# Cosine Calculator

Created by Hanna Pamuła, PhD candidate
Reviewed by Bogna Szyk
Last updated: Apr 06, 2022
Table of contents:

The cosine calculator is a twin tool of our sine calculator - add to them the tangent tool and you'll have a pack of the most popular trigonometric functions. Simply type the angle - in degrees or radians - and you'll find the cosine value instantly. Read on to understand what is a cosine and to find the cosine definition, as well as a neat table with cosine values for basic angles, such as cos 0°, cos 30° or cos 45°.

## What is cosine? Cosine definition

Cosine is one of the most basic trigonometric functions. It may be defined on the basis of right triangle or , in analogical way as the sine is defined:

The cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.

cos(α) = adjacent / hypotenuse = b / c

If you're not sure what the adjacent and hypotenuse is (and opposite, as well), check out the explanation in the .

A name cosine comes from a Latin prefix co- and sine function - so it literally means sine complement. And, indeed, the cosine function may be defined that way: as the sine of the complementary angle - the other non-right angle. The abbreviation of cosine is cos, e.g. cos(30°).

Important properties of a cosine function:

• Range (codomain) of a cosine is -1 ≤ cos(α) ≤ 1
• Cosine period is equal to 2π
• It's an even function (while sine is odd!), which means that cos(-α) = cos(α)
• Cosine definition is essential to understand the law of cosines - a very useful law to solve any triangle.

## Cosine graph and table (cos 0, cos 30 degrees, cos 45 degrees...)

The image below shows the cosine function in <-2π, 2π> range. Also, if you'd like to learn how to play around with it, make sure to check the phase shift calculator.

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Exact cosine value is particularly easy to remember and to define for certain angles - probably you learned that cos 0° = 1, cos 30° = √3/2 or cos 45° = √2/2. Other basic angles are shown in the table:

CSV To HTML using codebeautify.org
α (angle) sin(α)
Degrees Radians Exact Decimal
0 1 1
15° π/12 (√6 + √2) / 4 0.9659258263
30° π/6 √3/2 0.8660254038
45° π/4 √2/2 0.7071067812
60° π/3 0.5 0.5
75° 5π/12 (√6 - √2) / 4 0.2588190451
90° π/2 0 0
105° 7π/12 -(√6 - √2) / 4 -0.2588190451
120° 2π/3 -0.5 -0.5
135° 3π/4 -√2/2 -0.7071067812
150° 5π/6 -√3/2 -0.8660254038
165° 11π/12 -(√6 + √2) / 4 -0.9659258263
180° π -1 -1

Moreover, you can observe how the cosine function behaves according to the quadrant in which it lays. Remember about periodicity of the cosine function cos(α + 360°) = cos(α), if your angle is out of the range of the table below.

Quadrant / Border Degrees Radians Value Sign Monotony Convexity
0 1 maximum
1st Quadrant 0° < α < 90° 0 < α < π/2 0 < cos(α) < 1 + decreasing concave
90° π/2 0 root, inflection
2nd Quadrant 90° < α < 180° π/2 < α < π -1 < cos(α) < 0 - decreasing convex
180° π -1 minimum
3rd Quadrant 180° < α < 270° π < α < 3π/2 -1 < sin(α) < 0 - increasing convex
270° 3π/2 0 root, inflection
4th Quadrant 270° < α < 360° 3π/2 < α < 2π 0 < sin(α) < 1 + increasing concave

## Example: how to use a cosine calculator

Now you got the hang of what is cosine, using this cosine calculator is a piece of cake!

1. Enter the angle. Switch between the units by a simple click on the unit name. Let's take 40° as an example.
2. Keep calm and read the result - in our case, cos(40°) ≈ 0.766 (remember, it's an approximate, cosine exact value can be found only for specific cases).

Give this cosine calculator a go! Play around by typing the cosine value and finding the angle. The only thing to notice is that our tool will show you the angles in 0 - 180° range - as you know about the periodicity and that the cosine is an even function, it shouldn't be a problem for you to find other possible solutions.

Hanna Pamuła, PhD candidate
Angle α
deg
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