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## Kerala Plus Two Maths Notes Chapter 2 Inverse Trigonometric Functions

Introduction

Trigonometric functions are real functions which are not objective and thus its inverse does not exist. In this chapter we study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverse and observe its graphical peculiarities.

A. Concepts

I. Functions

sin^{-1} x : [-1, 1 ] → [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)]

cos^{-1} x: [-1, 1] → [0, π]

tan^{-1} x : R → \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

cosec^{-1} x : R – (-1, 1) → [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)] – {0}

sec^{-1} x : R -(-1, 1) → [0, π] – {\(\frac{\pi}{2}\)}

cot^{-1} x : R → (0, π)

II. Properties

1. sin (sin^{-1} x) = x, x ∈ [-1, 1]

sin^{-1}(sinx) = x, x ∈ [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)]

cos(cos^{-1} x) = x, x ∈ [-1, 1]

cos^{-1}(cosx) = x, x ∈ [o, π]

tan(tan^{-1} x) = x, x ∈ R

tan^{-1}(tan x) = x, x ∈ \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

2. sin^{-1}(-x) = -sin^{-1}(x), x ∈ [-1, 1]

tan^{-1}(-x) = -tan^{-1}(x), x ∈ R

cosec^{-1}(-x) = -cosec^{-1}(x), x ∈ R -(-1, 1)

cos^{-1}(-x) = π – cos^{-1}(x), x ∈ [-1, 1]

cot^{-1}(-x) = π – cot^{-1}(x), x ∈ R

sec^{-1}(-x) = π – sec^{-1}(x), x ∈ R -(-1, 1)

sin^{-1}(x) + cos^{-1}(x) = \(\frac{\pi}{2}\), x ∈ [-1, 1].

3. cosec^{-1}(x) + sec^{-1}(x) = \(\frac{\pi}{2}\), |x| ≥ 1

tan^{-1}(x) + cot^{-1}(x) = \(\frac{\pi}{2}\), x ∈ R

4. sin^{-1} x

5. cos^{-1} x

6. tan^{-1}(x) + tan^{-1}(y) =

7. tan^{-1}(x) – tan^{-1}(y) =

8. 2 tan^{-1} x

9. sin^{-1} x ± sin^{-1} y