# Inverse Trigonometric Functions

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

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

f(x) = y⇔ f$$^{-1}$$ (y) = x.

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

Note the difference between sin$$^{-1}$$x and sin θ: sin$$^{-1}$$x represents an angle while sin θ represents a pure number; again, for a given value of x (- 1 ≤ x ≤ 1) we may have infinitely many vales of sin$$^{-1}$$x i.e., sin$$^{-1}$$x is a multiple-valued function; but a given value of θ gives a definite finite value of sin θ i.e., sin θ is a single-valued function. Thus, if x is a real number lying between -1 and 1, then sin$$^{-1}$$ x is an angle between - $$\frac{π}{2}$$ and $$\frac{π}{2}$$ whose sine is x i.e.,

sin$$^{-1}$$x = θ

⇔ x = sin θ, where - $$\frac{π}{2}$$  ≤ x ≤ $$\frac{π}{2}$$ and - 1 ≤ x ≤ 1.

In the above discussion we have restricted the sine function to the interval [- $$\frac{π}{2}$$, $$\frac{π}{2}$$] to ake it a bijection. In fact we restrict the domain of sin θ to any of the interval [- $$\frac{π}{2}$$, $$\frac{π}{2}$$], [$$\frac{3π}{2}$$, $$\frac{5π}{2}$$], [- $$\frac{5π}{2}$$, -$$\frac{3π}{2}$$] etc. sin θ 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 sin$$^{-1}$$x is a function with domain [-1, 1] = {x ∈ R: - 1 ≤ x ≤ 1} and range [- $$\frac{π}{2}$$, $$\frac{π}{2}$$] or [$$\frac{3π}{2}$$, $$\frac{5π}{2}$$] or [- $$\frac{5π}{2}$$, -$$\frac{3π}{2}$$] and so on.

Similarly, if cos θ = x (- 1 ≤ x ≤ 1 ) then θ = cos$$^{-1}$$x i.e., cos$$^{-1}$$x (cos-inverse x) represents an angle and the cosine of this angle is equal to x. We have similar significances of the angles tan$$^{-1}$$x (tan-inverse x), cot$$^{-1}$$x (cot-inverse x), sec$$^{-1}$$x (sec-inverse x) and csc$$^{-1}$$x (csc-inverse x).

Therefore, if sin θ = x (- 1 ≤ x ≤ 1) then θ = sin$$^{-1}$$x;

if cos θ = x (- 1 ≤ x ≤ 1) then θ = cos$$^{-1}$$x ;

if tan θ = x (- ∞ < x < ∞) then θ = tan$$^{-1}$$x ;

if csc θ = x (I x I ≥ 1) then θ = csc$$^{-1}$$x.

if sec θ = x (I x I ≥ 1) then θ = sec$$^{-1}$$x ; and

if cot θ = x (- ∞ < x < ∞) then θ = cot$$^{-1}$$x ;

Conversely, sin$$^{-1}$$x = θ ⇒ sin θ = x;

cos$$^{-1}$$x = θ ⇒ cos θ = x

tan$$^{-1}$$x = θ ⇒ tan θ = x

csc$$^{-1}$$x = θ ⇒ csc θ = x

cot$$^{-1}$$x = θ ⇒ cot θ = x

The trigonometrical functions sin$$^{-1}$$x, cos$$^{-1}$$x, tan$$^{-1}$$x, cot$$^{-1}$$x, sec$$^{-1}$$x and csc$$^{-1}$$x are called Inverse Circular Functions.

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

Inverse Trigonometric Functions

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Arranging Numbers | Ascending Order | Descending Order |Compare Digits

Sep 15, 24 04:57 PM

We know, while arranging numbers from the smallest number to the largest number, then the numbers are arranged in ascending order. Vice-versa while arranging numbers from the largest number to the sma…

2. ### Counting Before, After and Between Numbers up to 10 | Number Counting

Sep 15, 24 04:08 PM

Counting before, after and between numbers up to 10 improves the child’s counting skills.

3. ### Comparison of Three-digit Numbers | Arrange 3-digit Numbers |Questions

Sep 15, 24 03:16 PM

What are the rules for the comparison of three-digit numbers? (i) The numbers having less than three digits are always smaller than the numbers having three digits as:

4. ### 2nd Grade Place Value | Definition | Explanation | Examples |Worksheet

Sep 14, 24 04:31 PM

The value of a digit in a given number depends on its place or position in the number. This value is called its place value.