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Domain and Range of Tangent and Cotangent Trigonometry Functions

The domain of a tangent or cotangent trig 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 tangent and cotangent are related not only by the fact that they’re reciprocals, but also by the behavior of their ranges. In reference to the coordinate plane, tangent is y/x, and cotangent is x/y. The domains of both functions are restricted, because sometimes their ratios could have 0s in the denominator, but their ranges are infinite.

Because x can’t equal 0 for the tangent function to work, this rule holds true:

image0.png

Both the tangent and secant functions have ratios with x in the denominator, making their domains the same.

In order for the cotangent function to work, y can’t equal 0.

image1.png

The ranges of both tangent and cotangent are infinite, which, when expressed in mathematical notation, looks like this:

image2.png

The range values for these functions get very small (toward negative infinity) or very large (toward positive infinity) whenever the denominator of the respective ratio gets close to 0. When you divide some number by a very small value, such as 0.0001, the result is large. The smaller the denominator, the larger the result.

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