Cosine Calculator

In a right triangle, the cosine of an acute angle θ is the ratio of the length of the leg adjacent to θ divided by the length of the hypotenuse: cos(θ) = adjacent / hypotenuse. For example, in a 3-4-5 right triangle where the angle opposite the side of length 4 is θ, the adjacent leg is 3 and the hypotenuse is 5, so cos(θ) = 3/5 = 0.6, which corresponds to θ ≈ 53.13°.

On the unit circle (radius 1 centered at the origin), a point at angle θ measured counter-clockwise from the positive x-axis has coordinates (cos θ, sin θ). So cos(θ) is literally the x-coordinate. This extends the definition to any real angle, including negatives and values greater than 360°. cos(−θ) = cos(θ) because reflecting a point across the x-axis keeps its x-coordinate unchanged — cosine is an even function.

The cosine wave oscillates between −1 and +1 for every real input: −1 ≤ cos(θ) ≤ 1. The function is periodic with period 2π (or 360°), meaning cos(θ + 360°) = cos(θ). It reaches its maximum value of 1 at θ = 0°, 360°, 720°, …, and its minimum of −1 at θ = 180°, 540°, …. The zeros occur at odd multiples of 90° (π/2): 90°, 270°, 450°, …

Enter the angle in the input field with the Degrees tab selected. The calculator converts the angle to radians internally (radians = degrees × π/180) and applies the cosine function, returning a value between −1 and +1 to 8 decimal places. For the common angles 0°, 30°, 45°, 60°, 90°, 180°, the exact form (such as √3/2 or 1/2) is shown alongside the decimal.

cos(90°) = 0 exactly. At 90° on the unit circle, the point is at (0, 1) — directly above the origin — so the x-coordinate, which equals cos(θ), is zero. In a right triangle, the two acute angles sum to 90°, so cos(90°) corresponds to a degenerate case where the adjacent leg has length 0. Similarly, cos(270°) = 0.

Yes. Cosine is negative whenever the angle lies in the second or third quadrant — roughly 90° < θ < 270° (modulo 360°). For example, cos(120°) = −1/2, cos(150°) = −√3/2, and cos(180°) = −1. The minimum value is −1, reached at 180° and at every 180° + 360°k.

Sine and cosine are co-functions: sin(θ) = cos(90° − θ). On the unit circle, sin(θ) is the y-coordinate while cos(θ) is the x-coordinate. Both oscillate between −1 and +1 with period 360°, but cosine starts at 1 when θ = 0° whereas sine starts at 0. The cosine wave is the sine wave shifted left by 90°.

A radian is the angle subtended at the center of a circle by an arc whose length equals the radius. One full turn equals 2π radians = 360°, so 1 rad ≈ 57.2957795°. To convert: radians = degrees × π/180, and degrees = radians × 180/π. For example, 60° = π/3 rad ≈ 1.04719755 rad. Select the Radians tab above to enter the angle directly in radians.

Cosine appears anywhere periodic or rotational behaviour shows up: the law of cosines for solving non-right triangles, Fourier analysis of signals and images, AC electrical circuits, projectile motion, the dot product of vectors (a·b = |a||b|cos θ), computer graphics rotation matrices, surveying, and astronomy. Engineers often use cos to decompose a force or velocity vector into its horizontal component.