Inverse Trigonometric Functions in Maths

JEE CRASH  Lite COURSE 2022

Inverse Trigonometric Functions

Inverse Trigonometric Functions in Maths is simply defined as the inverse of some basic trigonometric functions such as sine, cosine, tan, sec, cosec and cot. The other names of Inverse trigonometric functions are arcus function, anti-trigonometric function or cyclomatic function. We can understand the concept of inverse trigonometric function in a better way with the help of the triangle given below.


In the above triangle ABC, the basic trigonometric function will be defined as:


  • Sin θ = a/b

  • Cos θ = c/b

  • Tan θ = a/c


In the above basic trigonometric function, the angles are termed as input and the ratio of the sides of a triangle is considered as their outcomes. However, the inverse trigonometric function considers the ratio of the side of a triangle as input and obtains the value of angles.


  • Sin-1 (a/b) = θ

  • Cos-1 (c/b)  = θ

  • Tan-1 (a/c) = θ


It implies inverse trigonometric functions are beneficial whenever the sides of a triangle are known and we want to determine the angles of a triangle.


Note: The representation Sin-1 might create confusion because we usually use a negative exponent to represent reciprocals. Although in this situation, Sin-1θ =  1/Sin θ. We generally use Cosecant or csc to determine the reciprocal of sin. To avoid this confusion, some people choose to represent the inverse function by using arc as a prefix. For example.,  


  • Arcsin θ = Sin-1 θ


Domain and Principal Value of Inverse Trigonometric Functions

We know that the principal value of the trigonometric function at a point is the value of the inverse function at a point , which falls in the range of the principal values unit. The principal value of cos √(3/2) is π/6 as π/6 ∈ \[[0,\pi ]\].


Whenever any positive value and the negative values are given in a way that these two values are equal, then the principal value of the inverse trigonometric function will always be the positive value. Let us now discuss the domain and range of all the six inverse trigonometric functions.


Function

Domain

Range 

Y = Sin-1 x

-1 ≤ x ≤ 1

-π/2 ≤ y ≤ π/2

y= Cos-1 x

-1 ≤ x ≤ 1

0 ≤ y ≤ π

y = Tan-1 x

x ∈ R

-π/2 < x < π/2

y = Cosec-1 x

x ≤ -1 or x ≥ 1

-π/2 ≤  y ≤  π/2, y 0

y = Sec-1 x

x ≤ -1 or x ≥ 1

0 ≤  y ≤ π ; y π/2

y = Cot-1 x

x ∈ R 

0 < y < π

 

Inverse Trigonometric Function Formulas

Some basic inverse trigonometric functions formulas are discussed below:


  • Sin-1 (-α) = - Sin-1 (α)

  • Cos-1 (-α) = π - Cos-1 (α)

  • Sin-1 (α) + Cos-1 (α) = π/2

  • Tan-1 (α) + Tan-1 (β) = π + tan-1 (\[\frac{\alpha +\beta }{1-\alpha \beta }\])

  • 2 Sin-1 (α) = Sin-1 (2α\[\sqrt{1-\alpha ^2}\])

  • 3 Sin-1 (α) = Sin-1 (3α - 4α3)

  • Sin-1 (α) + Sin-1 (β) = Sin-1 (α\[\sqrt{1-\beta ^2}\] + β\[\sqrt{1-\alpha ^2}\]), if α, β ≥0 and α² + β² ≤ 1]

  •  Cos-1 (α) + Cos-1 (β) = Si (αβ -\[\sqrt{1-\alpha ^2}\]\[\sqrt{1-\beta ^2}\]), if α, β ≥ 0 and α² + β² ≤ 1]


Inverse Trigonometric Function Properties

Let us discuss now some inverse trigonometric function properties


  • Sin(arcsin α ) = α and arcsin (sin β)= β, where -1 ≤  α ≤ 1 and -π/2 ≤ β ≤ π/2

  • Tan(arctan α) = α and arctan (tan β)= β, where -π/2 ≤ β ≤ π/2

  • Sec(arcsec α ) = α and arcsec (sec β)= β, where α ≥ 1 and 0 ≤ β ≤ π/2

  • Sin-1  (1/θ) = Csc-1 θ

  • Cos-1  (1/θ) = Sec-1 θ

  • Tan-1  (1/θ) = Cot-1 θ

  • Sin-1 θ+  Cos-1 θ= π/2, θ ∈ \[[-1.1]\]

