Subscribe to our βΆοΈ YouTube channel π΄ for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
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) = yβ fβ1 (y) = x.
Consider the sine function. Clearly, sin: R β R given by sin ΞΈ = x for all ΞΈ β R is a many-one into function. So, its inverse does not exist. If we restrict its domain to the interval [- Ο2, Ο2] 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 sinβ1x which is read as sine inverse x or arc sin x. Therefore, the symbol sinβ1x represents an angle and the sine of this angle has the value x.
Note the difference between sinβ1x
and sin ΞΈ: sinβ1x represents an
angle while sin ΞΈ represents a
pure number; again, for a given value of x (- 1 β€ x β€ 1) we may have infinitely many vales of sinβ1x
i.e., sinβ1x 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 sinβ1 x is an angle between - Ο2
and Ο2 whose sine is x i.e.,
sinβ1x = ΞΈ
β x = sin ΞΈ, where - Ο2 β€ x β€ Ο2 and - 1 β€ x β€ 1.
In the above discussion we have restricted the sine function to the interval [- Ο2, Ο2] to ake it a bijection. In fact we restrict the domain of sin ΞΈ to any of the interval [- Ο2, Ο2], [3Ο2, 5Ο2], [- 5Ο2, -3Ο2] 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 sinβ1x is a function with domain [-1, 1] = {x β R: - 1 β€ x β€ 1} and range [- Ο2, Ο2] or [3Ο2, 5Ο2] or [- 5Ο2, -3Ο2] and so on.
Similarly, if cos ΞΈ = x (- 1 β€ x β€ 1 ) then ΞΈ = cosβ1x i.e., cosβ1x (cos-inverse x) represents an angle and the cosine of this angle is equal to x. We have similar significances of the angles tanβ1x (tan-inverse x), cotβ1x (cot-inverse x), secβ1x (sec-inverse x) and cscβ1x (csc-inverse x).
Therefore, if sin ΞΈ = x (- 1 β€ x β€ 1) then ΞΈ = sinβ1x;
if cos ΞΈ = x (- 1 β€ x β€ 1) then ΞΈ = cosβ1x ;
if tan ΞΈ = x (- β < x < β) then ΞΈ = tanβ1x ;
if csc ΞΈ = x (I x I β₯ 1) then ΞΈ = cscβ1x.
if sec ΞΈ = x (I x I β₯ 1) then ΞΈ = secβ1x ; and
if cot ΞΈ = x (- β < x < β) then ΞΈ = cotβ1x ;
Conversely, sinβ1x = ΞΈ β sin ΞΈ = x;
cosβ1x = ΞΈ β cos ΞΈ = x
tanβ1x = ΞΈ β tan ΞΈ = x
cscβ1x = ΞΈ β csc ΞΈ = x
cotβ1x = ΞΈ β cot ΞΈ = x
The trigonometrical functions sinβ1x, cosβ1x, tanβ1x, cotβ1x, secβ1x and cscβ1x are called Inverse Circular Functions.
Note: It should be noted that sinβ1x is not equal to (sin x)β1. Also noted that (sin x)β1is an angle whose sin is x. Remember that sinβ1x is a circular function but (sin x )β1 is the reciprocal of sin x i.e., (sin x)β1 = 1/sin x and it represents a pure number.
β Inverse Trigonometric Functions
11 and 12 Grade Math
From General solution of Trigonometric Equation to HOME PAGE
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.
Jul 25, 25 12:21 PM
Jul 25, 25 03:15 AM
Jul 24, 25 03:46 PM
Jul 23, 25 11:37 AM
Jul 20, 25 10:22 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.