Graphic Representation Of Inverse Trigonometric Function

While preparing for JEE Mains, JEE Advanced or 12th Boards, Inverse Trigonometric Function is one of the important topics, which when studied thoroughly is very interesting. Before we understand Graphic representation of Inverse Trigonometric Function, let us focus on Inverse Trigonometric function.

Inverse Trigonometric Function Formula

Function Domain Range of an Inverse Function
sin-1x(arcsinex) -1≤ x ≤1 -π/2≤y≤π/2
cos-1x(arcosinex) -1≤ x ≤1 0≤y ≤π
tan-1x(arctangentx) – ∞ < x <  ∞ -π/2<y<π/2
cot-1x(arcotangentx) – ∞ < x <  ∞ 0<y<π
sec-1x(arcsecantx) – ∞ ≤ x ≤-1 or 1≤x≤ ∞ 0≤y≤π,y≠ π/2
cosec-1x(arccosecantx) – ∞ ≤ x ≤-1 or 1≤x≤ ∞ -π/2≤y≤π/2, y≠0

Considering the domain and range of the inverse functions, following formulas are important to be noted:

  • sin(\(sin^{-1}x\)) = x, if -1 ≤ x ≤ 1 and \(sin^{-1}\)(siny) = y if -π/2 ≤ y ≤ π/2.
  • cos(\(cos^{-1}x\)) = x, if -1 ≤ x ≤ 1 and \(cos^{-1}(cosy) \)= y if 0 ≤ y ≤ π.
  • tan(\(tan^{-1}x\)) = x, if -∞ < x < ∞ and \(cos^{-1}(cosy) \)= y if -π/2 ≤ y ≤ π/2.
  • cot(\(cot^{-1}x\)) = x, if -∞ < x <  ∞ and \(cot^{-1}(coty)\) = y if 0 < y <π.
  • sec(\(sec^{-1}x\)) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ and \( sec^{-1}(secy) \)= y if -0 ≤ y ≤ π, y ≠ π/2.
  • cosec(\(cosec^{-1}x\)) = x, if -∞ ≤ x ≤- 1 or 1 ≤ x ≤ ∞ and \(cosec^{-1}(cosecy)\) = y if -π/2 ≤ y ≤ π/2, y ≠ 0.

NOTE:Inverse trigonometric functions are also called as “Arc Functions”, since for a given value of a trigonometric function, they produce the length of arc needed to obtain that particular value.

Graph of Inverse Trigonometric Functions:

Arcsine Function

What is arcsine (arcsinx) function?

Arcsine function is inverse of the sine function denoted by \(sin^{-1}x\). It is represented in the graph as shown below:

Trigonometric Function

Arccosine Function:

What is arccosine (arccosx) function?

Arccosine function is inverse of the cosine function denoted by \(cos^{-1}x\). It is represented in the graph as shown below:

Trigonometric Function

Arctangent Function:

What is arctangent (arctanx) function?

Arctangent function is inverse of the tangent function denoted by \(tan^{-1}x\). It is represented in the graph as shown below:

Trigonometric Function

Arccotangent Function:

What is arccotangent (arccotx) function?

Arccotangent function is inverse of the cotangent function denoted by \(cot^{-1}x\). It is represented in the graph as shown below:

Trigonometric Function

Arcsecant Function:

What is arcsecant (arcsecx) function?

Arcsecant function is inverse of the secant function denoted by \(sec^{-1}x\). It is represented in the graph as shown below:

Trigonometric Function

Arccosecant Function:

What is arccosecant (arccscx) function?

Arccosecant function is inverse of the cosecant function denoted by \( cosec^{-1}x \). It is represented in the graph as shown below:

Trigonometric Function

Circular representation of Inverse Trigonometric Functions

To solve most of the problems in Inverse Trigonometric Functions, it is very beneficial to understand the concept of circular representation of the trigonometric functions.

Let’s see an example for arcsinθ and arccosθ.

  • Here the frame of reference is important. Moving forward, we would be assuming clockwise direction to be positive, and anti-clockwise direction to be negative.
    With reference to the figure 7 and figure 8, point D is π/2, and E is -π/2. Point B is 0 and point C is π.
  • Hence for figure 7, \((sin^{-1}θ) = θ; ~and ~sin^{-1}(-θ) =-sin^{-1}(θ)\), taken in anticlockwise direction.Range ofarcsinθ isπ/2 ≤ θ ≤ π/2and is taken from OE to OD in anti-clockwise circular direction.

Trigonometric Function

Consider the inverse function \(cos^{-1}(-θ)\) = π– θ

(Range of cosine function is 0 ≤ θ≤ π)

Trigonometric Function

(-)θ is taken as the clockwise direction which is represented as OF.

A range of cosine function is0 ≤ θ ≤ π; from figure 8 –vertically opposite angles are equal that is Angle COG = Angle FOB.
The required distance traveled in the anti-clockwise direction to reach from OB to OG is π – θ. Hence cos-1(-θ) = π – θ.

Applications of Graph of Inverse Trigonometric Function:

Consider,\(y = sin^{-1}x + cos^{-1}x\)

Let, \(cos^{-1}x = t\);

cost = x;

sin ( – t) = x;

\(t = sin^{-1}(x)\)

Therefore,\(cos^{-1}x =sin^{-1}(x)\)

\(sin^{-1}x + cos^{-1}x = sin^{-1}x+ sin^{-1}(x) = \frac{\pi}{2}\)

Therefore graphical representation of \( [y = sin^{-1}x + cos^{-1}x] \)is,

Trigonometric Function

Similarly, \( y = tan^{-1}x + cot^{-1}x \)

\(tan^{-1}x + cot^{-1}x =tan^{-1}x+ tan^{-1}(x) = \frac{\pi}{2}\)

Therefore graphical representation of \([y = tan^{-1}x + cot^{-1}x] \)is,

Trigonometric Function

With the help of inverse trigonometric functions represented graqphically, we find that learning of topic becomes more easy, and more easy to explore.

As a matter of exercise, you should the following question:
Find the number of solutions of the combined equations, \( y = sin^{-1}x \),, and y = x.
Answer: One

The related subtopics will explain further details on the chapter of Inverse Trigonometric Functions & Trigonometry and will be as interesting and informative with Byju’s.


Practise This Question

If a1, a2,..., a15 are in A.P. and a1  + a8 + a15 = 15, then a2+a3+a8 +a13+a14 =