Free Cosine Calculator

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Cosine relates an angle to the ratio of the adjacent side to the hypotenuse in a right triangle, and extends naturally to any angle — positive, negative, or beyond 360° — through the unit circle, where cos(θ) is simply the x-coordinate of the point at angle θ. It is one of the two foundational trigonometric functions (alongside sine) that every other trig function, identity, and formula in the field is ultimately built from. Cosine appears throughout physics and engineering: it describes the horizontal component of a force or velocity vector, drives simple harmonic motion equations for springs and pendulums, models AC voltage and current waveforms in electrical engineering, and underlies the dot-product formula used to find the angle between two vectors. This calculator computes cos(θ) for any angle you enter, in either degrees or radians.

How it works

Picture a point moving around a circle of radius 1, centered at the origin, starting from the positive x-axis. For any angle θ measured counterclockwise from that starting point, cos(θ) is defined as the x-coordinate of the point the angle sweeps to. In a right triangle with an angle θ, this is equivalent to the ratio adjacent/hypotenuse. Because a full trip around the circle is 360° (2π radians), cosine repeats every 360° and its value always stays between -1 and 1 — it can never exceed the circle's radius of 1 in either direction. Enter an angle and choose degrees or radians, and the calculator returns cos(θ) directly.

Enter angle (θ).
Enter angle unit.
Click Calculate to see your results.

Examples

cos(60°)

cos(60°) = 0.5 exactly — one of the standard reference angles, since a 60° angle sits opposite a side that is exactly half the hypotenuse in a 30-60-90 triangle.

cos(0°)

cos(0°) = 1, the maximum value cosine ever reaches, corresponding to the starting point (1, 0) on the unit circle.

cos(180°)

cos(180°) = -1, the minimum value cosine ever reaches, corresponding to the point (-1, 0) directly opposite the start.

Common mistakes to avoid

Mixing up sine and cosine at 0° and 90° — cosine is 1 at 0° and 0 at 90°, exactly the reverse of sine.
Forgetting to switch the angle-unit selector to radians when entering a radian value, which silently produces a completely different (wrong) result.
Assuming cosine's output can exceed 1 or fall below -1 — it never can, by definition.
Confusing cos(θ) with its inverse, arccos — cosine converts an angle to a ratio, arccos converts a ratio back to an angle.

Frequently asked questions

Always between -1 and 1, inclusive, no matter what angle you enter — this is a direct consequence of cosine being an x-coordinate on a circle of radius 1.

They are "cofunctions": cos(θ) = sin(90° - θ), and sin(θ) = cos(90° - θ). Graphically, the cosine curve is just the sine curve shifted 90° to the left.

At 0°, the point on the unit circle is (1, 0). Cosine reads the x-coordinate (1), and sine reads the y-coordinate (0) — this is the single most common trig mix-up, so it's worth memorizing that pair explicitly.

Cosine is even: cos(-θ) = cos(θ). That means the cosine of a negative angle is always the same as the cosine of its positive counterpart, which is useful for quickly checking a result's sign.

Beyond right-triangle ratios, cosine drives the Law of Cosines for solving non-right triangles, the dot-product formula cos(θ) = (A·B)/(|A||B|) for finding the angle between two vectors, and the equations of oscillating systems like springs, pendulums, and AC electrical signals.

Yes — cosine is periodic with period 360° (2π radians), so cos(390°) gives the same result as cos(30°). The calculator handles angles of any size, including negative ones, without needing you to reduce them first.

Cosine takes an angle and returns a ratio between -1 and 1. Arccos (inverse cosine) does the reverse: it takes a ratio between -1 and 1 and returns an angle. They undo each other only within arccos's restricted 0°-180° output range.