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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.

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