How to use the ACOS function
What is the ACOS function?
The ACOS function calculates the arc-cosine of a given number.
Table of Contents
1. Introduction
What is arc-cosine?
The arc-cosine, denoted as cos-1, is the inverse function of the cosine trigonometric function. It takes a value between -1 and 1 and calculates the corresponding angle (in radians) whose cosine is equal to that value.
The relationship between the cosine function and the arc-cosine function is as follows:
In a right-angled triangle, where:
A is the angle (in radians)
b is the length of the adjacent side
c is the length of the hypotenuse
The cosine of the angle A can be expressed as:
cos(A) = b/c
By taking the arc-cosine (ACOS) of the ratio of b (adjacent) and c (hypotenuse), we can find the angle A:
A = arc-cos (b/c)
This means that the ACOS function calculates the angle (in radians) when given the ratio of the adjacent side to the hypotenuse.
You can find the cos-1 on many scientific calculators, the image to the right shows the Win 11 calculator in "scientific" mode.
Press with left mouse button on the image to see the full size.
What is the cosine ratio?
The cosine ratio is the adjacent side divided by the hypotenuse of a right triangle.
cosine = adjacent side / hypotenuse
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). The image above shows a triangle with angle C equal to π/2 radians (90°).
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 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
2. Syntax
ACOS(number)
| number | Required. A number equal to or larger than -1 or equal to or less than 1. -1 <= number <= 1. |
The arccosine is the angle whose cosine is number. The returned angle is given in radians in the range 0 (zero) to pi.
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 hypotenuse is 5 units?
C = π/2 radians (90°)
The argument is:
- number = adjacent / hypotenuse = b/c = 3/5 = 0.6
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 COS function:
This formula returns 0.6 which matches the ratio between the adjacent side (3) and the hypotenuse (5) which is 3/5=0.6
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?
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). We need to find the hypotenuse in order to calculate the ratio between the adjacent side and the hypotenuse.
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.
The argument is:
- number = adjacent / hypotenuse = b/c = 4/5 = 0.8
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 COS function:
This formula returns 0.8 which matches the ratio between the adjacent side (4) and the hypotenuse (5) which is 4/5=0.8
Functions in 'Math and trigonometry' category
The ACOS function function is one of 62 functions in the 'Math and trigonometry' category.
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