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
5. Function not working
The ACOT function returns
- #VALUE! error if you use a non-numeric input value.
- #NAME? error if you misspell the function name.
- propagates errors, meaning that if the input contains an error (e.g., #VALUE!, #REF!), the function will return the same error.
5.1 Troubleshooting the error value
When you encounter an error value in a cell a warning symbol appears, displayed in the image above. Press with mouse on it to see a pop-up menu that lets you get more information about the error.
- The first line describes the error if you press with left mouse button on it.
- The second line opens a pane that explains the error in greater detail.
- The third line takes you to the "Evaluate Formula" tool, a dialog box appears allowing you to examine the formula in greater detail.
- This line lets you ignore the error value meaning the warning icon disappears, however, the error is still in the cell.
- The fifth line lets you edit the formula in the Formula bar.
- The sixth line opens the Excel settings so you can adjust the Error Checking Options.
Here are a few of the most common Excel errors you may encounter.
#NULL error - This error occurs most often if you by mistake use a space character in a formula where it shouldn't be. Excel interprets a space character as an intersection operator. If the ranges don't intersect an #NULL error is returned. The #NULL! error occurs when a formula attempts to calculate the intersection of two ranges that do not actually intersect. This can happen when the wrong range operator is used in the formula, or when the intersection operator (represented by a space character) is used between two ranges that do not overlap. To fix this error double check that the ranges referenced in the formula that use the intersection operator actually have cells in common.
#SPILL error - The #SPILL! error occurs only in version Excel 365 and is caused by a dynamic array being to large, meaning there are cells below and/or to the right that are not empty. This prevents the dynamic array formula expanding into new empty cells.
#DIV/0 error - This error happens if you try to divide a number by 0 (zero) or a value that equates to zero which is not possible mathematically.
#VALUE error - The #VALUE error occurs when a formula has a value that is of the wrong data type. Such as text where a number is expected or when dates are evaluated as text.
#REF error - The #REF error happens when a cell reference is invalid. This can happen if a cell is deleted that is referenced by a formula.
#NAME error - The #NAME error happens if you misspelled a function or a named range.
#NUM error - The #NUM error shows up when you try to use invalid numeric values in formulas, like square root of a negative number.
#N/A error - The #N/A error happens when a value is not available for a formula or found in a given cell range, for example in the VLOOKUP or MATCH functions.
#GETTING_DATA error - The #GETTING_DATA error shows while external sources are loading, this can indicate a delay in fetching the data or that the external source is unavailable right now.
5.2 The formula returns an unexpected value
To understand why a formula returns an unexpected value we need to examine the calculations steps in detail. Luckily, Excel has a tool that is really handy in these situations. Here is how to troubleshoot a formula:
- Select the cell containing the formula you want to examine in detail.
- Go to tab “Formulas” on the ribbon.
- Press with left mouse button on "Evaluate Formula" button. A dialog box appears.
The formula appears in a white field inside the dialog box. Underlined expressions are calculations being processed in the next step. The italicized expression is the most recent result. The buttons at the bottom of the dialog box allows you to evaluate the formula in smaller calculations which you control. - Press with left mouse button on the "Evaluate" button located at the bottom of the dialog box to process the underlined expression.
- Repeat pressing the "Evaluate" button until you have seen all calculations step by step. This allows you to examine the formula in greater detail and hopefully find the culprit.
- Press "Close" button to dismiss the dialog box.
There is also another way to debug formulas using the function key F9. F9 is especially useful if you have a feeling that a specific part of the formula is the issue, this makes it faster than the "Evaluate Formula" tool since you don't need to go through all calculations to find the issue..
- Enter Edit mode: Double-press with left mouse button on the cell or press F2 to enter Edit mode for the formula.
- Select part of the formula: Highlight the specific part of the formula you want to evaluate. You can select and evaluate any part of the formula that could work as a standalone formula.
- Press F9: This will calculate and display the result of just that selected portion.
- Evaluate step-by-step: You can select and evaluate different parts of the formula to see intermediate results.
- Check for errors: This allows you to pinpoint which part of a complex formula may be causing an error.
The image above shows cell reference B3 converted to hard-coded value using the F9 key. The ACOT function requires numerical values which is not the case in this example. We have found what is wrong with the formula.
Tips!
- View actual values: Selecting a cell reference and pressing F9 will show the actual values in those cells.
- Exit safely: Press Esc to exit Edit mode without changing the formula. Don't press Enter, as that would replace the formula part with the calculated value.
- Full recalculation: Pressing F9 outside of Edit mode will recalculate all formulas in the workbook.
Remember to be careful not to accidentally overwrite parts of your formula when using F9. Always exit with Esc rather than Enter to preserve the original formula. However, if you make a mistake overwriting the formula it is not the end of the world. You can “undo” the action by pressing keyboard shortcut keys CTRL + z or pressing the “Undo” button
5.3 Other errors
Floating-point arithmetic may give inaccurate results in Excel - Article
Floating-point errors are usually very small, often beyond the 15th decimal place, and in most cases don't affect calculations significantly.
Functions in 'Math and trigonometry' category
The ACOT function function is one of 62 functions in the 'Math and trigonometry' category.
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