# Domain and Range of Sine and Cosine Functions

The sine and cosine functions are unique in the world of trig functions, because their ratios always have a value. No matter what angle you input, you get a resulting output. The value you get may be 0, but that’s a number, too. In reference to the coordinate plane, sine is *y*/*r*, and cosine is *x*/*r*.

The radius, *r*, is always some positive number (which is why these functions always have a value, because they don’t ask you to divide by 0), and *r* is always a number greater than (or equal to) the absolute value of *x* or *y*.

## Domains of sine and cosine

The domains of sine and cosine are infinite. In trig speak, you say something like this: If theta represents all the angles in the domain of the two functions

which means that theta can be any angle in degrees or radians — any real number.

## Ranges of sine and cosine

The output values for sine and cosine are always between (and including) –1 and 1. In trig speak, it goes something like this: If

represent the output values of the functions

The ratios *y*/*r* and *x*/*r* will never be improper fractions — the numerator can never be greater than the denominator — because the value of *r*, the radius, is always the biggest number. At best, if the angle theta has a terminal side on an axis (meaning that one of the sides is equal to *r*), then the value of those ratios is 1 or –1.