How to use the ATAN function
What is the ATAN function?
The ATAN function calculates the arctangent of a number. The returned angle is in radians between -pi/2 to pi/2.
Table of Contents
1. Introduction
What is the trigonometric tangent?
The tangent is a trigonometric function that relates an angle θ in a right triangle to the ratio of the length of the side opposite (a) the angle and the length of the adjacent side (b) of the triangle. A right triangle has one angle that measures 90° or π/2 radians which is approximately 1.5707963267949 radians.
The arc-tan is the inverse sine also written tan-1. The inverse tangent is used to find the angle θ when given the sine ratio.
The relationship between the sin function and the arc-tan 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 tan of the angle A can be expressed as:
tan(A) = a/b
By taking the arc-tan (ATAN) of the ratio of a (opposite) and b (adjacent), we can find the angle A:
A = arc-tan (a/b)
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.
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
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 a circle's radius?
The radius of a circle is the distance from the center point to any point on the circle's edge or circumference.
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.
2. Syntax
ATAN(number)
| number | Required. The tangent of the angle. |
3. Example 1
Find the angle A (in radians) between the hypotenuse and the adjacent side of a right-angled triangle, where the adjacent side is 3 units, and the hypotenuse is 5 units?
C = π/2 radians (90°)
The argument is:
- number = opposite / adjacent = a/b = 4/3 = 1.3333333
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 value using the SIN function:
This formula returns 1.33333 which matches the ratio between the opposite side (4) and the adjacent (3) which is 4/3=1.33333
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, use the ATAN 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 = opposite / adjacent= a/b = 3/4 = 0.75
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 TAN function:
This formula returns 0.75 which matches the ratio between the opposite side (3) and the adjacent side (4) which is 3/4=0.75
To calculate the hypotenuse we can use Pythagoras theorem which states that the squared hypotenuse is equal to the sum of the squared opposite side and the adjacent side. c2 = a2 + b2
c = (32+42)1/2 = (9+16)1/2 = 251/2 = 5.
'ATAN' function examples
The following article has a formula that contains the ATAN function.
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Functions in 'Math and trigonometry' category
The ATAN function function is one of 61 functions in the 'Math and trigonometry' category.
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