How to use the DEGREES function

What is the DEGREES function?

The DEGREES function calculates degrees based on radians. Degrees are denoted by °.

1. Introduction

What is degree?

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 radian?
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 the number pi?

Pi is a irrational number meaning it cannot be expressed as a ratio of two integers. In other words, it has an infinite number of decimal places with no repeating pattern.

It is calculated by the ratio of the circumference of a circle to its diameter. circumference = diameter * π or circumference = 2 * π * radius, this means that the diameter = 2 x radius.

When is the DEGREES function useful?

The trigonometric arc functions return a value in radians. If you need the value in degrees you need to convert it to degrees using the DEGREES function.

Excel Function	Right Triangle Relationship
ASIN	sin(θ) = opposite/hypotenuse
ACOS	cos(θ) = adjacent/hypotenuse
ATAN	tan(θ) = opposite/adjacent
ACOT	tan(θ) = opposite/adjacent

The trigonometric functions relate an angle expressed in radians or degrees to two out of three sides in a right triangle.

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 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)

The following table shows degrees from 0 to 360 in 15 degree increments along with the corresponding radian values as fractions of pi:

Degrees	Radians
15	1/12π
30	1/6 π
45	1/4 π
60	1/3 π
75	5/12π
90	1/2 π
105	7/12π
120	2/3 π
135	3/4 π
150	5/6 π
165	11/12π
180	1/1 π
195	13/12π
210	7/6 π
225	5/4 π
240	4/3 π
255	17/12π
270	3/2 π
285	19/12π
300	5/3 π
315	7/4 π
330	11/6 π
345	23/12π
360	2/1 π

How is a circle related to trigonometry?

The geometry and periodicity of circles interconnect deeply with the definitions, identities, and applications of trigonometry. The unit circle bridges the geometry of the circle with the behavior of trig functions.

Right angle is 90 degrees, straight angle is 180 degrees, and a full circle is 360 degrees.

What is the unit circle?

The unit circle is a circle with a radius of 1 unit that is centered at the origin (0, 0) on the Cartesian coordinate plane.

What is the circumference of a circle?

The circumference is the linear distance enclosing the circle.

What is the diameter of a circle?

The diameter of a circle is the straight line distance that passes through the center of the circle connecting one point on the circumference to another, going all the way across the circle.

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.

What is the math formula behind converting radians to degrees?

The formula to convert radians to degrees is:
degrees = (radians * 180) / π

• degrees is the angle measure in degrees
• radians is the angle measure in radians
• π (pi) is approximately equal to 3.14159...

The steps to derive this formula are:

1. Define a full circle/revolution as 360 degrees or 2π radians.
2. Set up a proportion: degrees/360 = radians/2π
3. Cross-multiply: degrees = (radians * 360)/2π
4. Simplify by dividing both sides by 2: degrees = (radians * 180)/π

To convert from radians to degrees you multiply the radian value by 180/π which is approximately 180/3.14159 = 57.2957795... This makes sense geometrically because π radians corresponds to 180 degrees (a straight angle), so 1 radian is (180/π) degrees.

The inverse formula to convert from degrees to radians is:
radians = (degrees * π) / 180

2. Syntax

DEGREES(angle)

angle

Required. The radian you want to convert into degrees.

3. Example 1

Convert the following values in radians [0, π/4, π/2, 3π/4, π, 5/4π, 3/2π, 7/4π, 2π] to degrees?

The radians are specified in cell range B16:B24.

Formula in cell E16:

The formula in cell E16 returns 0 degrees, this means that 0 (zero) radians is equal to 0 (zero) degrees.

The image above displays a chart that illustrates the relationship between angles expressed in radians and their corresponding values in degrees. The chart consists of two main sections: a graphical representation and a tabular data section.

The graphical representation shows a unit circle divided into quadrants, with angles represented in both radians and degrees. The angles are marked along the circumference of the circle, with their radian values indicated below and their degree values indicated above. For example, the angle π/4 radians is equivalent to 45 degrees, and the angle 3π/2 radians is equivalent to 270 degrees.

The tabular data section consists of two columns:

1. Radians: This column lists various angle values in radians, including 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, and 2π.
2. DEGREES function: This column shows the corresponding degree values for the radian values in the previous column, calculated using the DEGREES function in Excel. The DEGREES function converts an angle value from radians to degrees.

For example, the radian value π/4 corresponds to 45 degrees, π/2 corresponds to 90 degrees, and 2π corresponds to 360 degrees.

The chart serves as a visual aid and reference for understanding the conversion between radian and degree measurements of angles which is essential in various mathematical and scientific applications.

4. Example 2

The measure of angle N is 2π/3 radians. The measure of angle D is 5π/12 radians greater that the measure of angle N. What is the measure of angle D, in degrees?

Angle	Radians
N	2.0943951023932	2π/3
D	N + 1.30899693899575	N + 5π/12

Formula in cell E16:

Here is how to calculate degrees based on 2π/3 radians:

1. Divide 2π/3 by 2π to get the ratio: (2π/3)/2π = 1/3
2. Multiply the ratio by 360: (1/3)* 360 = 120 degrees

Formula in cell E17:

Here is how to calculate degrees based on 5π/12 radians:

1. Divide 5π/12 by 2π to get the ratio: (5π/12)/2π = 5/24
2. Multiply the ratio by 360: (5/24)* 360 = 75 degrees

Formula in cell E19:

To calculate angle N we need to add the angles to get a total: 120 + 75 = 195 degrees.

The image shows a graphical representation of angles in a coordinate plane:

• The angle N 2π/3 radians (approximately 120 degrees) is shown.
• The angle 5π/12 radians (approximately 75 degrees) is shown.
• The angle 13π/12 radians (approximately 195 degrees) is shown.

5. Function not working

The DEGREES function returns

• #VALUE! error if the argument is non-numeric.
• #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.
• #SPILL! error occurs in Excel 365 when a dynamic array formula returns an array that cannot fit into the available cells.

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.

1. The first line describes the error if you press with left mouse button on it.
2. The second line opens a pane that explains the error in greater detail.
3. The third line takes you to the "Evaluate Formula" tool, a dialog box appears allowing you to examine the formula in greater detail.
4. This line lets you ignore the error value meaning the warning icon disappears, however, the error is still in the cell.
5. The fifth line lets you edit the formula in the Formula bar.
6. 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:

1. Select the cell containing the formula you want to examine in detail.
2. Go to tab “Formulas” on the ribbon.
3. 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.
4. Press with left mouse button on the "Evaluate" button located at the bottom of the dialog box to process the underlined expression.
5. 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.
6. 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..

1. Enter Edit mode: Double-press with left mouse button on the cell or press F2 to enter Edit mode for the formula.
2. 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.
3. Press F9: This will calculate and display the result of just that selected portion.
4. Evaluate step-by-step: You can select and evaluate different parts of the formula to see intermediate results.
5. 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 B24 converted to hard-coded value using the F9 key. The DEGREES 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.

'DEGREES' function examples

Functions in 'Math and trigonometry' category

The DEGREES function function is one of 62 functions in the 'Math and trigonometry' category.

Excel function categories

Excel categories

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