# 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 , 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:

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.