# Cosine Calculator

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 unit circle, 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 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. Also, if you'd like to learn how to play around with it, make sure to check the phase shift calculator.

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:

α (angle) | sin(α) | ||
---|---|---|---|

Degrees | Radians | Exact | Decimal |

0° | 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° | 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!

**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.