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Trigonometric functions are used to find the measurements like heights of mountains and tall buildings without using measurement tools. In this article, we will be covering Derivative of Inverse Trigonometric Functions in detail. Trigonometric functions are sin, cos, tan, cot, sec and cosec. Just like the mathematical operations addition and subtraction, the inverse of each other is also similar.
Read Also: Some Applications of Trigonometry
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Key Terms: Trigonometric Functions, Inverse function, Inverse sine function, sin, cos, tan, cot, sec, cosec, Denominator
Derivative of Inverse Trigonometric Functions
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Inverse Trigonometric functions can be differentiated and the derivatives can be found. The derivatives of Inverse Trigonometric functions are as follows:
To find the derivative of a inverse function y=sin−1x, we have
y = sin−1x ………..(1)
x = siny
Differentiating equation 1 with respect to x, we get
dy/dx = 1/cosy
Substituting the value of y in the denominator gives
dy/dx = 1/cosy = 1/cos (sin−1x) ………..(2)
From equation 2 it can be inferred that the value of cos y cannot be equal to 0 or the function becomes undefined.
Sin−1x ≠ − π/2,π/2
i.e. x ≠ − 1,1
From (1) we have
y = sin−1x
sin y = sin (sin−1x)
Using property of trigonometric function,
cos 2y = 1 – sin2 y = 1 – (sin (sin−1x))2
= 1 – x2
cos y = √1 – x2 ……..(3)
Now substituting the value of (3) in (1), we get
dy/dx = 1/√1−x2
Hence the Derivative of Inverse sine function is
d/dx (sin−1x) = 1/√1−x2
Similarly when all the Trigonometric functions are differentiated, each of the functions has a derivative. The derivatives of the inverse Trigonometric functions are:
| Function | Derivative |
|---|---|
| arc(sin x) | \( 1/\sqrt1-x^2\) |
| arc(cos x) | \(-1/\sqrt1-x^2\) |
| arc(tan x) | \(1/1+x^2\) |
| arc(cot x) | \(-1/1+x^2\) |
| arc(cosec x) | \(-1/|x|\sqrt x^2-1\) |
| arc(sec x) | \(1/|x|\sqrt x^2-1\) |
Properties of Inverse Trigonometric Functions
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The properties of Inverse Trigonometric Functions has been tabulated below:
| Inverse sine function | Inverse cosine function | Inverse Tangent function | Inverse Cosecant function | Inverse secant function | Inverse Cot function |
|---|---|---|---|---|---|
| Domain: [-1,1] | Domain: [-1,1] | Domain: R | Domain: (-∞,-1] U [1,∞) | Domain: (-∞,-1] U [1,∞) | Domain: R |
| Range: [-π/2, π/2] | Range: [0,π] | Range: [-π/2, π/2] | Range: [-π/2, π/2]-{0} | Range: [0, π]-{π/2} | Range: (0, π) |
| Not a periodic function | Not a periodic function | Not a periodic function | Not a periodic function | Not a periodic function | Not a periodic function |
| Odd function | Neither even nor odd function | Odd function | Odd function | Neither even nor odd function | Neither even nor odd function |
| Strictly increasing function | Strictly decreasing function | Strictly increasing function | Strictly decreasing function with respect to its domain | Strictly decreasing function with respect to its domain | Strictly decreasing function |
| One to one function | One to one function | One to one function | One to one function | One to one function | One to one function |
Read More: Trigonometry Important Formulas
Relevant Formulae of Trigonometric Functions
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Some of the important formulas of the six main trigonometric functions are as follows:
- Sin−1x + cos−1x = π/2
When the above equation is differentiated with respect to x we get
d/dx(sin−1x) + d/dx(cos−1x) = 0
d/dx(sin−1x) = − d/dx(cos−1x)
- Sec−1x + tan−1x = π/2
d/dx(sec−1x) = − d/dx (tan−1x)
- Cosec−1x + cot−1x = π/2
d/dx (cosec−1x) = − d/dx(cot−1x)
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Things to Remember
- Inverse Trigonometric functions can be differentiated and the derivatives can be found.
- Trigonometric functions are used to find the measurements like heights of mountains and tall buildings without using measurement tools.
- Trigonometric functions are sin, cos, tan, cot, sec and cosec.
- Trigonometric functions are differentiated, each of the functions has a derivative.
- There are some of the important formulas of the six main trigonometric functions.
Previous Year Questions
- If cos−1(135)−sin−1(1312)=cos−1x, then xx is equal to….[JEE MAIN 2009]
- The value of cos[tan−1{sin(cot−1x)}] is...[KEAM 2009]
- The solutions set of inequation cos−1x<sin−1x is….[KEAM 2011]
- If cos−153+cos−11312=cos−1k, then the value of kk is...[UPSEE 2018]
- The value of cos−1x+cos−1(2x+213−3x2);21≤x≤1 is..[BITSAT 2010]
- If sin−1(x−2x2+4x3−8x4+...)−6π where ∣x∣<2 then the value of xx is….[WBJEE 2015]
- 2cos−1x=sin−1(2x1−x2) is valid for all values of xx satisfying...[KCET 2012]
- The simplified form of tan−1(x+yx−y) is equal to….[KCET 2016]
- Given 0≤x≤21 then the value of tan[sin−1{2x+21−x2}−sin−1x] is….[KCET 2014]
- The value of sin(2sin−10.8) is equal to….[KCET 2014]
Sample Questions
Ques. Find the derivative of tan-1(sin-1(2x)). (2 marks)
Ans. To find the derivative we need to differentiate the given equation on both sides with respect to x.
We get
d/dx(tan−1(sin−1(2x)))
=1/1+(sin−1(2x))2 ×( 1/√1–(2x)2) × 2
Ques. Find the derivative of a function Sin−1(1−x2/1+x2) (2 marks)
Ans. Given y = sin−1(1−x2/1+x2)
dy/dx=1/√1−((1−x2/1+x2)2 × d/dx(1−x2/1+x2)
dy/dx=1/√((1+x2)2-(1-x2)2)/(1+x2)2)) × d/dx(1−x2/1+x2)
dy/dx=1+x2/√(1+x4+2x2)-(1+x4-2x2) × ((1+x2)(-2x)-(1-x2)(2x)))/(1+x2)2))
dy/dx=1+x2/√4x2 × ((-2x-2x3-2x-+2x3)/(1+x2)2))
dy/dx=-2/1+x2
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