If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## 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 start fraction, start color #11accd, a, end color #11accd, divided by, start color #aa87ff, c, end color #aa87ff, end fraction, we can consider start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #11accd, a, end color #11accd, end fraction.
• Instead of start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #aa87ff, c, end color #aa87ff, end fraction, we can consider start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction.
• Instead of start fraction, start color #11accd, a, end color #11accd, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction, we can consider start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #11accd, a, end color #11accd, end fraction.
These new ratios are the reciprocal trig ratios, and we’re about to learn their names.

## The cosecant $(\csc)$left parenthesis, \csc, right parenthesis

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.
sine, left parenthesis, A, right parenthesis, equals, start fraction, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, divided by, start color #aa87ff, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #aa87ff, end fraction, equals, start fraction, start color #11accd, a, end color #11accd, divided by, start color #aa87ff, c, end color #aa87ff, end fraction
\csc, left parenthesis, A, right parenthesis, equals, start fraction, start color #aa87ff, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #aa87ff, divided by, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, end fraction, equals, start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #11accd, a, end color #11accd, end fraction

## The secant $(\sec)$left parenthesis, \sec, right parenthesis

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.
cosine, left parenthesis, A, right parenthesis, equals, start fraction, start color #ed5fa6, start text, a, d, j, a, c, e, n, t, end text, end color #ed5fa6, divided by, start color #aa87ff, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #aa87ff, end fraction, equals, start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #aa87ff, c, end color #aa87ff, end fraction
\sec, left parenthesis, A, right parenthesis, equals, start fraction, start color #aa87ff, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #aa87ff, divided by, start color #ed5fa6, start text, a, d, j, a, c, e, n, t, end text, end color #ed5fa6, end fraction, equals, start fraction, start color #aa87ff, c, end color #aa87ff, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction

## The cotangent $(\cot)$left parenthesis, cotangent, right parenthesis

The cotangent is the reciprocal of the tangent. It is the ratio of the adjacent side to the opposite side in a right triangle.
tangent, left parenthesis, A, right parenthesis, equals, start fraction, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, divided by, start color #ed5fa6, start text, a, d, j, a, c, e, n, t, end text, end color #ed5fa6, end fraction, equals, start fraction, start color #11accd, a, end color #11accd, divided by, start color #ed5fa6, b, end color #ed5fa6, end fraction
cotangent, left parenthesis, A, right parenthesis, equals, start fraction, start color #ed5fa6, start text, a, d, j, a, c, e, n, t, end text, end color #ed5fa6, divided by, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, end fraction, equals, start fraction, start color #ed5fa6, b, end color #ed5fa6, divided by, start color #11accd, a, end color #11accd, end fraction

## 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 descriptionMathematical relationship
cosecantThe cosecant is the reciprocal of the sine.\csc, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, sine, left parenthesis, A, right parenthesis, end fraction
secantThe secant is the reciprocal of the cosine.\sec, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, cosine, left parenthesis, A, right parenthesis, end fraction
cotangentThe cotangent is the reciprocal of the tangent.cotangent, left parenthesis, A, right parenthesis, equals, start fraction, 1, divided by, tangent, left parenthesis, A, right parenthesis, end fraction

## Finding the reciprocal trigonometric ratios

### Let's study an example.

In the triangle below, find \csc, left parenthesis, C, right parenthesis, \sec, left parenthesis, C, right parenthesis, and cotangent, left parenthesis, C, right parenthesis.

#### 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) &= \dfrac{\purpleC{\text{hypotenuse}}} {\blueD{\text{ opposite}}} \\\\ &= \dfrac{{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) &= \dfrac{\purpleC{\text{hypotenuse}}}{\maroonC{\text{adjacent}}} \\\\ &= \dfrac{{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) &= \dfrac{\maroonC{\text{adjacent}}}{\blueD{\text{opposite}}} \\\\ &= \dfrac{{8}}{{15}} \end{aligned}

## Try it yourself!

Problem 1
\csc, left parenthesis, X, right parenthesis, equals

Problem 2
\sec, left parenthesis, W, right parenthesis, equals

Problem 3
cotangent, left parenthesis, R, right parenthesis, equals

Challenge problem
What is the exact value of \csc, left parenthesis, 45, degrees, right parenthesis?