Why aren't the reciprocal functions taught with the normal three? Are they simply less used or are they harder to teach without sin, cos, and tan?

Both. They are less used and without the 3 foundational functions, they are a touch harder to teach. We often teach using SOH-CAH-TOA and using a right triangle, so sin/cos/tan are very well known.

Are cse, sec and cot in a calculator?

Sometimes. There are also sometimes inverses of all of them, AND hyperbolic versions of all of those!

what would be some applications for using the inverse functions? BW- they seem more intuitive then the sine, and cosine. Tangent seems more intuitive too.

Do you mean the "Reciprocal functions" like secant and cosecant. The inverse trigonometric functions (the cyclometric functions) are represented by arcosine, arcsine etc. Reciprocal functions were used in tables before computer power went up and there are some instances where calculating an inverse of a function is easier than the function. As to Inverse tringonometric functions they are used to calculate angles.

Is there an inverse for the reciprocal functions: cosecant, secant, and cotangent?

Yes, they're arccosecant, arcsecant, and arccotangent.

How do I solve an equation like this: csc theta=1/sin theta?

That's not a problem, its a statement. csc(Θ) = 1/sin(Θ), sec(Θ) = 1/cos(Θ), and cot(Θ) = 1/tan(Θ).

What is the difference between cosecant and arcsine?

Cosecant is a reciprocal function but arcsine is an inverse function. Maybe these two links would help clarify it further: Review of Trigonometric ratios : https://www.khanacademy.org/math/geometry-home/right-triangles-topic/reciprocal-trig-ratios-geo/a/trigonometric-ratios-review And Intro to arcsine https://www.khanacademy.org/math/precalculus/trig-equations-and-identities-precalc/inverse-trig-functions-precalc/v/inverse-trig-functions-arcsin

What are the inverse trig functions of the reciprocal trig functions?

Like the original inverse trig functions (arcsin, arccos, arctan) you have learned , the inverse trig functions of the reciprocal trig functions is arccot, arcsec, arccsc.

Do we have the inverse function of cosecant, secant and cotangent too?

If by inverse function you mean the arc functions (like arcsin, arccos, arctan), yes, you do have arccsc, arcsec and arccot. However, those are very rarely used, as are the csc, sec, and cot functions because those are normally just written as 1/sin, 1/cos and 1/tan.

Main content

Trigonometry

Course: Trigonometry > Unit 1

Lesson 7: The reciprocal trigonometric ratios

Trigonometric ratios review

Google Classroom

Review all six trigonometric ratios: sine, cosine, tangent, cotangent, secant, & cosecant.

What are the trigonometric ratios?


sine, left parenthesis, angle, 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 #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, end fraction
cosine, left parenthesis, angle, A, right parenthesis, equals	start fraction, start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff, divided by, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, end fraction
tangent, left parenthesis, angle, 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, a, d, j, a, c, e, n, t, end text, end color #aa87ff, end fraction
cotangent, left parenthesis, angle, A, right parenthesis, equals	start fraction, start color #aa87ff, start text, a, d, j, a, c, e, n, t, 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
\sec, left parenthesis, angle, A, right parenthesis, equals	start fraction, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, divided by, start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff, end fraction
\csc, left parenthesis, angle, A, right parenthesis, equals	start fraction, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, divided by, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, end fraction

Want to learn more about sine, cosine, and tangent? Check out this video.
Want to learn more about cotangent, secant, and cosecant? Check out this article.

Practice set 1: sine, cosine, and tangent

Problem 1.1

Current

sine, left parenthesis, angle, B, right parenthesis, equals

Use an exact expression.

Want to try more problems like this? Check out this exercise.

Practice set 2: cotangent, secant, and cosecant

Problem 2.1

Current

cotangent, left parenthesis, angle, B, right parenthesis, equals

Use an exact expression.

Want to try more problems like this? Check out this exercise.

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jaykumar.p
Posted 6 years ago. Direct link to jaykumar.p's post “Why aren't the reciprocal...”
Why aren't the reciprocal functions taught with the normal three?
Are they simply less used or are they harder to teach without sin, cos, and tan?
Button navigates to signup pageButton navigates to signup page
(7 votes)
Answer
- Ron Joniak
  Posted 6 years ago. Direct link to Ron Joniak's post “Both. They are less used ...”
  Both. They are less used and without the 3 foundational functions, they are a touch harder to teach. We often teach using SOH-CAH-TOA and using a right triangle, so sin/cos/tan are very well known.
  Button navigates to signup page
  (8 votes)
plpeaksterman67
Posted 7 years ago. Direct link to plpeaksterman67's post “what would be some applic...”
what would be some applications for using the inverse functions? BW- they seem more intuitive then the sine, and cosine. Tangent seems more intuitive too.
Button navigates to signup pageComment on plpeaksterman67's post “what would be some applic...”
(5 votes)
Answer
- VaeSapiens
  Posted 7 years ago. Direct link to VaeSapiens's post “Do you mean the "Reciproc...”
  Do you mean the "Reciprocal functions" like secant and cosecant. The inverse trigonometric functions (the cyclometric functions) are represented by arcosine, arcsine etc.
  
