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## Trigonometry

### Course: Trigonometry > Unit 1

Lesson 7: The reciprocal trigonometric ratios# Trigonometric ratios review

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

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

## Practice set 2: cotangent, secant, and cosecant

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

## Want to join the conversation?

- 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?(7 votes)- 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.(8 votes)

- 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.(5 votes)
- 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.(10 votes)

- Are cse, sec and cot in a calculator?(6 votes)
- Sometimes. There are also sometimes inverses of all of them, AND hyperbolic versions of all of those!(4 votes)

- What are the hyperbolic trig functions?(4 votes)
- Is there an inverse for the reciprocal functions: cosecant, secant, and cotangent?(2 votes)
- Yes, they're arccosecant, arcsecant, and arccotangent.(4 votes)

- 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?(2 votes)
- Arcsecant, arccosecant, and arctangent are all inverses of the reciprocal functions.(1 vote)

- How do I solve an equation like this: csc theta=1/sin theta?(1 vote)
- That's not a problem, its a statement. csc(Θ) = 1/sin(Θ), sec(Θ) = 1/cos(Θ), and cot(Θ) = 1/tan(Θ).(3 votes)

- What is the difference between cosecant and arcsine?(1 vote)
- 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(3 votes)

- What are the inverse trig functions of the reciprocal trig functions?(1 vote)
- 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.(3 votes)

- Do we have the inverse function of cosecant, secant and cotangent too?(1 vote)
- 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.(2 votes)