Domain and Range of Cosecant and Secant Trigonometry Functions

The domain of a cosecant or secant 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 cosecant and secant functions are closely tied to sine and cosine, because they’re the respective reciprocals. In reference to the coordinate plane, cosecant is r/y, and secant is r/x. The value of r is the length of the hypotenuse of a right triangle — which is always positive and always greater than x or y.

The only problem that arises when computing these functions is when either x or y is 0 — when the terminal side of the angle is on an axis. A function with a 0 in the denominator creates a number or value that doesn’t exist (in math speak, the result is undefined), so anytime x is 0, the angle doesn’t have a secant function, and if y is 0, the angle doesn’t have a cosecant function. The x is 0 when the terminal side is on the y-axis, and the y is 0 when the terminal side is on the x-axis.

Domains of cosecant and secant trig functions

The domains of cosecant and secant are restricted — you can only use the functions for angle measures with output numbers that exist.

Anytime the terminal side of an angle lies along the x-axis (where y = 0), you can’t perform the cosecant function on that angle. In trig speak, the rule looks like this:

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Anytime the terminal side of an angle lies along the y-axis (where x = 0), you can’t perform the secant function on that angle. So in trig speak, you’d say this:

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Ranges of cosecant and secant

The ratios of the cosecant and secant functions on the coordinate plane, r/y and r/x, have the hypotenuse, r, in the numerator. Because r is always positive and greater than or equal to x and y, these fractions are always improper (greater than 1) or equal to 1. The ranges of these two functions never include proper fractions (numbers between –1 and 1).

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