# Derivative of Inverse Trigonometric functions

In this section, we'll look at the derivatives of inverse trig functions. We'll need the formula from the previous section relating the derivatives of inverse functions to derive the derivatives of inverse trig functions. If f(x) and g(x) are inverse functions, then g'(x)=1f'(x)

## Introduction

Arcus functions, cyclometric functions, and anti-trigonometric functions are all names for inverse trigonometric functions. These functions determine the angle for a given trigonometric value. In engineering, geometry, navigation, and other fields, inverse trigonometric functions are useful.

### Inverse Sine

y=sin-1xsiny=x for -2y2

So evaluating an inverse trig function is equivalent to asking what angle (i.e. y) we plugged into the sine function to get x. The above restrictions on y are there to ensure that we get a consistent answer from the inverse sine. We know that there are an infinite number of possible angles, and we want a consistent value when working with inverse sine. Using the above range of angles yields all possible sine function values exactly once. If you're not sure, draw a unit circle and you'll see that that range of angles (the y's) will cover all possible sine values.

Note as well that since -1sin(y)1 we also have -1x1

The inverse sine function and the sine function have the following relationship.

sinsin-1x=x sin-1sinx=x

In other words, they are polar opposites. This means that we can use the preceding fact to calculate the derivative of inverse sine. To begin with,

f(x)=sinx

g(x)=sin-1x

g'(x)=1f'(g(x))=1Cossin-1x

This is an ineffective formula. Let us see if we can come up with a better formula. To begin, recall the definition of the inverse sine function.

y=sin-1(x) x=siny

Using the first part of this definition, the derivative's denominator becomes,

Cossin-1x=cos(y)

Cos2y+Sin2y=1 Cos2y=1-Sin2y

Now apply the second part of the inverse sine function definition. The common denominator is then,

Cossin-1x=1-Sin2y=1-x2

When all of this is added up, the following derivative is obtained.

ddxsin-1x=11-x2

### Inverse Cosine

y=Cos-1xCos y=x for 0y

We have a restriction on the angles, y, that we get from the inverse cosine function, just like we do with the inverse sine function. Again, if you want to double-check, a quick sketch of a unit circle should convince you that this range will cover all possible cosine values exactly once. In addition, we have,

-1x1 because -1Cos(y)1

Because the inverse cosine and cosine functions are inverses of each other, we have,

CosCos-1x=x Cos-1Cosx=x

We'll do the same thing we did with the inverse sine above to find the derivative. To begin with,

f(x)=Cosx

g(x)=Cos-1x

g'(x)=1f'(g(x))=1-sinCos-1x

Simplifying the denominator here is nearly identical to the work done for the inverse sine, so it isn't shown here. We get the following derivative after simplifying.

ddxCos-1x=11-x2

As a result, the inverse cosine derivative is nearly identical to the inverse sine derivative. The only distinction is the negative sign.

### Inverse Tangent

y=tan-1xtan y=x for -2y2

Again, we have a constraint on y, but this time we can't let it go. Because tangent isn't defined at those two points, y can be either of the two endpoints in the restriction above. To ensure that this range covers all possible tangent values, draw a quick sketch of the tangent function. We can see that this range does indeed cover all possible tangent values. Furthermore, there are no restrictions on x in this case because tangent can take on any value.

We can ask for the limits of the inverse tangent function as x approaches plus or minus infinity because there is no restriction on x. We'll need the graph of the inverse tangent function to do this. This is depicted below.

From this graph we can see that

xtan-1x=2 xtan-1x=-2

Because the tangent and inverse tangent functions are inverse,

tantan-1x=x tan-1tanx=x

As a result, we can begin by calculating the derivative of the inverse tangent function.

f(x)=tanx

g(x)=tan-1x

We have,

y=tan-1x x=tan y

The denominator is then,

sec2tan-1x=sec2y

Now,

Cos2y+Sin2y=1

And divide every term by cos2y we will get,

1+tan2y=sec2y

The denominator is then,

sec2tan-1x=sec2y=1+tan2y

Finally, applying the second part of the definition of the inverse tangent function yields,

sec2tan-1x=1+tan2y=1+x2

The inverse tangent's derivative is then,

ddxtan-1x=11+x2

There are three more inverse trig functions, but these are the most common. The remaining three formulas could be derived in the same manner as the previous three. The derivatives of all six inverse trig functions are shown below.

- ddxsin-1x=11-x2
- ddxcos-1x=11-x2
- ddxtan-1x=11+x2
- ddxcot-1x=11+x2
- ddxsec-1x=1xx2-1
- ddxcosec-1x=1xx2-1

### Alternate Notation

On occasion, an alternate notation is used to denote the inverse trig functions. This is the notation,

sin-1x=arcsin x cot-1x=arccot x

cos-1x=arccos x sec-1x=arcsec x

tan-1x=arctan x cosec-1x=arc cosec x

### Conclusion

Inverse trigonometry is the reciprocal of trigonometric normal functions. The angle of the trigonometric functions is used to find the value of sides in trigonometry. In inverse trigonometry, on the other hand, the sides of the triangle are used to calculate the angle.

