Cos Calculator
Need the cosine of 47.3° right now – or perhaps cos(2π/5)? A cos calculator gives you the answer in a fraction of a second, whether you work in degrees or radians. Below is the tool, followed by a full breakdown of how cosine works, the most common values, and the formulas behind the numbers.
Reference: common cosine values
| Angle (°) | Angle (rad) | Exact | Decimal |
|---|---|---|---|
| 0° | 0 | 1 | 1.0000 |
| 30° | π/6 | √3/2 | 0.8660 |
| 45° | π/4 | √2/2 | 0.7071 |
| 60° | π/3 | 1/2 | 0.5000 |
| 90° | π/2 | 0 | 0.0000 |
| 120° | 2π/3 | −1/2 | −0.5000 |
| 135° | 3π/4 | −√2/2 | −0.7071 |
| 150° | 5π/6 | −√3/2 | −0.8660 |
| 180° | π | −1 | −1.0000 |
| 270° | 3π/2 | 0 | 0.0000 |
| 360° | 2π | 1 | 1.0000 |
What is cosine?
Cosine is one of the three primary trigonometric functions, alongside sine and tangent. In a right-angled triangle, the cosine of an angle equals the ratio of the length of the adjacent side to the hypotenuse:
$$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$On the unit circle – a circle with a radius of 1 centered at the origin – the cosine of any angle is simply the x-coordinate of the point where the terminal side of the angle intersects the circle. This geometric interpretation makes cosine work for angles of any size, not just acute angles in a right triangle.
Key properties of the cosine function:
- Range: from −1 to 1 (inclusive)
- Period: 360° or 2π radians – the pattern repeats every full rotation
- Even function: cos(−x) = cos(x), meaning the graph is symmetric about the y-axis
- Pythagorean identity: cos²(θ) + sin²(θ) = 1
How does the cos calculator work?
The calculator above takes two inputs: an angle value and the unit (degrees or radians). It applies the standard cosine function – the same one used in scientific calculators and programming languages – and returns a decimal result rounded to a configurable number of decimal places.
If you enter an angle in degrees, the tool first converts it to radians using the formula:
$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$It then computes the cosine using a numerical algorithm (Taylor series or CORDIC), which provides accuracy to at least 10 decimal places.
Table of common cosine values
The table below lists cosine for the angles you will encounter most often in textbooks, exams, and engineering.
| Angle (degrees) | Angle (radians) | cos(θ) | Decimal |
|---|---|---|---|
| 0° | 0 | 1 | 1.0000 |
| 30° | π/6 | √3/2 | 0.8660 |
| 45° | π/4 | √2/2 | 0.7071 |
| 60° | π/2 | 1/2 | 0.5000 |
| 90° | π/2 | 0 | 0.0000 |
| 120° | 2π/3 | −1/2 | −0.5000 |
| 135° | 3π/4 | −√2/2 | −0.7071 |
| 150° | 5π/6 | −√3/2 | −0.8660 |
| 180° | π | −1 | −1.0000 |
| 270° | 3π/2 | 0 | 0.0000 |
| 360° | 2π | 1 | 1.0000 |
These values come directly from the unit circle and the special right triangles (30-60-90 and 45-45-90).
How to calculate cosine by hand
For standard angles (0°, 30°, 45°, 60°, 90°, and their multiples), you can derive cosine from known triangle ratios or the unit circle – no calculator needed.
For arbitrary angles, the main method is the Taylor series expansion:
$$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots$$where x is in radians. Each additional term improves accuracy. Taking just the first five terms gives a result accurate to about four decimal places for angles under 90°.
Example: cos(60°). Convert to radians: 60 × π/180 ≈ 1.0472.
- 1 = 1
- −(1.0472)² / 2 = −0.5483
- +(1.0472)⁴ / 24 = +0.0501
- −(1.0472)⁶ / 720 = −0.0025
- +(1.0472)⁸ / 40320 = +0.0001
Sum ≈ 0.4993, which rounds to 0.5 – the exact value of cos(60°).
Degrees vs. radians: how to convert
The two units measure the same angles differently:
- Degrees divide a full circle into 360 parts – intuitive for everyday use.
- Radians relate the angle to arc length on the unit circle – natural for mathematics and physics. A full circle equals 2π radians ≈ 6.2832.
Conversion formulas:
$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$$$\text{degrees} = \text{radians} \times \frac{180}{\pi}$$When using a cos calculator, always check which unit is selected. cos(90) in degrees is 0, but cos(90) in radians is approximately −0.4481 – a completely different result.
Real-world applications of cosine
Cosine appears throughout science and engineering:
- Physics – resolving vectors into horizontal and vertical components; analyzing wave motion and oscillations
- Electrical engineering – calculating power factor in AC circuits (cos φ determines the ratio of real power to apparent power)
- Navigation and surveying – computing distances between points using the law of cosines: c² = a² + b² − 2ab·cos(C)
- Computer graphics – rotation of objects in 2D and 3D space relies on cosine (and sine) transformation matrices
- Signal processing – the Discrete Cosine Transform (DCT) underlies JPEG image compression and MP3 audio encoding
The law of cosines
Beyond right triangles, cosine enables solving any triangle through the law of cosines:
$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$where a, b, c are side lengths and C is the angle opposite side c. When C = 90°, cos(C) = 0 and the formula reduces to the Pythagorean theorem. This generalization makes the law of cosines essential for surveying, navigation, and structural engineering.
This article is for educational purposes. For professional calculations in engineering or finance, verify results with certified tools.