Cosine Calculator

What this calculator does

So, you have an angle and you want its cosine. This tool takes the angle, in degrees or radians, and returns the cosine rounded to three decimal places.

One box holds the angle, and a dropdown says whether it is degrees or radians. Then it is a single button press.

How to use it

1. Type your angle.
2. Choose degrees or radians to match it.
3. Press Calculate.

Match the unit to your angle, since the same number means a different angle in each. When you choose degrees, the tool converts to radians internally before it computes.

What cosine actually is

If sine is a height, cosine is a width. In a right triangle, the cosine of an angle is the side next to that angle, the adjacent side, divided by the hypotenuse. That is the middle of the mnemonic, CAH: Cosine equals Adjacent over Hypotenuse. On the unit circle, cosine is the horizontal position of the point at that angle, its x-coordinate, exactly as sine is the vertical one. So sine and cosine are two halves of the same picture: for any angle, cosine tells you how far across the point is, and sine tells you how far up.

The "co" in cosine: the complement

The name is not decoration. Cosine is short for complementary sine, and it means exactly that: the cosine of an angle equals the sine of its complement, the angle that completes it to 90 degrees. So the cosine of 20 degrees is the sine of 70 degrees, and the cosine of 60 degrees is the sine of 30 degrees. This is why the two functions feel like mirror images. Anywhere you see a "co" in trigonometry, cosine, cotangent, cosecant, it is signalling this same complement relationship with the plain version of the function.

Its range, and the shifted wave

Like sine, cosine is the coordinate of a point on a circle of radius 1, so it also stays between -1 and 1 for every angle. It equals 1 at 0 degrees, where the point is all the way to the right, drops to 0 at 90 degrees, and reaches -1 at 180 degrees. Plotted against the angle, cosine draws the same rolling wave as sine, just shifted along by 90 degrees, starting at the crest instead of at zero. The two waves running a quarter-turn apart is what lets sine and cosine together describe any smooth oscillation or circular motion.

A worked example

Take 60 degrees. The point at 60 degrees on the unit circle sits at a horizontal position of one half, so the cosine of 60 degrees is 0.5, shown as 0.500. At 0 degrees the point is fully to the right, so the cosine is 1. At 90 degrees the point is straight up with no horizontal offset, so the cosine is 0.

Degrees or radians

Degrees divide a full turn into 360 parts, the familiar everyday scale. Radians measure the angle by the arc length swept around the unit circle, making a full turn 2 pi radians and a right angle pi over 2. Most higher mathematics prefers radians because the formulas are tidier, while degrees stay common in everyday use. Set the dropdown to whichever your angle uses.

Questions people ask

What does cosine tell me?

In a right triangle, the adjacent side over the hypotenuse. On the unit circle, the horizontal position of the point at that angle. It is a plain number set only by the angle.

How is cosine different from sine?

Sine is the height of the unit-circle point, cosine is its width. Their waves are identical in shape but shifted 90 degrees apart.

Why is it called cosine?

Because it is the complementary sine: the cosine of an angle equals the sine of the angle that completes it to 90 degrees.

Why does cosine stay between -1 and 1?

Because it is a coordinate of a point on a circle of radius 1, so it cannot exceed 1 or fall below -1.

Degrees or radians?

Whichever your angle is written in. Degrees for everyday angles, radians for most higher maths. Match the dropdown so the answer is correct.

References

On cosine as the horizontal coordinate and complement of sine. Cosine is the x-coordinate of the unit-circle point, and its name marks it as the sine of the complementary angle.

1. Eric W. Weisstein, "Trigonometry," from MathWorld, a Wolfram resource, on the unit-circle and right-triangle definitions of cosine and the SOHCAHTOA relations.
2. "Trigonometry: India and the Islamic world," Encyclopaedia Britannica, on the early history of the sine and cosine and their names.