# Reciprocity

We say that something is a reciprocal when we interchange the numerator of some expression with the denominator of some expression. For example, the reciprocal of 2/5 is 5/2.

## Example 1

**What is the reciprocal of the following expression?**

To solve this, we just simply interchange the numerator and denominator:

(2)# Reciprocal Trigonometric Ratios and Functions

For our general purposes, we sometimes use the reciprocal trigonometric ratios and functions: secant, cosecant, and cotangent. We will define all of these ratios in the following ways:

(3)Alternatively we can define secant, cosecant, and cotangent in the following manner:

(4)## Proof of Reciprocal Functions

We will only prove secant for our purposes. We first assume that:

(5)But we also know that:

(6)So if we define secant to be:

(7)Then the following holds when we substitute cosine for its ratio:

(8)Which when simplified we obtain:

(9)With how we have defined reciprocity above, we can say that secant is the reciprocal of cosine or cosine is the reciprocal of secant. The table below summarizes reciprocity of the other trigonometric ratios:

f(x) | CosΘ | SinΘ | TanΘ |
---|---|---|---|

Reciprocal | SecΘ | CscΘ | CotΘ |

f(x) | SecΘ | CscΘ | CotΘ |
---|---|---|---|

Reciprocal | CosΘ | SinΘ | TanΘ |