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 (gradians work too) — and you'll find the cosine value instantly.
Read on to understand what a cosine is 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 based on a right triangle or the unit circle, in an analogical way to how 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
The name cosine comes from the Latin prefix co- and the 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:
- The range (codomain) of cosine is −1 ≤ cos(α) ≤ 1;
- The cosine period is equal to 2π, so cos(α + 360°) = cos(α);
- It's an even function (while sine is odd!), which means that cos(−α) = cos(α); and
- The cosine definition is essential to understanding 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 cosine function is a smooth wave that oscillates between −1 and 1 with a period of 2π. It starts at its maximum value of 1 when the angle is 0, crosses zero at 90°, reaches its minimum of −1 at 180°, crosses zero again at 270°, and returns to 1 at 360°.
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 below.
| α (degrees) | α (radians) | cos(α) — exact | cos(α) — 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 | 1/2 | 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 | −1/2 | −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 the angle lies. Remember the periodicity of the cosine function — cos(α + 360°) = cos(α) — if your angle is not shown in the table above.
Example: How to Use the Cosine Calculator
- Enter the angle in the input field. For example, type 45.
- Choose the angle unit — degrees, radians, or gradians. Keep it on degrees for this example.
- Click Calculate.
- The calculator instantly shows the cosine value: cos(45°) = √2/2 ≈ 0.7071067812.
- It also shows the same angle converted to every unit, the related trig values (sin, tan, sec), and the quadrant the angle lands in.
Use the quick-value buttons to jump straight to standard angles like 0°, 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, and 360°.
FAQs
What is the cosine of 0, 30, 45, 60, and 90 degrees?
cos 0° = 1, cos 30° = √3/2 ≈ 0.8660, cos 45° = √2/2 ≈ 0.7071, cos 60° = 1/2 = 0.5, and cos 90° = 0. These five values are the most commonly used and worth memorizing.
Is cosine an even or odd function?
Cosine is an even function, meaning cos(−α) = cos(α). Its graph is symmetric about the y-axis. Sine, by contrast, is an odd function.
What is the range of the cosine function?
The cosine of any real angle always lies between −1 and 1 (inclusive): −1 ≤ cos(α) ≤ 1. There is no angle whose cosine is greater than 1 or less than −1.
How are cosine and sine related?
They are cofunctions: cos(α) = sin(90° − α). The cosine of an angle equals the sine of its complement. Together with the Pythagorean identity sin²(α) + cos²(α) = 1, this links the two functions at every angle.
How do I convert between degrees, radians, and gradians?
A full circle is 360° = 2π rad = 400 grad. To convert degrees to radians, multiply by π/180; to convert degrees to gradians, multiply by 10/9. This calculator shows all three units automatically.