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
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 sine calculator.
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:
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:
|15°||π/12||(√6 + √2) / 4||0.9659258263|
|75°||5π/12||(√6 - √2) / 4||0.2588190451|
|105°||7π/12||-(√6 - √2) / 4||-0.2588190451|
|165°||11π/12||-(√6 + √2) / 4||-0.9659258263|
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|
|1st Quadrant||0° < α < 90°||0 < α < π/2||0 < cos(α) < 1||+||decreasing||concave|
|2nd Quadrant||90° < α < 180°||π/2 < α < π||-1 < cos(α) < 0||-||decreasing||convex|
|3rd Quadrant||180° < α < 270°||π < α < 3π/2||-1 < sin(α) < 0||-||increasing||convex|
|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!
- Enter the angle. Switch between the units by a simple click on the unit name. Let's take 40° as an example.
- 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.