#### Related Links

- Derivative of Inverse Trigonometric functions
- Decimal Expansion Of Rational Numbers
- Cos 90 Degrees
- Factors of 48
- De Morgan’s First Law
- Counting Numbers
- Factors of 105
- Cuboid
- Cross Multiplication- Pair Of Linear Equations In Two Variables
- Factors of 100
- Factors and Multiples
- Derivatives Of A Function In Parametric Form
- Factorisation Of Algebraic Expression
- Cross Section
- Denominator
- Factoring Polynomials
- Degree of Polynomial
- Define Central Limit Theorem
- Factor Theorem
- Faces, Edges and Vertices
- Cube and Cuboid
- Dividing Fractions
- Divergence Theorem
- Divergence Theorem
- Difference Between Square and Rectangle
- Cos 0
- Factors of 8
- Factors of 72
- Convex polygon
- Factors of 6
- Factors of 63
- Factors of 54
- Converse of Pythagoras Theorem
- Conversion of Units
- Convert Decimal To Octal
- Value of Root 3
- XXXVII Roman Numerals
- Continuous Variable
- Different Forms Of The Equation Of Line
- Construction of Square
- Divergence Theorem
- Decimal Worksheets
- Cube Root 1 to 20
- Divergence Theorem
- Difference Between Simple Interest and Compound Interest
- Difference Between Relation And Function
- Cube Root Of 1728
- Decimal to Binary
- Cube Root of 216
- Difference Between Rows and Columns
- Decimal Number Comparison
- Data Management
- Factors of a Number
- Factors of 90
- Cos 360
- Factors of 96
- Distance between Two Lines
- Cube Root of 3
- Factors of 81
- Data Handling
- Convert Hexadecimal To Octal
- Factors of 68
- Factors of 49
- Factors of 45
- Continuity and Discontinuity
- Value of Pi
- Value of Pi
- Value of Pi
- Value of Pi
- 1 bigha in square feet
- Value of Pi
- Types of angles
- Total Surface Area of Hemisphere
- Total Surface Area of Cube
- Thevenin's Theorem
- 1 million in lakhs
- Volume of the Hemisphere
- Value of Sin 60
- Value of Sin 30 Degree
- Value of Sin 45 Degree
- Pythagorean Triplet
- Acute Angle
- Area Formula
- Probability Formula
- Even Numbers
- Complementary Angles
- Properties of Rectangle
- Properties of Triangle
- Co-prime numbers
- Prime Numbers from 1 to 100
- Odd Numbers
- How to Find the Percentage?
- HCF Full Form
- The Odd number from 1 to 100
- How to find HCF
- LCM and HCF
- Calculate the percentage of marks
- Factors of 15
- How Many Zeros in a Crore
- How Many Zeros are in 1 Million?
- 1 Billion is Equal to How Many Crores?
- Value of PI
- Composite Numbers
- 100 million in Crores
- Sin(2x) Formula
- The Value of cos 90°
- 1 million is equal to how many lakhs?
- Cos 60 Degrees
- 1 Million Means
- Rational Number
- a3-b3 Formula with Examples
- 1 Billion in Crores
- Rational Number
- 1 Cent to Square Feet
- Determinant of 4×4 Matrix
- Factor of 12
- Factors of 144
- Cumulative Frequency Distribution
- Factors of 150
- Determinant of a Matrix
- Factors of 17
- Bisector
- Difference Between Variance and Standard Deviation
- Factors of 20
- Cube Root of 4
- Factors of 215
- Cube Root of 64
- Cube Root of 64
- Cube Root of 64
- Factors of 23
- Cube root of 9261
- Cube root of 9261
- Determinants and Matrices
- Factors of 25
- Cube Root Table
- Factors of 28
- Factors of 4
- Factors of 32
- Differential Calculus and Approximation
- Difference between Area and Perimeter
- Difference between Area and Volume
- Cubes from 1 to 50
- Cubes from 1 to 50
- Curved Line
- Differential Equations
- Difference between Circle and Sphere
- Cylinder
- Difference between Cube and Cuboid
- Difference Between Constants And Variables
- Direct Proportion
- Data Handling Worksheets
- Factors of 415
- Direction Cosines and Direction Ratios Of A Line
- Discontinuity
- Difference Between Fraction and Rational Number
- Difference Between Line And Line Segment
- Discrete Mathematics
- Disjoint Set
- Difference Between Log and Ln
- Difference Between Mean, Median and Mode
- Difference Between Natural and whole Numbers
- Difference Between Qualitative and Quantitative Research
- Difference Between Parametric And Non-Parametric Tests
- Difference Between Permutation and Combination