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### Course: Trigonometry > Unit 1

Lesson 7: The reciprocal trigonometric ratios# Reciprocal trig ratios

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

These new ratios are the

**reciprocal trig ratios**, and we’re about to learn their names.## The cosecant $(\mathrm{csc})$

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.## The secant $(\mathrm{sec})$

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.## The cotangent $(\mathrm{cot})$

The

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

secant | The secant is the reciprocal of the cosine. | |

cotangent | The cotangent is the reciprocal of the tangent. |

## Finding the reciprocal trigonometric ratios

### Let's study an example.

In the triangle below, find $\mathrm{csc}(C)$ , $\mathrm{sec}(C)$ , and $\mathrm{cot}(C)$ .

#### 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.

##### 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.

##### 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.

## Try it yourself!

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