Cosine Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Dec 08, 2022

This cosine calculator is a twin tool to our sine calculator – add to them the tangent calculator, 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°.

Right triangle: illustration of the cosine definition.

What is cosine? Cosine definition

Cosine is one of the most basic trigonometric functions. It may be defined based on a right triangle or unit circle, in an 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

Right triangle: illustration of the cosine definition. Adjacent side over a hypotenuse.

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

The name cosine comes from the 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); and
  • Cosine definition is essential to understand the law of cosines – a very useful law to solve any triangle. Discover it with our law of cosines calculator!
Animation showing relationship between sine, cosine, unit circle and angle.
Lucas Vieira / Public domain

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.

Plot of cos(x) in <-2π, 2π> range

The exact cosine value is particularly easy to remember and to define for certain angles – you probably 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

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 the periodicity of the cosine function cos(α + 360°) = cos(α) if your angle is not shown in 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 with 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 value; an exact cosine 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
Angle α
deg
Plot of cos(x) in <-2π, 2π> range
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