How to use the ATAN2 function

What is the ATAN2 function?

The ATAN2 function calculates the arc-tangent of an angle using specific x- and y-coordinates. The returned angle is in radians between -pi to pi, including pi. The angle is between the x-axis and the line from (0,0) and point (x_num, y_num)

Atan2 is more useful when converting Cartesian (x,y) to polar (r,θ), it returns the correct angle θ regardless of quadrant.

1. Introduction

What is the difference between atan2 function and atan function?

Atan handles only angles in the first quadrant while atan2 covers all quadrants.

When to use the atan2 function?

The atan2 function is more useful than atan for converting from cartesian (x,y) coordinates to polar (r,θ) form because it correctly handles all quadrants and signs to determine the full angular direction θ.

If x_num is 1 and y_num is 1 then the angle between the x-axis and the line from (0,0) to (1,1) is 45 degrees or 1/4 pi radians. 1/4 pi is 0.785398163.

What is cartesian (x,y)?

The Cartesian coordinate grid has an x-axis running left-to-right and a y-axis running up-and-down meaning it has two axis or dimensions (2D). Cartesian coordinates are just a way to capture any point on a 2D grid using an x and y value.

What is a quadrant?

The Cartesian coordinate grid has 4 quadrants formed by the x-axis and the y-axis.

Each quadrant has a combination of positive or negative x and y values:

- Quadrant 1 is the upper right with positive x and positive y.
- Quadrant 2 is the upper left with negative x but positive y.
- Quadrant 3 is the lower left with negative x and negative y.
- Quadrant 4 is the lower right with positive x but negative y.

The signs of the (x,y) coordinates tell you which quadrant a given point is located in.

What is a sign?

The sign tells you if a numeral value is positive or negative. For example, if x is -5 and y is -10 then the coordinate is in the third quadrant.

What is polar (r,θ)?

Polar coordinates identify points using a distance r and angle θ. r is the distance of the point from the center origin. θ (theta) is the angle between the point and the positive x-axis.

For example, (5, 100°) in polar coordinates represents r=5 units from the origin or intersection of the x and y axis. The angle θ=100 degrees is related to the x-axis counterclockwise.

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 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

2. Syntax

ATAN2(x_num, y_num)

x_num	Required. The x-coordinate.
y_num	Required. The y-coordinate.

3. Example

Calculate the angles, both radians and degrees, for the following four coordinates using both the ATAN and ATAN2 functions: (4,3) , (-4,3) , (-4.-3) , (4, -3)?

First coordinate: (4,3) blue line. Here are the arguments:

• x_num: 4
• y_num: 3

Formula calculating the radians in cell E22:

The formula in cell E22 returns 0.643 radians which corresponds to approx. 36.87 degrees.

Second coordinate: (-4,3) red line. Here are the arguments:

• x_num: -4
• y_num: 3

Formula calculating the radians in cell E23:

The formula in cell E23 returns 2.498 radians which corresponds to approx. 143.13 degrees.

Third coordinate: (-4,-3) green line. Here are the arguments:

• x_num: -4
• y_num: -3

Formula calculating the radians in cell E24:

The formula in cell E24 returns -2.498 radians which corresponds to approx. -143.13 degrees.

Fourth coordinate: (4,-3) yellow line. Here are the arguments:

• x_num: 4
• y_num: -3

Formula calculating the radians in cell E25:

The formula in cell E25 returns -0.643 radians which corresponds to approx. -36.87 degrees.

The image above shows a chart containing four different lines all originating from coordinates (0,0). The different lines are based on these coordinates:

• (4,3) - blue line.
• (-4,3) - red line.
• (-4.-3) - green line.
• (4, -3) - yellow line.

Below the chart are the arguments and calculations for each of these coordinates, both ATAN and ATAN2 functions. The results from these calculations represent the radians between the positive x axis and the corresponding line.

The lines are in a quadrant each, the first quadrant is located between the positive y axis and the positive x axis and it contains the blue line with these coordinates: (4,3). The angle between the positive x-axis and the blue line is 0.643 radians or 36.87 degrees. The ATAN function returns the same angle because the blue line is in the first quadrant.

The second line is red and is located in the second quadrant between the negative x-axis and the positive y-axis. It has the following coordinates (-4,3). The angle between the positive x-axis and the red line is 2.498 radians or 143.13 degrees degrees. The ATAN function returns an angle that needs to be corrected, first identify which quadrant the line is located in which you can easily do by looking at the signs of the coordinates. Add pi to the angle -0.643 which is 2.498 radians to calculate the correct angle in the second quadrant.

The third line is green and is located in the third quadrant between the negative x-axis and the negative y-axis. It has the following coordinates (-4,-3). The angle between the positive x-axis and the green line is -2.498 radians or -143.13 degrees degrees. The ATAN function returns an angle that needs to be corrected, subtract pi and the angle 0.643 which is -2.498 radians to calculate the correct angle in the third quadrant.

The fourth line is yellow and is located in the fourth quadrant between the positive x-axis and the negative y-axis. It has the following coordinates (4,-3). The angle between the positive x-axis and the yellow line is -0.643 radians or -36.87 degrees. The ATAN function returns the same angle because the angle is between -π/2 and π/2.

4. Function not working

The ATAN" 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.

4.1 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.

4.2 The formula returns an unexpected value

To understand why a formula returns an unexpected value we need to examine the calculations steps in detail. 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 C2 converted to hard-coded value using the F9 key. The ATAN2 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

4.3 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 ATAN2 function function is one of 62 functions in the 'Math and trigonometry' category.

Excel function categories

Excel categories

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