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.

Functions in 'Math and trigonometry' category

The ATAN2 function function is one of 69 functions in the 'Math and trigonometry' category.

Excel function categories

Excel categories

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