# How to use the ACOT function

**What is the ACOT function?**

The ACOT function calculates the arc-cotangent of a given number which is an angle given in radians from 0 (zero) to pi.

#### Table of Contents

## 1. Introduction

**What is the trigonometric tangent?**

The tangent ratio is the opposite side divided by adjacent side of a right triangle.

tangent= opposite / adjacent side

**What is the cotangent?**

The trigonometric cotangent is a function that relates an angle of a right triangle to the ratio of adjacent side and the opposite side. It is also the inverse of the tangent, cot(θ) = 1/tan(θ).

**What is the arc-cotangent?**

The arc-cotangent is the inverse cotangent also written cot^{-1}. The inverse cotangent is used to find the angle θ when given the cotangent ratio.

The relationship between the cot function and the arc-cot function is as follows:

In a right-angled triangle, where:

A is the angle (in radians)

b is the length of the adjacent side

a is the length of the opposite side

The cotangent of the angle A can be expressed as:

cot(A) = b/a

By taking the arc-cotangent (ACOT) of the ratio of b (adjacent) and a (opposite), we can find the angle A:

A = arc-cot (b/a)

This means that the ACOT function calculates the angle (in radians) when given the ratio of the adjacent side to the opposite side.

**What is the opposite side?**

The opposite side is the side opposite to the angle being considered. The image above shows a right-angled triangle, it has three internal angles represented by **A**, **B**, and **C**. The opposite side is determined by the chosen angle **A**, **B** or **C**. **A** has the opposite side a, **B** - b, and **C** - c

**What is the adjacent side?**

The adjacent side is the side that is in contact with the angle being considered and the right angle.

**What is the hypotenuse?**

The hypotenuse is the longest side of the right-angled triangle. It is the side opposite to the right angle (90 degrees).

**What is the angle θ?**

The Greek letter theta (θ) is commonly used to represent an unknown angle in a right triangle. The ACOS function returns the angle θ expressed in radians.

**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 is an arc?**

An arc is a curved segment of a circle's circumference, it is a portion of the circle's curve, defined by two endpoints.

In other words, an arc is formed by two radii intersecting the circumference and the enclosed edge between them.

**What is radii?**

The plural form of the word "radius".

**What is the radius of a circle?**

The radius of a circle is the distance from the center point to any point on the circle's edge or circumference. The radius lets you calculate a circle's circumference and area.

**What are degrees?**

Degrees are a unit used to measure angles. It is based on dividing a full circle into 360 equal parts. Degrees are divided into fractional parts like minutes and seconds for more precision.

**What is the relationship between radians and degrees?**

The circumference of a circle is 360 degrees or 2π radians.

360 degrees = 2π radians

which is

degrees = radians x (180 / π)

Excel has two functions for converting between radians and degrees: RADIANS | DEGREES

**What is the difference between the COT function and the ACOT function?**

The COT function calculates the ratio of the adjacent side divided by the opposite side based on an angle expressed in radians.

COT(θ) = opposite / adjacent

The ACOT function calculates the angle expressed in radians based on a number representing the ratio of the adjacent side divided by the opposite side.

ACOT(opposite / adjacent) = θ

## 2. Syntax

ACOT(*number*)

number |
Required. The number is the cotangent of the angle you want. This must be a real number. |

### Comments

Use DEGREES function to convert radians to degrees.

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What is the DEGREES function? The DEGREES function calculates degrees based on radians. Degrees are denoted by °. Table of […]

## 3. Example 1

**Find the angle (in radians) between the hypotenuse and the adjacent side of a right-angled triangle, where the adjacent side is 3 units, and the opposite is 4 units?**

**C** = π/2 radians (90°)

The argument is:

- number = adjacent / hypotenuse = b/a = 3/4 = 0.75

Formula in cell C20:

The formula in cell C20 returns 0.927295218001612 radians which represents the angle for A in the image above. To get the result in degrees we can use the DEGREES function:

which returns approx. 53.13°

We can also calculate the ratio based on the angle using the COT function:

This formula returns 0.75 which matches the ratio between the adjacent side (3) and the opposite (4) which is 3/4=0.75

The image above shows a right-angled triangle in blue, the opposite side named a is equal to 4. The adjacent side named b is equal to 3, the hypotenuse named c is equal to 5. A right-angled triangle means that one of the internal angles is equal to π/2 radians (90°).

**C** = π/2 radians (90°)

**A** = 0.927 radians (53.13°)

**B** = 180° - 90° - 53.13° = 36.87°

## 4. Example 2

**Calculate the angle (in radians) between the horizontal and the line joining the points (0, 0) and (4, 3) in the Cartesian plane using the ACOT function?**

The Cartesian coordinate system specifies each point by a pair of real numbers called coordinates x and y (x,y). The question describes a line from (0,0) to (4,3), this means that x is equal to 4 and y is equal to 3.

This tells us that the opposite side in the triangle is 3 (a) and the adjacent side is 4 (b).

The argument is:

- number = adjacent / opposite= b/a = 4/3 = 1.33333

Formula in cell C20:

The formula in cell C20 returns 0.643501108793284 radians which represents the angle between the line (0,0) - (4,3) and the horizontal dashed black line, in the image above. To get the result in degrees we can use the DEGREES function:

which returns approx. 36.87°

We can also calculate the ratio based on the angle using the COT function:

This formula returns 1.33333333 which matches the ratio between the adjacent side (4) and the opposite (3) which is 4/3=1.333333

### Functions in 'Math and trigonometry' category

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

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