The inverse trigonometric functions, also referred to as arcus functions, antitrigonometric functions or cyclometric functions, are the inverse functions of the basic trigonometric functions, which include sine, cosine, tangent, cotangent, secant, and cosecant. These inverse functions in trigonometry are used to find the angle with any of the trigonometry ratios. The inverse trigonometry functions are widely used in engineering, physics, geometry, and navigation.

Table of Contents:

Definition

Formulas

Graphs

Table

Derivatives

Properties

Examples

Inverse trigonometric functions are functions that can be used to reverse the effects of trigonometric functions. They are used to calculate the angle or the length of a side of a right triangle when the other two sides are known.

Inverse trigonometric functions, also known as Arc Functions, are used to find the angle measure in a right triangle when two sides of the triangle are known. This is because they produce the length of arc needed to obtain a particular value of the trigonometric functions, such as sine, cosine, tangent, cosecant, secant, and cotangent, which are especially applicable to the right angle triangle. These functions perform the opposite operation of the trigonometric functions.

Formulas

The basic inverse trigonometric formulas are:

• $\sin^{-1} x = \theta$
• $\cos^{-1} x = \theta$
• $\tan^{-1} x = \theta$

| Arcsine | $\sin^{-1}(-x) = -\sin^{-1}(x), x \in [-1, 1]$ |

| Arccosine | $$\cos^{-1}(-x) = \pi - \cos^{-1}(x), x \in [-1, 1]$$ |

| Arctangent | tan^-1(-x) = -tan^-1(x), x ∈ R |

| Arccotangent | cot^-1(-x) = π – cot^-1(x), x ∈ R |

| Arcsecant | sec^-1(x) = π - sec^-1(x), |x| ≥ 1 |

| Arccosecant | cosec^-1(-x) = -cosec^-1(x), |x| ≥ 1 |

#Inverse Trigonometric Functions Graphs

The inverse of the six important trigonometric functions are:

• Inverse Sine (arcsin)
• Inverse Cosine (arccos)
• Inverse Tangent (arctan)
• Inverse Secant (arcsec)
• Inverse Cosecant (arccsc)
• Inverse Cotangent (arccot)

These inverse trigonometric functions are related to the trigonometric ratios.

Arcsine

Arccosine

Arctangent

Arccotangent

Arcsecant

Arccosecant

Let us discuss the six important types of inverse trigonometric functions, including their definitions, formulas, graphs, properties, and solved examples.

Arcsine Function

The arcsine function, denoted by sin^-1x, is the inverse of the sine function and is represented in the graph as shown below:

| Domain | -1 \leq x \leq 1 |

| Range | -π/2 ≤ y ≤ π/2 |

Arccosine Function

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

y = arccos x

Domain of arcsine function: -1 ≤ x ≤ 1

Range of arcsine function: -π/2 ≤ y ≤ π/2

| Domain | -1 ≤ x ≤ 1 |

| Range | 0 ≤ y < 2π |

Arctangent Function

The arctangent function, denoted by tan^-1x, is the inverse of the tangent function. It is represented in the graph as shown below:

y = arctan x

Domain & Range of Arctangent:

Domain: All Real Numbers Range: [-π/2, π/2]

| Domain | -∞ < x < ∞ |

| Range | -π/2 < y < π/2 |

Arccotangent (Arccot) Function

The arccotangent function, denoted by $\cot^{-1}(x)$, is the inverse of the cotangent function.

y = arccot(x)

Domain of Arccotangent: $(-\infty, \infty)$
Range of Arccotangent: $(-\frac{\pi}{2}, \frac{\pi}{2})$

| Domain | $$-\infty < x < \infty$$ |

| Range | 0 < y < π |

Arcsecant Function

What is the arcsecant (arcsec) function?

The arcsecant function, denoted by $\sec^{-1}(x)$, is the inverse of the secant function. It is represented in the graph as shown below:

y = arcsec x

Domain of Arcsecant: $(-\infty, -1] \cup [1, \infty)$

Range of Arcsecant: $(-\frac{\pi}{2}, \frac{\pi}{2})$

| Domain | -∞ < x < -1 or 1 < x < ∞ |

| Range | 0 ≤ y < π, y ≠ π/2 |

Arccosecant Function

What is arccosecant (arccsc x)?

Arccosecant (arccsc x) is the inverse of the cosecant function, denoted by cosec^-1x. It is represented in the graph as shown below:

![Arccsc Function Graph]()

y = arccosecant x

Domain of Arccosecant: $(-\infty, 0) \cup (0, \infty)$

Range of Arccosecant: $(-\pi/2, \pi/2)$

| Domain | $-\infty \leq x \leq -1$ or $1 \leq x \leq \infty$ |

| Range | -π/2 ≤ y < π/2 |

#Inverse Trigonometric Functions Table

Let us rewrite here all the inverse trigonometric functions with their notation, definition, domain, and range:

• Inverse Sine (sin^-1): Notation: sin^-1; Definition: the angle whose sine is a given number; Domain: [-1, 1]; Range: [-π/2, π/2]

• Inverse Cosine (cos^-1): Notation: cos^-1; Definition: the angle whose cosine is a given number; Domain: [-1, 1]; Range: [0, π]

• Inverse Tangent (tan^-1): Notation: tan^-1; Definition: the angle whose tangent is a given number; Domain: all real numbers; Range: [-π/2, π/2]

• Inverse Cotangent (cot^-1): Notation: cot^-1; Definition: the angle whose cotangent is a given number; Domain: all real numbers except 0; Range: [0, π]

• Inverse Secant (sec^-1): Notation: sec^-1; Definition: the angle whose secant is a given number; Domain: [-1, 1] and all real numbers except 0; Range: [0, π]

• Inverse Cosecant (csc^-1): Notation: csc^-1; Definition: the angle whose cosecant is a given number; Domain: [-1, 1]; Range: [-π/2, π/2]

| Function Name | Notation | Definition | Domain of x | Range |

| Arcsine (Inverse Sine) | y = sin^-1(x) | x = sin y | -1 ≤ x ≤ 1 | * -π/2 ≤ y ≤ π/2 |

-90° ≤ y ≤ 90°

| Arccosine or inverse cosine | y = cos^-1(x) | x = cos y | -1 ≤ x ≤ 1 | * 0 ≤ y ≤ π

0° ≤ y ≤ 180°

This is a table

| Arctangent |