Inverse Trigonometric Functions: Formula, Table & Derivatives

Inverse Trigonometric Functions are the inverse functions of the basic trigonometric functions.

• The basic trigonometric functions include sine, cosine, tangent, cotangent, secant, and cosecant.
• These functions are also known as arcus functions, anti-trigonometric functions, or cyclometer functions.
• Inverse trigonometry is applied in various fields such as physics, engineering, geometry, navigation, geometry, aviation, marine biology, etc.
• Inverse Trigonometry is useful for obtaining angles of a triangle from any trigonometric function. 

Key Terms: Inverse Trigonometric Functions, Functions, Perpendicular,cotangent, secant, Tangent, trigonometric, Traingle, sine, cosine, tangent, cotangent, secant, cosecant

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Definition of Inverse Trigonometric Functions

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Inverse trigonometric Functions are known as the anti-trigonometric functions as they are the inverse of trigonometric functions.

• They are also known as the ''Arc Functions''.
• Inverse Trigonometric Functions perform opposite functions of the trigonometric Functions.
• These functions are useful for finding angles when any two sides of a right-angled triangle are given.
• The inverse trigonometric functions are written with the prefix 'arc'.

Also Read:

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Formulas Of Inverse Trigonometric Functions

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Inverse Trigonometric Formulas are necessary to solve the questions.

Some basic trigonometric formulas are-:

• 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
• sin^-1x + cos^-1x = π/2, x ∈ [-1,1]
• tan^-1x+ cot^-1x = π/2, x ∈ R
• sec^-1x + cosec^-1x = π/2, x ∈ R - [-1,1]
• sin^-1x = cosec^-11/x, x ∈ R - (-1,1)
• cos^-1x = sec^-11/x, x ∈ R - (-1,1)
• tan^-1x = cot^-11/x, x > 0
• tan^-1x = - π + cot^-1 x, x < 0
• sin^-1x + sin^-1y = sin^-1(x.√(1 - y²) + y√(1 - x²))
• sin^-1x - sin^-1y = sin^-1(x.√(1 - y²) - y√(1 - x²))
• cos^-1x + cos^-1y = cos^-1(xy - √(1 - x²).√(1 - y²))
• cos^-1x - cos^-1y = cos^-1(xy + √(1 - x²).√(1 - y²))
• tan^-1x + tan^-1y = tan^-1(x + y)/(1 - xy), if xy < 1
• tan^-1x + tan^-1y = tan^-1(x - y)/(1 + xy), if xy > - 1

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Graphs Of Inverse Trigonometric Functions

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The graphs of inverse trigonometric functions are given below

sin^-1x Graph

sin^-1x is the inverse of the sine function. It is denoted by the graph as shown below-:

• Domain = -1 ≤ x ≤ 1
• Range = -π/2 ≤ y ≤ π/2

cos^-1(x) Graph

cos^-1(x) is the inverse of the cosine function. It is denoted by the graph as shown below-:

• Domain = -1≤ x ≤1
• Range = 0 ≤ y ≤ π

tan^-1(x) Graph

 tan^-1(x) is the inverse of the tangent function. It is denoted by the graph as shown below-:

• Domain = -∞ < x < ∞
• Range =-π/2 < y < π/2

cosec^-1(x) Graph

cosec^-1(x) is the inverse of the Cosecant function. It is denoted by the graph as shown below-:

• Domain = -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
• Range = -π/2 ≤ y ≤ π/2, y ≠ 0

sec^-1(x) Graph

sec^-1(x) is the inverse of the Secant function. It is denoted by the graph as shown below-:

• Domain = -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
• Range 0 ≤ y ≤ π, y ≠ π/2

cot^-1(x) Graph

cot^-1(x) is the inverse of the Cotangent function. It is denoted by the following graph-:

• Domain = -∞ < x < ∞
• Range = 0 < y < π

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Domain and Range Of Inverse Trigonometric Functions

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The graphs help in comprehending and comparing different functions.

• The domain of a function is shown along the x-axis of a graph, while the range of a function is denoted by the y-axis of the graph.
• Domain and range give us the principle value of the inverse trigonometric function.

The domain and range of different functions are as follows-:

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Derivative Of Inverse Trigonometric Function

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Inverse Trigonometric Functions are integral in Calculus.

• The derivatives of inverse trigonometric functions are used to solve many questions.
• The derivatives of these functions are first-order derivatives.

The Derivative of the functions are as follows-:

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Things to Remember

• Inverse Trigonometric Functions are the inverse of the basic trigonometric functions like sin x, cosx, tan x, cosec x, sec x, and cot x.
• Inverse Trigonometry is used to find the angle of a right-angled triangle when two sides are given.
• Inverse Trigonometric functions are also known as arcus functions, anti-trigonometric functions, or cyclometer functions.
• The domain of a function is represented along the x-axis, while the Range of a function is represented along the y-axis.
• Derivatives of the Inverse Trigonometric Functions are also an important part of calculus.
• They are used in solving numerous problems.

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Previous Year Questions

1. If tan−1xtan−1⁡x + tan−1ytan−1⁡y = 2π32π3 , then cot−1xcot−1⁡x + cot−1ycot−1⁡y is equal t
2. If sin−1x+sin−1y=π2 , then cos−1x+cos−1y is equal to
3. cos−1(−12)−2sin−1(12)+3cos−1(−1√2)−4tan−1(−1) equals
4. The period of sin4x+cos4x is
5. If 2tan−1(cosx)=tan−1(2cosecx) then sinx+cosx is equal to
6. The period of tan3θ is
7. ∫ex(1+sinx)1+cosxdx=….[COMEDK UGET 2011]
8. If tan(x+y)=33 and x=tan−13, then y is…..[COMEDK UGET 2015]
9. cot−1(21)+cot−1(13)+cot−1(−8)=…...[COMEDK UGET 2015]
10. tan(cos−1(15√2)−sin−1(4√17)) is….….​.[COMEDK UGET 2015]
11. If 12≤x≤1,12≤x≤1, then cos−1x+cos−1(x2+√3−3x22)=...[COMEDK UGET 2010]
12. If cos−1x+cos−1y+cos−1z=3π, then xy+yz+zx is...[COMEDK UGET 2011]
13. Define f (x) = min{x^2 + 1, x + 1] for.xe∈R. Then 1∫−1f(x)dx is...[COMEDK UGET 2005]
14. The value of the integral π/4∫01+sin2cos3xdx is​….[COMEDK UGET 2005]
15. tan−1(1x+y)+tan−1(yx2+xy+1)=​…...[COMEDK UGET 2011]
16. tan[12sin−1(2x1+x2)+12cos−1(1−x21+x2)]=….[COMEDK UGET 2007]
17. The value of tan−1√2+√3−√2−√3√2+√3+√2−√3….[COMEDK UGET 2008]
18. The value of cot−1{√1−sinx+√1+sinx√1−sinx−√1+sinx}(0<x<π2) is​
19. In a ΔABC, if A=tan−12 and B=tan−13 , then C=​
20. sin−1(1√e)>tan−1(1√π)sin−1(1e)>tan−1(1π) sin−1x>tan−1ysin−1x>tan−1y for x>y,∀x,y∈(0,1)​