  • Tan-1θ  +  Cot1 θ = π/2, θ ∈ R

  • Sec-1 θ+  Cosec1 θ = π/2, |x| ≤ 1

 

Inverse Circular Functions Graph

The six inverse circular function graphs are discussed below:


1. ArcSin x

Let y = sin-1 x,|x| ≤ 1 , y ∈ \[[(-\frac{\pi }{2},\frac{\pi }{2})]\]

Important Points To be Considered


Sin-1 x is an odd function (symmetric about its origin) and it falls in the range of (-π/2,π/2)


The maximum value of sin-1 x is π/2, and it lies at x = 1


The minimum value of sin-1 x is -π/2, and it lies at x = -1


Sin-1 x is a function that is decreasing in its domains


Sin-1 x is also a periodic function.


2. ArcCos x

Let y = cos-1 x,|x| ≤ 1 , y ∈ \[[0,\pi ]\]


Important Points To be Considered


  • Cos-1 x is neither an odd nor an even function and it falls in the range of (0,π)

  • The maximum value of cos-1 x is π, and it lies at x = -1

  • The minimum value of cos-1 x is 0, and it lies at x = 1

  • Cos-1 x is a function that is decreasing in its domains

  • Cos-1 x is also a periodic function.


3. ArcTan x

Let y = tan-1 x, x ∈ R, y ∈ \[[(-\frac{\pi }{2},\frac{\pi }{2})]\]


Important Points To be Considered


  • Tan-1 x is an odd function (symmetric about its axis) and it falls in the range of (-π/2,π/2)

  • The maximum and minimum value of tan-1 x cannot be defined.

  • Tan-1 x is a function that is decreasing in its domains

  • Tan-1 x is also a periodic function.


4. Arc Cot x

Let y = cot-1 x, x ∈ R, y ∈ \[[0,\pi ]\] 


Important Points To be Considered


  • Cot-1 x is neither odd nor even function and it falls in the range of 0,π

  • The maximum and minimum value of cot-1 x cannot be defined.

  • Cot-1 x is a function that is decreasing in its domains

  • Cot-1 x is also a periodic function.


5. ArcSec x

Let y = sec-1 x, |x| ≥ 1, y ∈ (0, π/2) ∪ (π/2,π )


Important Points To be Considered


  • Sec-1 x is neither an odd nor an even function.

  • Sec-1 x falls in the range of \[[0,π]\]

  • The maximum value of sec-1 x is π, and it lies at x = 1

  • The minimum value of sec-1 x is -π, and it lies at x = -1

  • Sec-1 x is a function that is decreasing in two different intervals

  • Sec-1 x is also a periodic function.


6. Arc Cosec x

Let y = cosec-1 x, |x| ≥ 1, y ∈ (–π/2,0) ∪ ( 0,π/2)


Important Points To be Considered


  • cosec-1 x is an odd function that falls in the range of –π/2, π/2.

  • The maximum value of cosec-1 x is π/2, and it lies at x = 1

  • The minimum value of cosec-1 x is -π/2, and it lies at x = -1

  • cosec-1 x is a function that is decreasing in two different intervals cosec-1 x is also a periodic function.

  

Some Significant Properties Of Inverse Trigonometric Functions

The domain and range of the functions decide the properties of the inverse trigonometric functions. These properties are very important for solving the questions and for a clear understanding of the concepts. Following are the inverse trigonometric functions properties:


  • Property I

Sin−1 (x) = cosec−1 (1/x), x ∈ [−1, 1] − {0}


Cos−1 (x) = sec−1 (1/x), x ∈ [−1, 1] − {0}


Tan−1 (x) = cot−1 (1/x), if x > 0 and cot−1 (1/x) −π, if x < 0


Cot−1(x) = tan−1 (1/x), if x > 0 and tan−1 (1/x) + π, if x < 0


  • Property II

Sin−1 (−x) = −Sin−1(x)


Tan−1 (−x) = −Tan−1(x)


Cos−1 (−x) = π − Cos−1(x)


Cosec−1 (−x) = − Cosec−1(x)


Sec−1 (−x) = π − Sec−1(x)


Cot−1 (−x) = π − Cot−1(x)


  • Property III

Sin−1 (1/x) = cosec−1x, x ≥ 1 or x ≤ −1


Cos−1 (1/x) = sec−1x, x ≥ 1 or x ≤ −1


Tan−1 (1/x) = −π + cot−1(x)


