CBSE Class 12 Mathematics Notes Chapter 2 Inverse Trigonometric Functions

Inverse Trigonometric Functions, also known as Arc functions, are used to represent the inverse functions of basic trigonometric functions. These functions are used to determine the length of the arc to obtain a particular value. 

• Inverse Trigonometric Functions are used in trigonometry to get the angle with any of the trigonometric ratios.
• Six basic trigonometric functions include sine, cosine, tangent, secant, cosecant, and cotangent. 
• These inverse functions are also known as Anti trigonometric functions and cyclometric functions.
• Inverse Trigonometric Functions have wide applications in the fields of engineering, physics, geometry and navigation.
• They are also used to solve problems in calculus.

According to the CBSE Syllabus 2023-24, the chapter on Inverse Trigonometric Functions comes under Unit 1 of Relations and Functions. NCERT Class 12 Mathematics Chapter Inverse Trigonometric Functions holds a weightage of around 4-8 marks. 

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Class 12 Mathematics Chapter 2 Notes – Inverse Trigonometric Functions

Range and Domain of Inverse Trigonometric Functions

• Range for a given data set is defined as the difference between the lowest and the highest values. 
• Domain is defined as the set of all possible values that is accepted by the functions
• The range and domain of a function are based on the possibility of the function being defined in the real set. 
• These values can be used to determine the principal value of the inverse trigonometric function.
• Domain is represented along the x-axis of the graph.
• Range is represented on the y-axis of the graph.
• Domain of inverse sine function is a set of real numbers with a range of [–1, 1].

Range and Domain of Inverse Trigonometric Functions

Formula of Inverse Trigonometric Functions

Some important formula of inverse trigonometric functions are as follows:

• sin^-1(-x) = -sin^-1(x), x ∈ [-1, 1]
• cos^-1(-x) = π -cos^-1(x), x ∈ [-1, 1]
• tan^-1(-x) = -tan^-1(x), x ∈ R
• cot^-1(-x) = π – cot^-1(x), x ∈ R
• sec^-1(-x) = π -sec^-1(x), |x| ≥ 1
• cosec^-1(-x) = -cosec^-1(x), |x| ≥ 1

Inverse Trigonometric Function Graphs

• There are six inverse trigonometric functions for each of the trigonometric ratios. 
• The graph of each trigonometric ratio varies from function to function.
• Each function ranges from -∞ to ∞.

Arcsine Function

• Arcsine Function is defined as the inverse of the sine function. 
• The inverse of sine function is expressed as y = sin^-1x (arcsin x)
• It is denoted by sin^-1x, which is represented by the graph below:

Sin^-1 x graph

Arccosine Function

• Arccosine Function is defined as the inverse of the cos function. 
• The inverse of cosine function is expressed as y = cos^-1x (arccosin x)
• It is denoted by cos^-1x, which is represented by the graph below:

Cos^-1x graph

Arctangent Function

• Arctangent Function is defined as the inverse of the tan function. 
• The inverse of tan function is expressed as y = tan^-1x (arctangent x)
• It is denoted by tan^-1x, which is represented by the graph below:

Tan^-1x graph

Arccotangent Function

• Arccotangent Function is defined as the inverse of the cot function. 
• The inverse of cot function is expressed as y = cot^-1x (arccotangent x)
• It is denoted by cot^-1x, which is represented by the graph below:

cot^-1x graph

Arcsecant Function

• Arcsecant Function is defined as the inverse of the sec function. 
• The inverse of sec function is expressed as y = sec^-1x (arcsecant x)
• It is denoted by sec^-1x, which is represented by the graph below:

sec^-1x graph

Arccosecant Function

• Arccosecant Function is defined as the inverse of the cosec function. 
• The inverse of cosec function is expressed as y = cosec^-1x (arcocsecant x)
• It is denoted by cosec^-1x, which is represented by the graph below:

cosec^-1x graph

Principal Value of Inverse Trigonometric Functions

• The principal value is the value of the inverse trigonometric function that lies in the principal value branch.
• It is equivalent to the range of each of the inverse functions.
• The value of the principal branch will satisfy the trigonometric function at x.

