Derivative of Inverse Trigonometric Functions: Overview, Questions, Preparation

Inverse Trigonometric Functions 2021 ( Maths Inverse Trigonometric Functions )

Rajdeep Das
Updated on Jul 29, 2021 03:56 IST

By Rajdeep Das

Table of content
  • Derivative of Inverse Trigonometric Functions
  • Inverse Trigonometric Functions with their Domains and Ranges
  • Properties of Inverse Trigonometric Functions
  • Derivatives of Inverse Trigonometric Functions
  • Derivative of Inverse Trigonometric Functions in Class 12
  • Illustrated Examples
  • FAQs
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Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions are often referred to as arcus functions, anti-trigonometric functions, or cyclometric functions. These functions are often used to produce an angle for a trigonometric value. Inverse trigonometric functions have diverse uses in engineering, geometry, navigation, etc.

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Inverse Trigonometric Functions with their Domains and Ranges

sin–1 : [–1, 1] → [-π/2, π/2]    

cos –1 : [–1, 1] → [0, π] 

cosec–1 : R – (–1,1) → [-π/2, π/2]  – {0} 

sec –1 : R – (–1, 1) → [0, π] – { π/2 } 

tan–1 : R → [-π/2, π/2]     

cot–1 : R → (0, π)

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Properties of Inverse Trigonometric Functions

(i) sin–1 1/x = cosec–1 x, x ≥ 1 or x ≤ – 1 

(ii) cos–1 1/x = sec –1 x, x ≥ 1 or x ≤ – 1

(iii) tan–1 1/x = cot–1 x, x > 0

(iv) sin–1 (–x) = – sin–1 x, x ∈ [– 1, 1] 

(v) tan–1 (–x) = – tan–1 x, x ∈ R 

(vi) cosec–1 (–x) = – cosec–1 x, | x | ≥ 1

(vii) cos–1 (–x) = π – cos–1 x, x ∈ [– 1, 1] 

(viii) sec–1 (–x) = π – sec–1 x, | x | ≥ 1 

(ix) cot–1 (–x) = π – cot–1 x, x ∈ R

(x) sin–1 x + cos–1 x = π/2 , x ∈ [– 1, 1] 

(xi) tan–1 x + cot–1 x = π/2 , x ∈ R 

(xii) cosec–1 x + sec–1 x = π/2 , | x | ≥ 1

(xiii) tan–1 x + tan–1 y = tan–1 x + y / 1– xy , xy

(xiv) tan–1 x – tan–1 y = tan–1 x – y/1 + xy , xy > – 1

(xv) 2tan–1 x = sin–1 2x/1+x2, | x | ≤ 1 

(xvi) 2tan–1 x = cos–1 1-x2/1+x2, x ≥ 0 

(xvii) 2 tan–1 x = tan–1 2x/1-x2, – 1

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Derivatives of Inverse Trigonometric Functions

arcsin x

1/1-x2

arccos x

-1/1-x2

arctan x

1/1+x2

arccot x

-1/1+x2

arcsec x

1/|x| x2 - 1

arccsc x

-1/|x| x2 - 1

Example:

Differentiate the function f(x) = cos-1x using the first principle.

limh->0 {f(x + h) – f(x)} / h

cos-1x + sin-1x = pi/2

cos-1x = pi/2 – sin-1x

f(x) = cos-1x

f(x + h) = cos-1(x + h)

limh->0 {cos-1(x + h ) – cos-1(x)} / h

limh->0 {pi/2 – sin-1(x + h) – (pi/2 – sin-1x) } / h

limh->0 {pi/2 – sin-1(x + h) – pi/2 + sin-1x } / h

– limh->0 {sin-1(x + h) – sin-1x} / h

limh->0 { sin-1(x + h) – sin-1x } / h = 1 / √(1 – x2)

– 1 / √(1 – x2)

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Derivative of Inverse Trigonometric Functions in Class 12

This concept is taught under the chapter Derivative of Inverse Trigonometric Functions

in class 12. In this chapter, you will learn about the nature of inverse trigonometric functions and their derivatives and use this knowledge to solve questions. The weightage of this chapter is four marks. 

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Illustrated Examples

1. Find y’ if y=arctanx3

2. Find f′( x) if f( x) = cos −1(5 x).

f’(x) = -1/1-(5x)2 . 5

= -5/1-25x2

3. Differentiate y=5x6−sec−1(x)

dy/dx=30x5−1/x√x2−1

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FAQs

Q: What are the six inverse trig functions?

A: Sine, cosine, tangent, cotangent, secant, and cosecant.

Q: What is the derivative of an inverse function?

A: If g is the inverse of f, the derivative of f(x) is 1/g'(f(x)), not 1/f'(x).  g'(x)=e^x, so the derivative of ln(x) is 1/[e^(ln(x))]=1/x.

Q: What is the derivative of tan inverse?

A: y = arcsin(x) -1 x 1.

Q: Is arctan inverse tan?

A: The arctan function is the inverse of the tangent function.

Q: What is the formula for derivatives?

A: The derivative of the function y = f(x) of the variable x calculates the rate at which the value y of the function varies with respect to the shift in the variable x.
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Maths Inverse Trigonometric Functions Exam

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