Cosine Calculator
Quickly compute the cosine of any angle in degrees. Ideal for resolving horizontal force components, solving triangles, and working with periodic functions in physics or engineering.
About this calculator
The cosine function is a fundamental trigonometric ratio. In a right triangle, cos(θ) equals the length of the side adjacent to the angle divided by the hypotenuse: cos(θ) = adjacent / hypotenuse. Because calculators and programming languages work in radians internally, the degree value is first converted: cos(θ) = cos(angle × π / 180). Like sine, cosine output is always bounded between −1 and 1. Key reference values are cos(0°) = 1, cos(60°) = 0.5, cos(90°) = 0, and cos(180°) = −1. Cosine is essential in the Law of Cosines for solving non-right triangles, in dot-product calculations in vector math, and in describing the horizontal component of any angled force or displacement.
How to use
Suppose you want the cosine of 60°. Enter 60 in the Angle field. The calculator converts this to radians: 60 × π / 180 ≈ 1.0472 radians. It then evaluates cos(1.0472) = 0.5. This result tells you that in a right triangle with a 60° angle, the adjacent side is exactly half the hypotenuse length. You can confirm this with the classic 30-60-90 triangle, where the side adjacent to 60° is always 0.5 times the hypotenuse.
Frequently asked questions
What does the cosine of an angle represent geometrically?
Geometrically, cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle. On the unit circle, cos(θ) is the x-coordinate of the point on the circle at angle θ from the positive x-axis. This makes cosine a measure of horizontal projection. When an object moves or a force acts at an angle, the cosine of that angle gives the fraction of the motion or force acting in the horizontal direction.
How is cosine different from sine and when should I use it?
While sine relates to the side opposite an angle, cosine relates to the side adjacent to it. Use cosine when you know an angle and the hypotenuse and need the adjacent side, or when decomposing a vector into its horizontal component. For example, a 100 N force applied at 30° to the horizontal has a horizontal component of 100 × cos(30°) ≈ 86.6 N. Sine would give the vertical component instead. In practice, choose cosine for horizontal or along-the-axis components and sine for perpendicular or cross-axis components.
Why does cosine equal 1 at 0 degrees and 0 at 90 degrees?
At 0°, the angle points directly along the adjacent side, meaning the adjacent side equals the hypotenuse in length, giving a ratio of 1. As the angle increases toward 90°, the adjacent side shrinks relative to the hypotenuse until it reaches zero length at exactly 90°, making the ratio 0. On the unit circle, this corresponds to the x-coordinate moving from 1 (at angle 0°, pointing right) to 0 (at angle 90°, pointing straight up). This behavior makes cosine a natural model for horizontal oscillations and wave phenomena.