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 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') "Add Hy" ('add height') "Opp Add" (like an 'op ed' article) It's easy to start the rhythm with: "Sine Opp Hy" (sounds like 'Sign up high') easy 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.

So what is the exact difference between cosecant, secant, cotangent and cos-1, sin-1, tan-1?

cosecant, secant and tangent are the reciprocals of sine, cosine and tangent. sin-1, cos-1 & tan-1 are the inverse, NOT the reciprocal. That means sin-1 or inverse sine is the angle θ for which sinθ is a particular value. For example, sin30 = 1/2. sin-1 (1/2) = 30. For more explanation, check this out. https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles/trig-solve-for-an-angle/a/inverse-trig-functions-intro

My Question is ... Why do we use trigonometric function and reciprocal ? What is the useful things from using these methods in mathematics? .....................................................

Trigonometry is the foundation of physics, engineering, and architecture, especially if they have a periodic cycle. Later in your mathematical career you will be supplied with many uses of trigonometric functions and it's inverses — calculus is a prime example of such.

Are there any other ratios we will learn in the future?

Well, the textbook answer is that there are only 6 trig ratios, which we have already covered. However, if you really want to devel into the topic, the historical answer would be that there are at least 12 ratios, which include the ones we've learned and some new ones which are versine, haversine, coversine, hacoversine, exsecant, and excosecant. They can be expressed as the following: `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) – 1 excosecant(θ)= csc(θ) – 1`

Main content

Trigonometry

Course: Trigonometry > Unit 1

Lesson 7: The reciprocal trigonometric ratios

Reciprocal trig ratios

Google Classroom

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 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 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 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 description	Mathematical relationship
cosecant	The 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
secant	The 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
cotangent	The 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}

[I'd like to see more examples, please.]

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?

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Nayan Bansal
Posted 7 years ago. Direct link to Nayan Bansal's post “What are these new ratios...”
What are these new ratios used for?
Button navigates to signup pageComment on Nayan Bansal's post “What are these new ratios...”
(63 votes)
Answer
- Anthony Natoli
  Posted 7 years ago. Direct link to Anthony Natoli's post “Eventually, in calculus, ...”
  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
  Comment on Anthony Natoli's post “Eventually, in calculus, ...”
  (95 votes)
Shambhavi
Posted 7 years ago. Direct link to Shambhavi's post “What is cosh, sinh, and t...”
What is cosh, sinh, and tanh? I saw these functions on the calculator.
Button navigates to signup pageComment on Shambhavi's post “What is cosh, sinh, and t...”
(28 votes)
Answer
- T H
  Posted 7 years ago. Direct link to T H's post “Those are hyperbolic func...”
  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))
  Comment on T H's post “Those are hyperbolic func...”
  (33 votes)
Joaquin Butial
Posted 7 years ago. Direct link to Joaquin Butial's post “Why do you need these fun...”
Why do you need these functions if you already have sine, cosine, and tangent?
Button navigates to signup pageComment on Joaquin Butial's post “Why do you need these fun...”
(13 votes)
Answer
- Matthew Daly
  Posted 7 years ago. Direct link to Matthew Daly's post “Strictly speaking, we don...”
  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.
  Comment on Matthew Daly's post “Strictly speaking, we don...”
  (61 votes)
rebecca hu
Posted 7 years ago. Direct link to rebecca hu's post “How would you find the si...”
How would you find the sin, cosine, or tangent of 90 degrees?
Button navigates to signup pageButton navigates to signup page
(8 votes)
Answer
- doctorfoxphd
  Posted 7 years ago. Direct link to doctorfoxphd's post “Well, sin of 90 degrees m...”
  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.
  Button navigates to signup page
  (26 votes)
Alexis
Posted 7 years ago. Direct link to Alexis's post “I created Cho Sha Cao. Ho...”
I created Cho Sha Cao. However I don't recommend using it because you can easily mix Cho and Cao up.
Button navigates to signup pageComment on Alexis's post “I created Cho Sha Cao. Ho...”
(16 votes)
Answer
creationtribe
Posted 4 years ago. Direct link to creationtribe's post “I don't like using Soh Ca...”
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')
"Add Hy" ('add height')
"Opp Add" (like an 'op ed' article)

It's easy to start the rhythm with:
"Sine Opp Hy" (sounds like 'Sign up high') easy
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
Button navigates to signup pageComment on creationtribe's post “I don't like using Soh Ca...”
(8 votes)
Answer
- Polina Vitić
  Posted 4 years ago. Direct link to Polina Vitić's post “I think your alternate mn...”
  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.
  Comment on Polina Vitić's post “I think your alternate mn...”
  (7 votes)
Ibrahim Yaser
Posted 3 years ago. Direct link to Ibrahim Yaser's post “My Question is ... Why do...”
My Question is ...
Why do we use trigonometric function and reciprocal ?
What is the useful things from using these methods in mathematics?
.....................................................
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- A/V
  Posted 3 years ago. Direct link to A/V's post “Trigonometry is the found...”
  Trigonometry is the foundation of physics, engineering, and architecture, especially if they have a periodic cycle. Later in your mathematical career you will be supplied with many uses of trigonometric functions and it's inverses — calculus is a prime example of such.
  Button navigates to signup page
  (4 votes)
Adhya Anil Kumar
Posted 6 years ago. Direct link to Adhya Anil Kumar's post “Are there any other ratio...”
Are there any other ratios we will learn in the future?
Button navigates to signup pageComment on Adhya Anil Kumar's post “Are there any other ratio...”
(2 votes)
Answer
- harsh ♡
  Posted 6 years ago. Direct link to harsh ♡'s post “Well, the textbook answer...”
  Well, the textbook answer is that there are only 6 trig ratios, which we have already covered. However, if you really want to devel into the topic, the historical answer would be that there are at least 12 ratios, which include the ones we've learned and some new ones which are versine, haversine, coversine, hacoversine, exsecant, and excosecant. They can be expressed as the following:
  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) – 1 excosecant(θ)= csc(θ) – 1
  Comment on harsh ♡'s post “Well, the textbook answer...”
  (6 votes)
Mahati
Posted 7 years ago. Direct link to Mahati's post “So what is the exact diff...”
So what is the exact difference between cosecant, secant, cotangent and cos-1, sin-1, tan-1?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Namitha Suresh
  Posted 7 years ago. Direct link to Namitha Suresh's post “cosecant, secant and tang...”
  cosecant, secant and tangent are the reciprocals of sine, cosine and tangent.
  sin-1, cos-1 & tan-1 are the inverse, NOT the reciprocal. That means sin-1 or inverse sine is the angle θ for which sinθ is a particular value. For example, sin30 = 1/2. sin-1 (1/2) = 30.
  For more explanation, check this out.
  https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles/trig-solve-for-an-angle/a/inverse-trig-functions-intro
  Button navigates to signup page
  (4 votes)
G junior
Posted 7 years ago. Direct link to G junior's post “who came up with the term...”
who came up with the terms 'sine' and 'cosine' and all of the other terms, and is there like a specific etymology with them or a literal definition of some sort or it's literally "sine - n. 'The ratio of a triangle's opposite side to its hypotenuse from a specific angle within the triangle'"?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- 🚀The knowledge Hunter🔭
  Posted 6 years ago. Direct link to 🚀The knowledge Hunter🔭's post “This website could be a g...”
  This website could be a great help http://www.algebralab.org/lessons/lesson.aspx?file=Trigonometry_TrigNameOrigins.xml
  Since It has a clear visual representation too.
  Button navigates to signup page
  (1 vote)