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Inverse Trigonometric Functions

We will discuss here about Inverse trigonometric Functions or inverse circular functions.

The inverse of a function f: A β†’ B exists if and only if f is one-one onto (i.e., bijection) and given by

f(x) = y⇔ fβˆ’1 (y) = x.

Consider the sine function. Clearly, sin: R β†’ R given by sin ΞΈ = x for all ΞΈ ∈ R is a many-one into function. So, its inverse does not exist. If we restrict its domain to the interval [- Ο€2, Ο€2] then we may have infinitely many values of the angle ΞΈ which satisfy the equation sin ΞΈ = x i.e., sine of any one of these angles is equal to x. Here angle ΞΈ is represented as sinβˆ’1x which is read as sine inverse x or arc sin x. Therefore, the symbol sinβˆ’1x represents an angle and the sine of this angle has the value x.

Note the difference between sinβˆ’1x and sin ΞΈ: sinβˆ’1x represents an angle while sin ΞΈ represents a pure number; again, for a given value of x (- 1 ≀ x ≀ 1) we may have infinitely many vales of sinβˆ’1x i.e., sinβˆ’1x is a multiple-valued function; but a given value of ΞΈ gives a definite finite value of sin ΞΈ i.e., sin ΞΈ is a single-valued function. Thus, if x is a real number lying between -1 and 1, then sinβˆ’1 x is an angle between - Ο€2 and Ο€2 whose sine is x i.e.,

sinβˆ’1x = ΞΈ

⇔ x = sin ΞΈ, where - Ο€2  β‰€ x ≀ Ο€2 and - 1 ≀ x ≀ 1.

In the above discussion we have restricted the sine function to the interval [- Ο€2, Ο€2] to ake it a bijection. In fact we restrict the domain of sin ΞΈ to any of the interval [- Ο€2, Ο€2], [3Ο€2, 5Ο€2], [- 5Ο€2, -3Ο€2] etc. sin ΞΈ is one-one onto function with range [-1, 1]. We therefore conclude that each of these intervals we can define the inverse of sine function. Thus sinβˆ’1x is a function with domain [-1, 1] = {x ∈ R: - 1 ≀ x ≀ 1} and range [- Ο€2, Ο€2] or [3Ο€2, 5Ο€2] or [- 5Ο€2, -3Ο€2] and so on.

Similarly, if cos ΞΈ = x (- 1 ≀ x ≀ 1 ) then ΞΈ = cosβˆ’1x i.e., cosβˆ’1x (cos-inverse x) represents an angle and the cosine of this angle is equal to x. We have similar significances of the angles tanβˆ’1x (tan-inverse x), cotβˆ’1x (cot-inverse x), secβˆ’1x (sec-inverse x) and cscβˆ’1x (csc-inverse x).

Therefore, if sin ΞΈ = x (- 1 ≀ x ≀ 1) then ΞΈ = sinβˆ’1x; 

if cos ΞΈ = x (- 1 ≀ x ≀ 1) then ΞΈ = cosβˆ’1x ; 

if tan ΞΈ = x (- ∞ < x < ∞) then ΞΈ = tanβˆ’1x ;

if csc ΞΈ = x (I x I β‰₯ 1) then ΞΈ = cscβˆ’1x.

if sec ΞΈ = x (I x I β‰₯ 1) then ΞΈ = secβˆ’1x ; and

if cot ΞΈ = x (- ∞ < x < ∞) then ΞΈ = cotβˆ’1x ;

Conversely, sinβˆ’1x = ΞΈ β‡’ sin ΞΈ = x;

 cosβˆ’1x = ΞΈ β‡’ cos ΞΈ = x

tanβˆ’1x = ΞΈ β‡’ tan ΞΈ = x

cscβˆ’1x = ΞΈ β‡’ csc ΞΈ = x

cotβˆ’1x = ΞΈ β‡’ cot ΞΈ = x

The trigonometrical functions sinβˆ’1x, cosβˆ’1x, tanβˆ’1x, cotβˆ’1x, secβˆ’1x and cscβˆ’1x are called Inverse Circular Functions.

Note: It should be noted that sinβˆ’1x is not equal to (sin x)βˆ’1. Also noted that (sin x)βˆ’1is an angle whose sin is x. Remember that sinβˆ’1x is a circular function but (sin x )βˆ’1 is the reciprocal of sin x i.e., (sin x)βˆ’1 = 1/sin x and it represents a pure number.

● Inverse Trigonometric Functions






11 and 12 Grade Math

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