Seshu cheera

Chapter 2: Inverse Trigonometric Functions (Class 12 CBSE Math) ✅ A. Key Concepts & Formulas Definition & Principal Values Inverse trig functions are the restricted inverses of trig functions: sin ⁡ − 1 x \sin^{-1} x : domain x ∈ [ − 1 , 1 ] x \in [-1,1] ; range [ − π / 2 , π / 2 ] [-π/2, π/2] cos ⁡ − 1 x \cos^{-1} x : domain [ − 1 , 1 ] [-1,1] ; range [ 0 , π ] [0, π] tan ⁡ − 1 x \tan^{-1} x : domain all real; range ( − π / 2 , π / 2 ) (-π/2, π/2) Similar definitions apply for cot ⁡ − 1 , sec ⁡ − 1 , csc ⁡ − 1 \cot^{-1}, \sec^{-1}, \csc^{-1} . Properties & Identities Odd–even behavior: sin ⁡ − 1 ( − x ) = − sin ⁡ − 1 x \sin^{-1}(-x)=-\sin^{-1}x , cos ⁡ − 1 ( − x ) = π − cos ⁡ − 1 x \cos^{-1}(-x)=π-\cos^{-1}x , etc. Complementary relations: sin ⁡ − 1 x + cos ⁡ − 1 x = π / 2 \sin^{-1}x + \cos^{-1}x = π/2 ; tan ⁡ − 1 x + cot ⁡ − 1 x = π / 2 \tan^{-1}x + \cot^{-1}x = π/2 . Reciprocal links: sin ⁡ − 1 ( 1 / a ) = csc ⁡ − 1 ( a ) \sin^{-1}(1/a)...

   Unit I – Relations & Functions from Class 12 CBSE Maths , including deeper insights , conceptual clarity , exercise solutions , and top-scoring CBSE answers : 🧠 1. In-Depth Concept Review 📌 Relations A relation on a set A A is any subset of A × A A \times A . Key types include : Empty relation : no pair included. Universal relation : all possible pairs included. ✅ Reflexive : ∀ x x , ( x , x ) ∈ R (x,x) \in R . ✅ Symmetric : if ( x , y ) ∈ R (x,y) \in R , then ( y , x ) ∈ R (y,x) \in R . ✅ Transitive : if ( x , y ) , ( y , z ) ∈ R (x,y), (y,z) \in R , then ( x , z ) ∈ R (x,z)\in R . ✅ Equivalence relation : satisfies all three; partitions A A into equivalence classes . 📌 Functions A relation is a function if each element in the domain corresponds to exactly one value in the codomain. Types of functions: Injective (one‑to‑one) : distinct inputs yield distinct outputs. Surjective (onto) : every element in the codomai...