Reciprocal Trigonometric Ratios


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?

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

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

\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:

\begin{align} \quad sec(\theta) = \frac{hypotenuse}{adjacent} \quad , \quad csc(\theta) = \frac{hypotenuse}{opposite} \quad , \quad cot(\theta) = \frac{adjacent}{opposite} \end{align}

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

\begin{align} \quad sec(\theta) = \frac{1}{cos(\theta)} \quad , \quad csc(\theta) = \frac{1}{sin(\theta)} \quad , \quad cot(\theta) = \frac{cos(\theta)}{sin(\theta)} \end{align}

Proof of Reciprocal Functions

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

\begin{align} \quad sec(\theta) = \frac{hypotenuse}{adjacent} \end{align}

But we also know that:

\begin{align} \quad cos(\theta) = \frac{adjacent}{hypotenuse} \end{align}

So if we define secant to be:

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

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

\begin{align} \quad sec(\theta) = \frac{1}{\frac{adjacent}{hypotenuse}} \end{align}

Which when simplified we obtain:

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

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
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