In this section, the method of finding the derivative of an inverse trigonometric function is introduced.

It is known that trigonometric functions are not one-to-one function, and thus they do not have inverse function. However, trigonometric functions can be converted to one-to-one functions by restricting their domains. For example, function f(x) = sin x is not a one-to-one function from -∞ to +∞ and has no inverse function. This can be seen on the left figure. However, it is a one-to-one function from -π/2 to +π/2, and thus f(x) has inverse function in this range.

The inverse function of f(x) = sin x is called the arcsine function. According to the definition of the inverse function, the following two equations are obtained and they are reciprocal inverse functions.

     sin^-1x = y

     sin y = x         for -π/2 ≤ y ≤ +π/2

The cancellation equation y = f(f ^-1(y)) and x = f ^-1(f(x)) for arcsine function becomes

     sin^-1 (sin x) = x          for -π/2 ≤ x ≤ +π/2

     sin (sin^-1 x) = x          for -1≤ x ≤ +1

The derivative of an arcsine function can be calculated as

where -1< x < +1

The inverse cosine function is denoted by cos^-1 or arccos. Its definition states:

     cos^-1x = y

     cos y = x      

are reciprocal inverse functions where 0 ≤ y ≤ π.

The cancellation equations for inverse cosine function are

     cos^-1 (cos x) = x          for 0 ≤ x ≤ π

     cos (cos^-1 x) = x          for -1 ≤ x ≤ +1

The derivative of the inverse cosine function is given by

The inverse tangent function is defined as the inverse function of tangent function and denoted as tan^-1 or atan. Its domain is the interval (-π/2, +π/2). Its definition states:

     tan^-1x = y

     tan y = x       

are reciprocal inverse functions where -π/2 ≤ y ≤ +π/2.

The arctangent function is horizontal asymptotes because when x approaches +∞, 1/x is extremely close to +π/2 and when x approaches -∞, 1/x is extremely close to -π/2. This can be expressed with mathematics equations:

and

The derivative of the inverse arctangent function is given by

The definition of csc^-1 is given by

     sec^-1x = y         for |x| ≥ 1

     sec y = x         for y ∈[0, π/2] ∪ [π/2, 0]

The definition of csc^-1 states

     csc^-1x = y         for |x| ≥ 1

     csc y = x         for y ∈(0, π/2] ∪ [π, 3π/2]

The definition of cot^-1states

     cot^-1x =& y     

     cot y = x         for y ∈(0, π)

Inverse trigonometric functions also have integration and derivation. Some of the formulas are listed in the table.