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# Theory Part 1

**DEFINITION **

__If a function is one to one and onto from A to B, then function g which associates each element y ∈ B to one and only one element x ∈ A, such that y = f(x), then g is called the inverse function of f denoted by x = g(y). __

∴ x = f^{-1} (y).

If cos θ = x, then θ may be any angle whose cosine is x, and we write θ = cos^{-1} x. It means that θ is an angle whose cosine is x.

Thus, sin^{-1} (1) is an angle, whose sine is (1), i.e. θ = sin^{-1} (1) = n π + (-1)^{n} (π/2)

Where, (π/2) is the least positive value of θ.

The functions sin^{-1} x, cos^{-1} x, tan^{-1} x, cot^{-1} x, cosec^{-1} and sec^{-1} x are called **inverse circular or inverse trigonometric functions.**

__Every inverse circular function is multi-valued. __To make each inverse circular function single valued, we define **principal value**.

**Function Domain Range (Principal Values)**

sin^{-1 }x [-1, 1] [(-π/2), (π/2)]

cos^{-1 }x [-1, 1] [0, π]

tan^{-1 }x **R **[(-π/2), (π/2)]

cot^{-1 }x** R** (0, π)

sec^{-1 }x **R**-(-1, 1). [0, π] - { π/2}

cosec^{-1 }x **R**-(-1, 1) [(-π/2), (π/2)] - {0}

**Note:**

(a) 1^{st} quadrant is common to the range of all the inverse functions.

(b) 3^{rd} quadrant is not used in inverse functions.

(c) 4^{th} quadrant is used in the clockwise direction i.e. -π/2 ≤y ≤0

(d) No inverse function is periodic.

(e) The inverse trigonometric functions are also written arc sinx, arc cosx etc.

**GRAPHS OF INVERSE TRIGONOMETRIC FUNCTIONS**

1. θ = sin^{-1} x, where θ ∈ [(-π/2), (π/2)] and x ∈ [-1, 1]

2. θ = cosec^{-1} x, where θ ∈ and x ∈ (-∞, -1] υ [1, ∞)

**Note: **cosec^{-1} x is a decreasing function in (-∞, -1]. It also decreases in [1, ∞)

3.** θ = cos ^{-1} x, where θ ∈ [0, π] and x ∈ [-1, 1]**

**Note: **cos^{-1} x is a decreasing function in [-1, 1]

4. θ = sec^{-1} x, here θ ∈ and x ∈ (-∞, -1] υ [1, ∞)

**Note: **sec^{-1} x is an increasing function in (-∞, -1]. It also increases in [1, ∞)

5. θ = tan^{-1} x, where θ ∈ [(-π/2), (π/2)] and x ∈ [-∞, ∞]

**Note: **tan^{-1} x is an increasing function in R.

6. θ = cot^{-1} x, where θ ∈ (0, π) and x ∈ [-∞, ∞]

**Note: **tan^{-1} x is an decreasing function in R.

**PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS **

**Property I: **

**(i) **sin^{-1} (sin x) = x; for all x ∈ [(-π/2), (π/2)]

Let y = sin^{-1} (sin x), x ∈ R, y ∈ [(-π/2), (π/2)], periodic with period 2π

**(ii) **cos^{-1} (cos x) = x; for all x ∈ [0, π] :

Let y = cos^{-1} (cos x), x ∈ R, y ∈ [0, π], periodic with period 2π

**(iii) **tan^{-1} (tan x) = x; for all x ∈ [(-π/2), (π/2)] :

Let y = tan^{-1} (tan x), x ∈ R –, periodic with period π

**(iv) **cosec^{-1} (cosec x) = x; for all x ∈ [(-π/2), (π/2)], x ≠ for other values of x see the graph:

Let y = cosec^{-1} (cosec x), x ∈ R, -{nπ, n ∈ I}, y ∈ y is periodic with period 2π

**vii)** sec^{-1} (sec x) = x; for all x ∈ [0, π], x ≠ (π/2) for other values of x see the graph:

Let y = sec^{-1} (sec x), y is periodic; x ∈ R –, with period 2π

**(vi) **cot^{-1} (cot x) = x; for all x ∈ (0, π), for other values of x see the graph:

Let y = cot^{-1} (cot x), x ∈ R - {nπ}, y ∈ (0, π), periodic with π