Cosine Calculator

What is Cosine?

In a right triangle, the cosine of an angle θ is the ratio of the adjacent side to the hypotenuse: cos(θ) = adjacent / hypotenuse. By definition, these values never leave the range between −1 and 1.

Worth memorizing are the notable values: cos(0°) = 1, cos(30°) ≈ 0.8660, cos(45°) ≈ 0.7071, cos(60°) = 0.5 and cos(90°) = 0.

Unit conversions: 1° = π/180 rad ≈ 0.01745 rad. In reverse, 1 rad = 180°/π ≈ 57.296°. And 400 gradians make 360°.

Cosine function: definition and example

In a right triangle, cos(θ) = adjacent / hypotenuse. Look at the unit circle instead and cos(θ) becomes the x-coordinate of the point sitting at angle θ. It's periodic with period 2π and even, meaning cos(-θ) = cos(θ), with amplitude 1. The values worth knowing by heart are cos(0) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0. So cos(60°) = 0.5.

Think of cosine as the sine running 90° ahead: cos(θ) = sin(θ + π/2). The Pythagorean identity sin²θ + cos²θ = 1 is nothing more than x² + y² = 1 read off the unit circle. And the law of cosines c² = a² + b² - 2ab·cos C is Pythagoras with a correction term: set C = 90°, cos C drops to 0, and the extra term disappears.

Applications

Through the dot product, cosine tells you the angle between vectors: u·v = |u||v|·cos θ. That's exactly why cosine similarity shows up everywhere in semantic search and NLP, where embeddings get compared by the cosine of their angle and magnitude is set aside. JPEG and MP3 compression lean on a discrete cosine transform (DCT) to strip out redundancy. The haversine formula behind spherical navigation uses cosines of latitudes to work out great-circle distances. And in AC circuits, cosine fixes the phase relations, since the power factor is cos φ.