# How to use the COT function

**What is the COT function?**

The COT function calculates the cotangent of an angle specified in radians.

## 1. Introduction

**What is the cotangent?**

The cotangent is one of the trigonometric functions closely related to the tangent function.

The cot function is defined as the ratio between the the length of adjacent side and length of the opposite side of a right triangle.

cot(θ) = adjacent / opposite

or

defined as the reciprocal of tangent:

cot(θ) = 1 / tan(θ)

The cotangent equals the ratio of the cos and sin:

cot(θ) = cos(θ) / sin(θ)

**What is the angle θ?**

The Greek letter theta (θ) is commonly used to represent an unknown angle in a right triangle.

**What is a right triangle?**

A right triangle is a type of triangle that contains one internal angle measuring 90 degrees or π/2 radians (a right angle).

**What are radians?**

Radians are a unit used to measure angles. An angle of 1 radian has an arc length equal to the circle's radius.

**What is the relationship between the number pi and radians?**

Radians measure angles by the length of the arc they make in a circle rather than degrees. The full circumference of any circle is 2π multiplied by the circle's radius (2πr).

Since the circumference goes all the way around a circle, that means the full circle measures 2π radians. Half a circle would be π radians (half of 2π). A quarter circle is 2π/4 = π/2 radians. An eighth of a circle is 2π/8 = π/4 radians.

Excel has a function that returns the number pi: PI function

**What are the main trigonometric functions, their domain and range?**

Function |
Domain (input) |
Range (output) |

sin(x) | All real numbers | (-1, 1) |

cos(x) | All real numbers | (-1, 1) |

tan(x) | All real numbers except multiples of π/2 | (-∞, ∞) |

sec(x) | All real numbers except multiples of π | (1, ∞) U (-∞, -1) |

csc(x) | All real numbers except integer multiples of π | (-∞, -1) U (1, ∞) |

cot(x) | All real numbers except integer multiples of π | (-∞, ∞) |

**What are the main trigonometric arcfunctions, their domain and range?**

Function |
Domain (input) |
Range (output) |

sin^{-1}(x) |
[-1, 1] | (-π/2, π/2) |

cos^{-1}(x) |
[-1, 1] | (0, π) |

tan^{-1}(x) |
All real numbers | (-π/2, π/2) |

sec^{-1}(x) |
[-1, 1] | (0, π/2) U (π/2, π) |

csc^{-1}(x) |
[-1, 1] | (-π/2, -0) U (0, π/2) |

cot^{-1}(x) |
All real numbers | (0, π) |

**What is the trigonometric domain?**

The domain of a trigonometric function refers to the set of input values it is defined and valid for. The secant, cosecant, and cotangent functions have restricted domains due to their asymptotes.

**What is the trigonometric range?**

The range of a trigonometric function refers to the set of possible output values it returns.

**What are asymptotes in terms of the cotangent function?**

Asymptotes are vertical lines that the function approaches but never reaches. Cotangent has vertical asymptotes at integer multiples of π.

## 2. Syntax

COT(*number*)

number |
Required. An angle in radians. |

Use the RADIANS function to convert degrees to radians.

Recommended articles

What is the RADIANS function? The RADIANS function converts degrees to radians. Table of Contents Introduction Syntax Example 1 Example […]

## 3. Example 1

*Calculate the cotangent of **π/4**? *

The argument is:

- number: C18 which contains π/4 radians or 0.785398163397448 radians.

Formula in cell C20:

Cell C20 returns the cotangent ratio which in this example is 1.

**Determine opposite side (a) if the adjacent side (b) is equal to 1 and the angle (A) is equal to π/4 radians?**

Cotangent A = b/a

Cotangent π/4 = 1

1 = b/a

If the ratio 1 is equal to b/a then 1 = 1/a

a = 1/1

a = 1

**Determine hypotenuse if the adjacent side (b) is 1 and the opposite side (a) is 1?**

You can calculate the hypotenuse using Pythagoras theorem which states that the squared hypotenuse is equal to the sum of the squared opposite side and the adjacent side.

c^{2} = a^{2} + b^{2
}b = 1

a = 1

c^{2} = 1^{2} + 1^{2}

c^{2} = 2

c = √2

c = 1.41421356237309

## 4. Example 2

**In a right triangle, if one acute angle measures 30 degrees and the opposite is 10 units long, find the length of the side adjacent to that angle?
**

The argument is:

- number: C18 which represents the angle in radians. Cell C18 contains 0.523598775598299 radians or π/6 radians 30 degrees.

Formula in cell C20:

Cell C20 returns the cotangent ratio which in this example is 1.73205080756888.

The question tells us that the opposite side in the triangle is 10 (a) and the cotangent ratio is 1.73205080756888.

number = adjacent / opposite= b/a = b/10 = 1.73205080756888

b = 10 * 1.73205080756888

b = 17.3205080756888

The length of the adjacent side (b) is 17.3205080756888

## 5. Example 3

**Calculate the cotangent ratio if the hypotenuse is 5 and the opposite side is 4?**

The arc-sine (ASIN function) can help us calculate the angle **A** based on the length of the hypotenuse and opposite side. The arc-sine of 4/5 equals 0.927295218001612 radians.

We now have the argument for the COT function so we can calculate the cotangent ratio.

The argument is:

- number: Cell reference C18 which represents the angle in radians. Cell C18 contains 0.927295218001612 radians.

Formula in cell C20:

Cell C20 returns the cotangent ratio which in this example is 0.75.

b/a = 0.75

b = 4 * 0.75

b = 3

It is also possible to calculate the adjacent side b using the Pythagoras theorem which states that the squared hypotenuse is equal to the sum of the squared opposite side and the adjacent side. c^{2} = a^{2} + b^{2}

b = (c^{2} - a^{2})^{(1/2)}

b = (5^{2} -4^{2})^{(1/2)}

b = (25 -16)^{(1/2)}

b = (9)^{(1/2)}

b = 3

The cotangent ratio is 3/4 = 0.75

## 6. Example 4

*Calculate the cotangent ratio if angle B is 30 degrees?*

We know that the sum of the angles in a right-angle triangle is 180 degrees or π radians. 180 = A + B + C

180 = A + 30 + 90

A = 180 - 30 - 90 = 60 degrees

The COT function requires radians in the number argument and we have 60 degrees. The RADIANS function allows us to easily convert the degrees to radians.

Formula in cell C20:

Cell C20 returns 0.577350269189626 which represents the cotangent ratio b/a or the adjacent side divided by the opposite side.

RADIANS(60) converts 60 degrees to 1.0471975511966 radians, the COT function calculates the cotangent ratio based on 1.0471975511966 radians.

### Functions in 'Math and trigonometry' category

The COT function function is one of 61 functions in the 'Math and trigonometry' category.

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