What is Cosine (Cos)?
In trigonometry, Cosine is one of the three main trigonometric functions. In a right-angled triangle, the cosine of an acute angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
On a Unit Circle (a circle with radius 1 centered at the origin), the cosine of an angle $\theta$ represents the x-coordinate of the point on the circle. This means that while sine tracks the height (vertical), cosine tracks the horizontal distance from the center.
Mathematical Definitions
Usage Areas
Physics
Calculating "Work" involves cosine: Work = Force × Distance × cos(θ). It filters out the force that doesn't contribute to movement.
Electrical Engineering
The "Power Factor" in AC circuits is the cosine of the phase difference between voltage and current.
Frequently Used Values
| Angle (Degree) | Angle (Radian) | cos(θ) |
|---|---|---|
| 0° | 0 | 1 |
| 30° | π/6 | √3/2 (≈0.866) |
| 45° | π/4 | √2/2 (≈0.707) |
| 60° | π/3 | 0.5 |
| 90° | π/2 | 0 |
How to Use This Cosine Calculator
Enter an angle in degrees or radians. The Cosine (Cosinus) Calculator returns cos(θ), the horizontal coordinate of the point on the unit circle at that angle. Cosine ranges from -1 to 1. Use it for trigonometry, physics, engineering, or any problem involving right triangles or periodic functions. cos(0°) = 1, cos(90°) = 0, cos(180°) = -1. For the vertical component use sine; for the ratio sin/cos use tangent.
When Cosine Is Used
Cosine appears in right triangles (adjacent/hypotenuse), waves, Fourier analysis, and vector projection. Ensure your calculator is in the correct angle mode (degrees vs radians). This tool is for educational and reference use. For inverse cosine (arccos), use a dedicated inverse trig tool.
Cosine vs sine and tangent
In a right triangle, cos(θ) = adjacent / hypotenuse, while sin(θ) = opposite / hypotenuse and tan(θ) = opposite / adjacent. For the unit circle, cosine gives the x-coordinate and sine the y-coordinate. Use cosine when you need the horizontal component (e.g. force along an axis, displacement in phase); use sine for the vertical component. For slope or the ratio sin/cos, use tangent. This calculator is for cosine only; for sine or tangent, use the corresponding tools.
On the unit circle, cosine is the x-coordinate of the point at angle θ; sine is the y-coordinate. Cosine is an even function (cos(-θ) = cos(θ)) and has period 2π. Common values: cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0. In physics and engineering, cosine models horizontal displacement, AC voltage, and projection onto an axis. This calculator is for informational use only.
Limitations
Ensure the angle is in degrees or radians as selected; mixing units gives wrong results. For very large angles, reduce modulo 360° or 2π to get the equivalent angle. For the inverse (angle from a cosine value), use an arccos calculator. This tool is for educational and reference use only; verify critical values with your curriculum or authority.
When to use cosine vs sine vs tangent
Use cosine when you have an angle and need the horizontal component (e.g. adjacent side in a right triangle, or the x-coordinate on the unit circle). Use sine for the vertical component. Use tangent when you need the ratio of opposite to adjacent (e.g. slope or angle of elevation). For the inverse (you have the ratio and need the angle), use arccos, arcsin, or arctan. This calculator only computes cos(θ); for sin or tan use the corresponding tools.
How Cosine Is Computed
The calculator converts the angle to radians if needed (degrees × π/180), then uses the standard math library cosine function. Cosine is defined on the unit circle as the x-coordinate of the point at that angle, and for right triangles as adjacent/hypotenuse. The function is periodic with period 2π (or 360°) and even (cos(-θ) = cos(θ)). For very large angles, the implementation typically reduces the angle to a standard range for numerical stability. This tool is for educational and reference use only; verify angle mode (degrees or radians) before use.
Tips for Use
Always check whether your problem uses degrees or radians; mixing them gives wrong results. In many calculators and programming languages, the default is radians. For right-triangle problems, identify the adjacent side and hypotenuse to apply cos(θ) = adjacent/hypotenuse. For the inverse (finding the angle from a ratio), use an arccos calculator. This tool is for informational use only.
When to Use This Calculator
Use it for trigonometry, geometry, or physics when you need the cosine of an angle. Enter the angle in degrees or radians as required. Verify angle mode for your application. This tool is for educational and reference use only.
Disclaimer
This tool is for informational use only. Verify angle mode and units for your application.
Related Tools
About the Cosine Calculator
A professional cosine tool designed for vector calculations, phase differences, and geometric projections.
Cosine Calculator How to Use?
- Enter the angle value in the box
- Determine the unit as degree or radian
- Click the Calculate button
- Review the visualized solution breakdown
Frequently Asked Questions
What is cosine?
Cosine (cos) is a trigonometric function that represents the ratio of the length of the adjacent side of an angle in a right-angled triangle to the length of the hypotenuse.
How is cosine calculated?
To calculate cosine, you input the angle (degree or radian) into the cosine function. On a unit circle, it corresponds to the x-coordinate of the point where the angle intersects the circle.
What is the Law of Cosines?
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles (c² = a² + b² - 2ab cos(C)). It generalizes the Pythagorean theorem to non-right triangles.
What is the inverse of cosine?
The inverse function is arccosine (arccos or cos⁻¹), which returns the angle whose cosine is a given number.
Cosine Calculator Features
- Fast trigonometric conversion
- High-precision results
- Comprehensive conceptual explanations
- User-friendly mobile interface