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Course: Geometry (all content) > Unit 13

Lesson 5: The reciprocal trigonometric ratios

Reciprocal trig ratios

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Learn how cosecant, secant, and cotangent are the reciprocals of the basic trig ratios: sine, cosine, and tangent.

We've already learned the basic trig ratios:

But there are three more ratios to think about:

Instead of $\frac{a}{c}$ ‍, we can consider $\frac{c}{a}$ ‍.
Instead of $\frac{b}{c}$ ‍, we can consider $\frac{c}{b}$ ‍.
Instead of $\frac{a}{b}$ ‍, we can consider $\frac{b}{a}$ ‍.

These new ratios are the reciprocal trig ratios, and we’re about to learn their names.

The cosecant $(\csc)$ ‍

The cosecant is the reciprocal of the sine. It is the ratio of the hypotenuse to the side opposite a given angle in a right triangle.

\sin (A) = \frac{opposite}{hypotenuse} = \frac{a}{c}

\csc (A) = \frac{hypotenuse}{opposite} = \frac{c}{a}

The secant $(\sec)$ ‍

The secant is the reciprocal of the cosine. It is the ratio of the hypotenuse to the side adjacent to a given angle in a right triangle.

\cos (A) = \frac{adjacent}{hypotenuse} = \frac{b}{c}

\sec (A) = \frac{hypotenuse}{adjacent} = \frac{c}{b}

The cotangent $(\cot)$ ‍

The cotangent is the reciprocal of the tangent. It is the ratio of the adjacent side to the opposite side in a right triangle.

\tan (A) = \frac{opposite}{adjacent} = \frac{a}{b}

\cot (A) = \frac{adjacent}{opposite} = \frac{b}{a}

How do people remember this stuff?

For most people, it's easiest to remember these new ratios by relating them to their reciprocals. The table below summarizes these relationships.

	Verbal description	Mathematical relationship
cosecant	The cosecant is the reciprocal of the sine.	$\csc (A) = \frac{1}{\sin (A)}$ ‍
secant	The secant is the reciprocal of the cosine.	$\sec (A) = \frac{1}{\cos (A)}$ ‍
cotangent	The cotangent is the reciprocal of the tangent.	$\cot (A) = \frac{1}{\tan (A)}$ ‍

Finding the reciprocal trigonometric ratios

Let's study an example.

In the triangle below, find

\csc (C)

\sec (C)

, and

\cot (C)

Solution

Finding the cosecant

We know that the cosecant is the reciprocal of the sine.

Since sine is the ratio of the opposite to the hypotenuse, cosecant is the ratio of the hypotenuse to the opposite.

\begin{aligned} \csc (C) & = \frac{hypotenuse}{opposite} \\ = \frac{17}{15} \end{aligned}

Finding the secant

We know that the secant is the reciprocal of the cosine.

Since cosine is the ratio of the adjacent to the hypotenuse, secant is the ratio of the hypotenuse to the adjacent.

\begin{aligned} \sec (C) & = \frac{hypotenuse}{adjacent} \\ = \frac{17}{8} \end{aligned}

Finding the cotangent

We know that the cotangent is the reciprocal of the tangent.

Since tangent is the ratio of the opposite to the adjacent, cotangent is the ratio of the adjacent to the opposite.

\begin{aligned} \cot (C) & = \frac{adjacent}{opposite} \\ = \frac{8}{15} \end{aligned}

Try it yourself!

Problem 1

\csc (X) =

Problem 2

\sec (W) =

Problem 3

\cot (R) =

Challenge problem

What is the exact value of $\csc (45^{\circ})$ ‍?

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