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## 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?

## Want to join the conversation?

• What are these new ratios used for?
• Eventually, in calculus, you will need sec(x), csc(x), and cot(x) for the derivative (rate of change) of some of the trigonometric functions. In particular, the first derivative of tan(x) is (sec(x) )^2
• What is cosh, sinh, and tanh? I saw these functions on the calculator.
• Those are hyperbolic functions:
sinh x = ½(e^x - e^(-x))
cosh x = ½(e^x + e^(-x))
tanh x = sinh x / cosh x = (e^x - e^(-x)) / (e^x + e^(-x))
• Why do you need these functions if you already have sine, cosine, and tangent?
• Strictly speaking, we don't these days. Historically speaking, finding trig values and reciprocals were much much harder than pressing two buttons on a scientific calculator. So people wanted to have separate tables for looking up 1/sin x and so on. In fact, those weren't the only "extra" trig tables people had back then. Check out this fun article!
http://blogs.scientificamerican.com/roots-of-unity/10-secret-trig-functions-your-math-teachers-never-taught-you/
So, why do we still hold on to secant, cosecant, and cotangent when we dropped stuff like havercosine and excosecant? A good reason is that they make the trig formulas in calculus a little easier to remember and use, and also because the geometric meaning of the secant can be valuable at times. But other than that, they totally take a back seat to the three principal trig functions.
• How would you find the sin, cosine, or tangent of 90 degrees?
• Well, sin of 90 degrees means that you are trying to find the opposite over hypotenuse of a triangle with a measure of 90 degrees. But the opposite side to a 90 degree angle IS the hypotenuse. We usually tackle these angles when we have moved to the unit circle because they don't fit the Soh Cah Toa rule. Sin of 90 degrees is one. Cos of 90 degrees is 0, because if the angle has rotated through 90 degrees, there is nothing left of the adjacent side. Tangent is the sine divided by the cosine, so if Sine of 90 degrees is one and Cosine of 90 degrees is zero, you have 1/0 which is undefined

In fact, you get an error message if you ask a calculator to give you tangent of 90.
• I created Cho Sha Cao. However I don't recommend using it because you can easily mix Cho and Cao up.
• I don't like using Soh Cah Toa. First, the 'h' is silent in two of them, so that's a first layer of decoding it in your head. Then, they're single letters that you have to plug into their original term, which is another layer of decoding, and then you have to go through and put it all together. To me it's a messy mnemonic.
I prefer:
"Opp Hy" (sounds like 'up high')
"Opp Add" (like an 'op ed' article)

It's easy to start the rhythm with:
then you just know that cosine comes after sine, and tangent is last because we all know that Op Ed articles can go off on tangents.

I dunno - it works a billion times better for me than Soh Cah Toa shrugs
• I think your alternate mnemonic works great! The whole point is to help you remember facts (in this case, the relationships between the trig functions and the sides of a right triangle).

The best mnemonics are the ones you make up that are meaningful to you, because they will be easier to remember and work better.
• Are there any other ratios we will learn in the future?
versine(θ) = 2 sin2(θ/2) = 1 – cos(θ)haversine(θ)= sin2(θ/2)coversine(θ)= 1 – sin(θ)hacoversine(θ)= 1/2(1-sin(θ))exsecant(θ)= sec(θ) – 1excosecant(θ)= csc(θ) – 1