Cosine Calculator

The cosine function, cos(x)

The cosine of an angle tells you how far along the horizontal axis you are after turning through that angle. In a right triangle with acute angle θ it is the length of the leg that touches θ (the adjacent side) divided by the hypotenuse.

cos θ = adjacent ÷ hypotenuse

On the unit circle the point reached after turning θ from the positive x-axis is (cos θ, sin θ), so the cosine is that point's x-coordinate. It starts at 1 for a zero angle, reaches 0 at 90°, −1 at 180°, 0 again at 270° and is back to 1 at 360°.

• Domain: all real numbers.
• Range: −1 to 1 inclusive.
• Period: 360° or 2π radians.
• Symmetry: cosine is an even function, cos(−θ) = cos θ, so its graph is a mirror image about the vertical axis.

Using the calculator

Enter the angle, choose whether it is in degrees (the default) or radians, and press Calculate. The result line reads cos(angle) = value, with the angle echoed back so you can confirm what was computed.

Answers are rounded to six decimals. A result that is a whole number appears without decimals (cos 90° gives 0 and cos 180° gives −1). A result with at most two decimals is shown with two, such as cos 60° = 0.50 and cos 120° = −0.50; one with three or four decimals is shown with four; any other keeps six, for example cos 1 rad = 0.540302.

Only plain decimal numbers are accepted in the Angle box: 45, -45.5 and 0.001 work, while .5, 5., +5, 1,5 and 1e3 are rejected with an error message.

Exact cosines of common angles

The cosines of the angles below can be written without decimals. Notice that they are the sines read in reverse order: cos 60° = sin 30°, because the two angles add up to 90°.

Computing a cosine by hand

Because cosine repeats every 360° and is symmetric, any angle can be brought into the range 0° to 90° with a sign. Reduce the angle modulo 360°, take the reference angle and apply the sign for the quadrant. Example: cos 300° has reference angle 60° and lies in the fourth quadrant, where cosine is positive, so cos 300° = cos 60° = 0.5.

For angles that are not special, work in radians and use the power series:

cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …x in radians; the terms shrink quickly when x is small.

Example: for x = 1 radian the terms are 1, −0.5, +0.041667, −0.001389 and +0.000025. Their running totals are 0.5, 0.541667, 0.540278 and 0.540303, closing in on the calculator's value of 0.540302.

Law of cosines: a worked example

The cosine also gives the third side of a triangle when you know two sides and the angle between them. This is the law of cosines, a generalisation of the Pythagorean theorem to triangles without a right angle.

c² = a² + b² − 2ab · cos C

Example: two sides of 7 and 9 enclose an angle of 40°. With cos 40° = 0.766044, c² = 49 + 81 − 2 × 7 × 9 × 0.766044 = 130 − 96.5216 = 33.4784, and c = √33.4784 ≈ 5.786. When the included angle is 90°, cos C = 0 and the formula collapses to c² = a² + b².

Example: splitting a force into components

A cosine is what you multiply by to find the part of a length or force that lies along a chosen direction. A rope pulls a crate with 50 N at 30° above the horizontal while the crate moves 8 m along the floor. Only the horizontal part of the pull does work:

work = F × d × cos θ = 50 × 8 × 0.866025 ≈ 346.4 J

The same idea gives the horizontal run of a ramp: a 12 m ramp rising at 8° covers 12 × cos 8° = 12 × 0.990268 ≈ 11.88 m along the ground.

Related functions and common slips

• The sine calculator gives the height instead of the horizontal position; the two satisfy sin²θ + cos²θ = 1.
• Arccosine works backwards: it turns a cosine value between −1 and 1 into an angle.
• Dividing sine by cosine yields the tangent; the trigonometry calculator solves complete right triangles.

The most frequent slip is the unit. cos(60) is 0.50 in degrees but −0.952413 when 60 is read as radians. Another is writing cos⁻¹ and treating it as 1 ÷ cos; the superscript means the inverse function, whereas the reciprocal is the secant.