Cosine Calculator
Calculate the cosine of an angle in degrees, radians or gradians. Precise result with up to 10 significant digits.
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.
FAQ
Why is cos(0) = 1? Angle 0 on the unit circle lands on the positive x-axis at (1, 0), and the x-coordinate there is 1.
Cosine vs sine? It's the same wave, just 90° out of phase. At θ = 0 cosine is already at its peak of 1 while sine is still at 0. Everywhere along the way, sin²θ + cos²θ = 1 holds.
When does the law of cosines reduce to Pythagoras? The moment C reaches 90°. Then cos(90°) = 0, the correction term drops, and you're left with the familiar c² = a² + b².
What is cosine similarity? It's the cosine of the angle between two vectors, running from -1 when they point opposite ways to 1 when they share a direction, with 0 marking orthogonality. Recommendation systems and search engines lean on it constantly.
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Calculate the cosine of an angle
Alongside the sine, the cosine is one of the foundations of trigonometry and turns up in everything involving angles, projections and oscillations. This calculator finds the cosine of any angle, with precision of up to ten significant digits.
It works with the angle in degrees, radians or gradians, and setting the correct unit is what separates a right result from a wrong one. You type the angle, choose the unit and see the cosine; no need to recall the table of notable values or open the scientific calculator.
The calculation is done in the browser. A practical reference for maths and physics studies, engineering and any application using the cosine.