  • Property IV

Sin−1 (cos θ) = π/2 − θ, if θ ∈ [0,π]


Cos−1 (sin θ) = π/2 − θ, if θ ∈ [−π/2, π/2]


Tan−1 (cot θ) = π/2 − θ, θ ∈ [0,π]


Cot−1 (tan θ) = π/2 − θ, θ ∈ [−π/2, π/2]


Sec−1 (cosec θ) = π/2 − θ, θ ∈ [−π/2, 0]∪[0, π/2]


Cosec−1 (sec θ) = π/2 − θ, θ ∈ [0,π]−{π/2}


  • Property V

Sin−1x + Cos−1x = π/2


Tan−1x + Cot−1(x) = π/2


Sec−1x + Cosec−1x = π/2


  • Property VI

tan−1(x) – tan−1(y) = tan−1 [(x−y)/ (1+xy)], xy > −1


tan−1(x) + tan−1(y) = tan− 1[(x + y)/ (1-xy)], xy < 1

FAQs on Inverse Trigonometric Functions in Maths

1. How can students prepare the Chapter "Inverse Trigonometric Functions" for their Maths examination?

With the help of the best study materials, students can easily prepare the Chapter "Inverse Trigonometric Functions" for their Maths examination. They can download the Vedantu app or access the official website of the Vedantu to get these course materials. There they can find the notes of the detailed explanation of each topic of this chapter. Along with the notes, they will also discover the sample papers and important questions. By practising these papers, students can make a strong grip over the lesson. Vedantu experts will help them in clarifying their doubts.

2. What is the meaning of the term "inverse trigonometric formulas"?

The formulas which are helpful in converting one function to another, in finding the principal angle values of the functions and in performing the numerous arithmetic operations over various trigonometric functions are known as inverse trigonometric formulas. All the fundamental trigonometric functions have been changed into inverse trigonometric functions. These functions are grouped under the four sets of formulas. the name of the sets are as follows:

  • Arbitrary Values

  • Complementary and Reciprocal Functions

  • Difference and Sum of Functions

  • Double and Triple of a function

3. What are the different types of inverse trigonometric functions?

The different types of inverse trigonometric functions are listed below:

  • Arcsine - It is the inverse of the sine function and is denoted by \[sin^{-1} x\].

  • Arccosine - The inverse of the cos function is called arccosine. The symbol used for this function is \[cos^{-1} x\].

  • Arctangent - It is the inverse of the tangent function. Denoted by \[tan^{-1} x\].

  • Arccotangent - This function is the inverse cotangent function and is written as \[cot^{-1} x\].

  • Arcsecant - It is the inverse secant function resembled by \[sec^{-1} x\].

  • Arccosecant - This function is the inverse of the cosecant function and is denoted by \[cosec^{-1} x\].

4. Evaluate cos-1 (1/2) + 2 sin-1 (1/2).

Let y = cos-1 (1/2)


Therefore,


cos y = (1/2)


cos y = cos (π /3).


As the principal value of cos-1 is [0, π ]


Hence, the principal value of cos-1 (1/2) is π/3.


Similarly, consider  y = sin-1 (1/2)


Therefore,


sin y = ½


sin y = sin ( π/6)


As the principal value of sin-1 is [(-π)/2, π/2 ]


Thus, the principal value of  sin-1 (1/2) is π/6.


Now cos-1 (1/2) = π/3 and sin-1 (1/2) = π/6


Substituting values in the given question,


cos-1 (1/2) + 2sin-1 (1/2)


= π /3 + 2( π/6)


= π/3 + π/3


= 2π /3


Hence, the value of cos-1 (1/2) + 2 sin-1 (1/2) is  2π /3.

5. What is the value of  tan-1 √3 – sec-1(–2) ?

Let y = tan-1√3


Therefore, tan y = √3


Now,


tan y = tan (π/3)


As the principal value of tan-1 is (−π/2, π/2)


Thus, the principal value of tan-1√3 is π/3.


Consider, y = sec-1 (–2)


Then, sec y = -2


sec y = sec (2π/3)


As the principal value range of sec-1 is [0,π] – {π/2}


Hence, the principal value of sec-1 (–2) = 2π/3


Now,


tan-1(√3) = π/3


sec-1 (–2) = 2π/3


Substituting the values in the given question


tan-1 √3 – sec-1 (−2)


= π/3 − (2π/3)


= π/3 − 2π/3


= (π − 2π)/3


= – π/3

Comment