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Domain and Range of Sine and Cosine Functions

The domain of a sine or cosine trigonometry function consists of all the input values that the function can handle — the way the function is defined. Of course, you want to get output values (which make up the range) when you enter input values.

The sine and cosine functions are unique in the world of trig functions, because their ratios always have a value. The value may be zero, but that’s a number, too. In reference to the coordinate plane and a circle with its center at the origin, 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 it’s always a number greater than the absolute value of x or y.

The domains of sine and cosine are infinite. In trig speak, you say something like this:


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


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.

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