  Reciprocal functions were used in tables before computer power went up and there are some instances where calculating an inverse of a function is easier than the function.
  
  As to Inverse tringonometric functions they are used to calculate angles.
  Comment on VaeSapiens's post “Do you mean the "Reciproc...”
  (10 votes)
Elijah Daniels
Posted 6 years ago. Direct link to Elijah Daniels's post “Are cse, sec and cot in a...”
Are cse, sec and cot in a calculator?
Button navigates to signup pageComment on Elijah Daniels's post “Are cse, sec and cot in a...”
(6 votes)
Answer
- Haeleon
  Posted 6 years ago. Direct link to Haeleon's post “Sometimes. There are also...”
  Sometimes. There are also sometimes inverses of all of them, AND hyperbolic versions of all of those!
  Comment on Haeleon's post “Sometimes. There are also...”
  (4 votes)
Exodus37
Posted 3 years ago. Direct link to Exodus37's post “What are the hyperbolic t...”
What are the hyperbolic trig functions?
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(4 votes)
Answer
owenashbeck
Posted 4 years ago. Direct link to owenashbeck's post “Is there an inverse for t...”
Is there an inverse for the reciprocal functions: cosecant, secant, and cotangent?
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(2 votes)
Answer
- kubleeka
  Posted 4 years ago. Direct link to kubleeka's post “Yes, they're arccosecant,...”
  Yes, they're arccosecant, arcsecant, and arccotangent.
  Button navigates to signup page
  (4 votes)
Durgen
Posted 2 years ago. Direct link to Durgen's post “So there are sine, cosine...”
So there are sine, cosine, tangent, arcsine, arccosine, arctangent, cosecant, secant, and cotangent. My calculator says there also seems to be arcsecant, arccosecant, and arccotangent. Is that correct? Are they called by different names?
Button navigates to signup pageComment on Durgen's post “So there are sine, cosine...”
(2 votes)
Answer
- sgohil26
  Posted 4 months ago. Direct link to sgohil26's post “Arcsecant, arccosecant, a...”
  Arcsecant, arccosecant, and arctangent are all inverses of the reciprocal functions.
  Button navigates to signup page
  (1 vote)
Jasmine J
Posted 5 years ago. Direct link to Jasmine J's post “How do I solve an equatio...”
How do I solve an equation like this: csc theta=1/sin theta?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Madd Sam
  Posted 5 years ago. Direct link to Madd Sam's post “That's not a problem, its...”
  That's not a problem, its a statement. csc(Θ) = 1/sin(Θ), sec(Θ) = 1/cos(Θ), and cot(Θ) = 1/tan(Θ).
  Comment on Madd Sam's post “That's not a problem, its...”
  (3 votes)
ditchdigger03
Posted 5 years ago. Direct link to ditchdigger03's post “What is the difference be...”
What is the difference between cosecant and arcsine?
Button navigates to signup pageComment on ditchdigger03's post “What is the difference be...”
(1 vote)
Answer
- Scott Freeman
  Posted 5 years ago. Direct link to Scott Freeman's post “Cosecant is a reciprocal ...”
  Cosecant is a reciprocal function but arcsine is an inverse function.
  Maybe these two links would help clarify it further:
  
  Review of Trigonometric ratios : https://www.khanacademy.org/math/geometry-home/right-triangles-topic/reciprocal-trig-ratios-geo/a/trigonometric-ratios-review
  
  And
  
  Intro to arcsine https://www.khanacademy.org/math/precalculus/trig-equations-and-identities-precalc/inverse-trig-functions-precalc/v/inverse-trig-functions-arcsin
  Button navigates to signup page
  (3 votes)
randomname19911
Posted 7 years ago. Direct link to randomname19911's post “What are the inverse trig...”
What are the inverse trig functions of the reciprocal trig functions?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Kevin Tan
  Posted 6 years ago. Direct link to Kevin Tan's post “Like the original inverse...”
  Like the original inverse trig functions (arcsin, arccos, arctan) you have learned , the inverse trig functions of the reciprocal trig functions is arccot, arcsec, arccsc.
  Button navigates to signup page
  (3 votes)
Mohammed Taqee
Posted 6 years ago. Direct link to Mohammed Taqee's post “Do we have the inverse fu...”
Do we have the inverse function of cosecant, secant and cotangent too?
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(1 vote)
Answer
- Fai
  Posted 6 years ago. Direct link to Fai's post “If by inverse function yo...”
  If by inverse function you mean the arc functions (like arcsin, arccos, arctan), yes, you do have arccsc, arcsec and arccot. However, those are very rarely used, as are the csc, sec, and cot functions because those are normally just written as 1/sin, 1/cos and 1/tan.
  Comment on Fai's post “If by inverse function yo...”
  (2 votes)