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.

Main content

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 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 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.
csc(C)=hypotenuse opposite=1715\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.
sec(C)=hypotenuseadjacent=178\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.
cot(C)=adjacentopposite=815\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
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4

Problem 2
\sec, left parenthesis, W, right parenthesis, equals
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4

Problem 3
cotangent, left parenthesis, R, right parenthesis, equals
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4

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

Want to join the conversation?

  • aqualine ultimate style avatar for user Nayan Bansal
    What are these new ratios used for?
    (56 votes)
    Default Khan Academy avatar avatar for user
  • male robot donald style avatar for user Shambhavi
    What is cosh, sinh, and tanh? I saw these functions on the calculator.
    (27 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Joaquin Butial
    Why do you need these functions if you already have sine, cosine, and tangent?
    (14 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Matthew Daly
      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.
      (54 votes)
  • duskpin ultimate style avatar for user rebecca hu
    How would you find the sin, cosine, or tangent of 90 degrees?
    (8 votes)
    Default Khan Academy avatar avatar for user
    • purple pi purple style avatar for user doctorfoxphd
      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.
      (25 votes)
  • starky tree style avatar for user Alexis
    I created Cho Sha Cao. However I don't recommend using it because you can easily mix Cho and Cao up.
    (13 votes)
    Default Khan Academy avatar avatar for user
  • male robot johnny style avatar for user creationtribe
    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
    (6 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Polina Vitić
      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.
      (5 votes)
  • aqualine ultimate style avatar for user Adhya Anil Kumar
    Are there any other ratios we will learn in the future?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • starky ultimate style avatar for user harsh ♡
      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
      (6 votes)
  • duskpin ultimate style avatar for user Mahati
    So what is the exact difference between cosecant, secant, cotangent and cos-1, sin-1, tan-1?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • starky sapling style avatar for user G junior
    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'"?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user Ayush Barik
    what is the exact difference between cosecant, secant, cotangent and cos-1, sin-1, tan-1?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • sneak peak blue style avatar for user Lupus
      cosecant, secant, and cotangent are basically flipping the fractions which is called reciprocal. E.g: 3/5 is turned into 5/3 when reciprocated.

      cos-1, sin-1, and tan-1 are when you use the same fractions but reverse their purpose. E.g: you use cos (55°) to find the side length and in another case you use cos-1(2/4) to find a missing angle.
      (1 vote)