How to use the COS function
What is the COS function?
The COS function calculates the cosine of an angle.
Table of Contents
1. Introduction
What is the cosine?
A, B and C are angles, C is 90° or π/2 radians. One of the angles must be 90° (right angle triangle), in the example above C is 90° or π/2 radians.
d, e, and f are lengths of the sides of the triangle. Side d and e are perpendicular to each other, meaning the angle between them are 90° or π/2 radians.
COSINE A is equal to the ratio between sides e and f, this is only true if the triangle is a right-angled triangle.
COSINE A = e/f
What is the cosine ratio?
The cosine ratio is the adjacent side divided by the hypotenuse of a right triangle: cosine = adjacent side / hypotenuse
The following cosine ratio is based on the image above:
COSINE A = e/f
Side e is the adjacent side based on angle A. Side f is the hypotenuse.
COSINE B = d/f
Side f is the adjacent side based on angle B. Side f is the hypotenuse.
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 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
COS(number)
| number | Required. The radian angle you want to know the cosine of. |
3. Example
Calculate the cosine of π/4?
The argument is:
- number: C18 which contains π/4 radians or 0.785398163397448 radians.
Formula in cell C3:
Cell C3 returns the cosine ratio which in this example is 0.707106781186548.
Determine the hypotenuse if the adjacent side is 1 and the angle is π/4?
Cosine A = b/c
Cosine π/4 = 0.707106781186548
0.707106781186548 = b/c
If the ratio 0.707106781186548 is equal to b/c then 0.707106781186548=1/c
c=1/0.707106781186548
c=1.41421356237309
c=√2
You can also calculate the hypotenuse using 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
If b = 1 then a must also be equal to 1, because the angle is π/4 or 45 degrees.
c2 = 12 + 12
c2 = 2
c = √2
c=1.41421356237309
4. How to plot a cosine wave
The image above demonstrates a scatter chart containing a cosine graph. Here is how to create a cosine plot:
The formula in cell C15 creates radians starting from -2*pi and adds (1/4)*pi for each cell below.
Formula in cell C15:
Formula in cell D15:
4.1 Explaining formula in cell C15
Step 1 - Calculate -2*pi
The PI function returns the number pi, it has no arguments.
-2*PI()
becomes
-2*3.14 (approx.)
and returns approx. -6.28
Step 2 - Subtract with pi/4
The ROW function returns a number representing the row in a cell reference.
ROW(reference)
A1 is a relative cell reference meaning it changes when the cell is copied to cells below, this creates a sequential list of numbers starting from 1 to n.
(ROW(A1)-1)*PI()/4
becomes
(1-1)*PI()/4
becomes
0*PI()/4
and returns 0 (zero).
Step 3 - Calculate radian
-2*PI()+(ROW(A1)-1)*PI()/4
becomes
-6.28 + 0 equals approx. -6.28
4.2 How to insert a Scatter chart with Smooth Lines and Markers
- Select cell range B3:B31.
- Press and hold CTRL.
- Select cell range D3:D13.
- Go to tab "Insert" on the ribbon.
- Press with left mouse button on the "Insert Line or Area chart" button, and a pop-up menu appears.
- Press with left mouse button on the "Line with Markers" button, see the image above.
- A new chart appears on the worksheet, see the image below.
Double press with left mouse button on the chart to open the "Format Chart Area Pane".
Select the line on the chart, press with left mouse button on the checkbox "Smoothed Line" on the "Format Chart Area Pane".
Double-press with left mouse button on the y-axis values, the "Format Axis Pane" shows up.
Press with mouse on the radio button "Axis value:" and type -1.5 (Horizontal Axis crosses)
Press with right mouse button on on x-axis values. Press with mouse on "Add Major gridlines
5. How to change the cosine wave amplitude
The general form of the COSINE wave is y = A*COS(B*(x-C))+D
Constant A in the formula above changes the amplitude or height of the cosine wave. The image above shows two different cosine waves, a green and a yellow one.
The yellow cosine wave has an amplitude or height of 2 and the green one has an amplitude of 1.
Formula in cell D14:
Formula in cell E14:
6. How to change the cosine wave period
The general form of the cosine wave is y = A*COS(B*(x-C))+D
Constant B changes the period, the chart above shows two cosine waves. The green cosine wave completes a cycle in 2π or 360°, however, the yellow cosine wave completes two cycles in 2π or 360°.
Formula in cell C3:
Formula in cell D3:
7. How to change the cosine wave mid-line
The general form of the Sin wave is y = A*Sin(B*(x-C))+D
Constant D lets you change the mid-line which is the center-line in which the cosine wave oscillates back and forth. The mid-line is horizontal and is right in between the minimum and maximum cosine function values.
Formula in cell D14:
The green cosine wave in column D has a maximum value of 1 and a minimum value of -1, the mid-line is 0 (zero).
Formula in cell E14:
The yellow cosine wave in column E has a maximum value of 2 and a minimum value of 0, the mid-line is 1 (zero).
Useful resources
COS function - Microsoft
COS function
'COS' function examples
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Functions in 'Math and trigonometry' category
The COS function function is one of 62 functions in the 'Math and trigonometry' category.
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