of basic trigonometry by defining the inverses of sine, cosine, tangent, and other trigonometric functions. Students learn about domain and principal value branches, which are essential because trigonometric functions are many-to-one and need restricted domains to be invertible. This chapter is fundamental for calculus, particularly in evaluating complex integrals and derivatives. In CBSE board exams, it carries a weightage of around 4 to 6 marks, featuring questions ranging from basic evaluation of principal values to solving complex algebraic equations involving inverse functions.
Start Learning FreeKey Concepts
Restriction of Domain
Trigonometric functions are not one-to-one over their natural domains, so we must restrict their domains to specific intervals (principal value branches) to make them invertible.
Principal Value Branch
The restricted range of values that an inverse trigonometric function can output, such as [-π/2, π/2] for sin⁻¹(x).
Properties of Inverse Trigonometric Functions
Identities and formulas relating reciprocal arguments (like cosec⁻¹(x) = sin⁻¹(1/x)) and negative arguments (like sin⁻¹(-x) = -sin⁻¹(x)).
Sum and Difference Identities
Formulas that combine two inverse trigonometric functions into a single function, such as tan⁻¹(x) + tan⁻¹(y) = tan⁻¹((x+y)/(1-xy)).
Important Formulas
Board Exam Info
In the CBSE Class 12 Mathematics board exam, Inverse Trigonometric Functions typically carries 4 to 6 marks. Questions usually include 1-mark multiple-choice questions (MCQs) or short-answer questions based on finding the principal value, and 3 to 4-mark long-answer questions requiring the simplification of expressions or solving equations using inverse trigonometric identities.
Frequently Asked Questions
Why do we need principal value branches for inverse trigonometric functions?
Trigonometric functions are periodic and many-to-one, meaning a single output value corresponds to infinitely many inputs. Restricting the domain makes the function one-to-one and onto, ensuring a unique inverse.
How do I know whether to use π - cos⁻¹(x) or -cos⁻¹(x) for negative inputs?
For cos⁻¹, sec⁻¹, and cot⁻¹, the principal value branch lies in the second quadrant (0 to π), which is why we subtract the positive value from π when the input is negative.
Are inverse trigonometric functions the same as reciprocal functions like (sin x)⁻¹?
No, sin⁻¹(x) represents the angle whose sine is x (arcsine), whereas (sin x)⁻¹ equals 1/sin(x), which is cosecant (cosec x).
Learn Inverse Trigonometric Functions with Your AI Tutor
10 different ways to study this chapter. Free for 3 chapters per day.
Lecture
Key Points
Interactive
Quiz
Flashcards