Odisha State Board CHSE Odisha Class 12 Math Notes Chapter 2 Inverse Trigonometric Functions will enable students to study smartly.

## CHSE Odisha 12th Class Math Notes Chapter 2 Inverse Trigonometric Functions

**Definitions:**

Let us first consider the function

Sin : R → [-1, 1]

Let y = Sin x, x ∈ R. Look at the graph of sin x. For y ∈ [-1, 1], there is a unique number x in each of the intervals…, \(\left[-\frac{3 \pi}{2},-\frac{\pi}{2}\right],\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right], \ldots\) such that y = sin x.

Hence any one of these intervals can be chosen to make sine function bijective.

We usually choose \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) as the domain of sine function. Thus sin: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) → [-1, 1] is bijective and hence admits of an inverse function with range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) denoted by sin^{-1} or arcsin (see footnote).

Each of the above-mentioned intervals as range gives rise to different branches of sin^{-1} function. The function sin^{-1} with the range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) is called the principal branch which is defined below.

sin^{-1} : [-1, 1] → \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) defined by y = sin^{-1} x ⇔ x = sin y.

The values of y (=sin^{-1} x) in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) are called principal values of sin^{-1}.

Similar considerations for other trigonometric functions give rise to respective inverse functions. We define below the principal branches of cos^{-1}, tan^{-1} and cot^{-1}.

cos^{-1} : [-1, 1] → [0, π] defined by

y = cos^{-1} x ⇔ x = cos y

tan^{-1} : R → \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) defined by

y = tan^{-1} x ⇔ x = tan y

cot^{-1} : R → (0, π) defined by

y = cot^{-1} x ⇔ x = cot y

**Important Properties**

Property – I

We know that when

f : X → Y is invertible then fof^{-1} = I_{y} and f^{-1}of = I_{x}.

Applying this we have.

(i) sin (sin^{-1} x) = x, x ∈ [-1, 1]

cos (cos^{-1} x) = x, x ∈ [-1, 1]

tan (tan^{-1} x) = x, x ∈ R

cot (cot^{-1} x) = x, x ∈ R

sec (sec^{-1} x) = x, x ∈ R -(-1, 1)

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

Property – II

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

(ii) cosec^{-1}(-x) = -cosec^{-1} x , |x| ≥ 1

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