Reciprocal Trigonometric Ratios

# 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?

(1)
\begin{align} \frac{1 + x^2}{-4} \end{align}

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

(2)
\begin{align} \frac{-4}{1 + x^2} \end{align}

# 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)
\begin{align} \quad sec(\theta) = \frac{1}{cos(\theta)} \end{align}

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

(8)