Principal Value of Inverse Trigonometric Functions

Properties of Inverse Trigonometric Functions

The properties of Inverse Trigonometric Functions are divided into various sets, which are as follows:

Property 1

• sin^-1(x) = cosec^-1(1/x) θ for all x ∈ [-1, 1] −{0}
• cos^-1(x) = sec^-1(1/x) θ for all x ∈ [-1, 1] −{0}
• tan^-1(x) = cot^-1 (1/x) − π, if x < 0 or cot^-1(1/x) for x > 0
• cot^-1(x) = tan^-1(1/x) + π for x < 0 or tan^-1(1/x), if x > 0 

Property 2

• sin (sin⁻¹ x) = x for all x ∈ [-1, 1]
• cos (cos⁻¹ x) = x for all x ∈ [-1, 1]
• tan (tan⁻¹ x) = x for all x ∈ R
• cosec (cosec⁻¹ x) = x for all x ∈ (∞, -1] ∪ [1, ∞)
• sec (sec⁻¹ x) = x for all x ∈ (-∞, -1] ∪ [1, ∞)
• cot (cot⁻¹ x) = x for all x ∈ R

Property 3

• sin⁻¹ (1/x) = cosec⁻¹ x for all x ∈ (∞, -1] ∪ [1, ∞)
• cos⁻¹ (1/x) = sec⁻¹ x for all x ∈ (-∞, -1] ∪ [1, ∞)
• tan⁻¹ (1/x) = cot⁻¹ x, for x > 0 and -π + cot⁻¹ x, for x < 0

Property 4

• sin⁻¹ (-x) = - sin⁻¹ x for all x ∈ [-1, 1]
• cos⁻¹(-x) = π - cos⁻¹ x for all x ∈ [-1, 1]
• tan⁻¹ (-x) = - tan⁻¹ x for all x ∈ R
• cosec⁻¹ (-x) = - cosec⁻¹ x for all x ∈ (∞, -1] ∪ [1, ∞)
• sec⁻¹ (-x) = π - sec⁻¹ x for all x ∈ (-∞, -1] ∪ [1, ∞)
• cot⁻¹ (-x) = π - cot⁻¹ x for all x ∈ R

Property 5

• sin^-1 x + cos^-1x = π/2 for all x ∈ [-1, 1]
• tan^-1 x + cot^-1 x = π/2 for all x ∈ R
• sec^-1 x + cosec^-1 x = π/2 for all x ∈ (-∞, -1] ∪ [1, ∞)

Property 6

If x and y are greater than 0

• tan^-1 x + tan^-1 y = π + tan^-1 (x + y/ 1-xy) , if xy > 1
• tan^-1 x + tan^-1 y = tan^-1 (x + y/ 1-xy) , if xy < 1

If x and y are less than 0

• tan^-1 x + tan^-1 y = tan^-1 (x + y/ 1-xy) , if xy > 1
• tan^-1 x + tan^-1 y = - π + tan^-1 (x + y/ 1-xy) , if xy < 1

Property 7

• sin^-1(cos θ) = π/2 − θ, for θ ∈ [0,π]
• cos^-1(sin θ) = π/2 − θ, for θ ∈ [−π/2 , π/2]
• tan^-1(cot θ) = π/2 − θ, for θ ∈ [0,π]
• cot^-1(tan θ) = π/2 − θ, for θ ∈ [−π/2, π/2]
• sec^-1(cosec θ) = π/2 − θ, for θ ∈ [−π/2, 0] ∪ [0, π/2]
• cosec^-1(sec θ) = π/2 − θ, for θ ∈ [0,π] − {π/2}
• sin^-1(x) = cos^-1 [√(1−x²)], for 0≤x≤1= −cos^-1 [√(1−x²)], for −1≤x<0

Property 8 

• sin^-1 x + sin^-1 y = sin^-1 { x √(1−y²) + y √(1- x²)}
• cos^-1 x + cos^-1 y = cos^-1 { xy - √(1−y²) √(1- x²)}

Property 9

• 2 sin^-1 x = sin^-1 (2x √(1- x²)
• 3 sin^-1 x = sin^-1 (3x - 4x³)
• 2 cos^-1 x = cos^-1 (2x² - 1)
• 3 cos^-1 x = 2 π + cos-1 (2x² - 1)
• 2 tan^-1x = tan^-1(2x / 1- x²)
• 3 tan^-1 x = tan^-1(3x - x³/ 1- 3x²)
• 2 tan^-1 x = sin ^-1(2x / 1 + x²), if -1 ≤ x ≤ 1
• 2 tan^-1 x = cos^-1(1- x² / 1 + x²), if 0 ≤ x < ∞

There are Some important List Of Top Mathematics Questions On Inverse Trigonometric Functions Asked In CBSE CLASS XII

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