# 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:

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:

## 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).