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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