# Inverse Trigonometric Functions

**Inverse trigonometric functions** are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. The definition of trigonometry can be given through inverse trigonometric functions. It should be noted that there are 6 inverse trigonometric functions which are:

- Arcsine
- Arccosine
- Arctangent
- Arccotangent
- Arcsecant
- Arccosecant

## Inverse Trigonometric Functions Formulas

Inverse trigonometric functions are also called “**Arc Functions**” since, for a given value of trigonometric functions, they produce the length of arc needed to obtain that particular value.

### Arcsine Function

Arcsine function is an inverse of the sine function denoted by sin^{-1}*x*. It is represented in the graph as shown below:

Domain |
-1 ≤ x ≤ 1 |

Range |
-π/2 ≤ y ≤ π/2 |

### Arccosine Function

Arccosine function is the inverse of the cosine function denoted by cos^{-1}*x*. It is represented in the graph as shown below:

Therefore, the inverse of cos function can be expressed as; **y = cos ^{-1}x **

**(arccosine**

*x*)**Domain & Range of**

**arcsine**function:Domain |
-1≤x≤1 |

Range |
0 ≤ y ≤ π |

### Arctangent Function

Arctangent function is the inverse of the tangent function denoted by tan^{-1}*x*. It is represented in the graph as shown below:

Therefore, the inverse of tangent function can be expressed as; **y = tan ^{-1}x **

**(arctangent**

*x*)**Domain & Range of Arctangent:**

Domain |
-∞ < x < ∞ |

Range |
-π/2 < y < π/2 |

### Arccotangent (Arccot) Function

Arccotangent function is the inverse of the cotangent function denoted by cot^{-1}*x*. It is represented in the graph as shown below:

Therefore, the inverse of cotangent function can be expressed as; **y = cot ^{-1}x **

**(arccotangent**Domain & Range of Arccotangent:

*x*)Domain |
-∞ < x < ∞ |

Range |
0 < y < π |

### Arcsecant Function

**What is arcsecant (arcsec)function?** Arcsecant function is the inverse of the secant function denoted by sec^{-1}*x*. It is represented in the graph as shown below:

Therefore, the inverse of secant function can be expressed as; **y = sec ^{-1}x **

**(arcsecant**

*x*)**Domain & Range of Arcsecant:**

Domain |
-∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ |

Range |
-π/2 < y < π/2 ; y≠0 |

### Arccosecant Function

What is arccosecant (arccsc *x*) function? Arccosecant function is the inverse of the cosecant function denoted by cosec^{-1}*x*. It is represented in the graph as shown below:

Therefore, the inverse of cosecant function can be expressed as; **y = cosec ^{-1}x **

**(arccosecant**

*x*)**Domain & Range of Arccosecant**

Domain |
-∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞ |

Range |
-π/2 < y < π/2 ; y≠0 |

### Inverse Trigonometric Functions Properties

The inverse trigonometric functions are also known as Arc functions. Inverse Trigonometric Functions are defined in a certain interval (under restricted domains). Read More on Inverse Trigonometric Properties here.

### Trigonometry Basics

Trigonometry basics include the basic trigonometry and trigonometric ratios such as sin x, cos x, tan x, cosec x, sec x and cot x. The following article from CT’S discusses the basic definition of another tool of trigonometry – Inverse Trigonometric Functions.

### Inverse Trigonometric Functions Examples

**Example 1:** Find the value of x, for sin(x) = 2.

**Solution:**

Given: sinx = 2 x =sin^{-1}(2), which is not possible.

Hence, there is no value of x for which sin x = 2; since the domain of sin^{-1}x is -1 to 1 for the values of x.

**Example 2:** Find the value of sin^{-1}(sin (π/6))

**Solution**:

sin^{-1}(sin (π/6) = π/6 (Using identity sin^{-1}(sin (x) ) = x)

**Example 3:** Find sin (cos^{-1} 3/5)

**Solution**:

Suppose that, cos^{-1} 3/5 = x

So, cos x = 3/5

We know, sin x = \(\sqrt{1 – cos^2 x}\)

So, sin x = \(\sqrt{1 – \frac{9}{25}}\) = 4/5

This implies, sin x = sin (cos^{-1} 3/5) = 4/5

### Practice Problems

**Problem 1: ** Solve tan(arcsin 12/13)

**Problem 2:** Find the value of x, cos(arccos 1) = cos x