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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 ac, we can consider ca.
  • Instead of bc, we can consider cb.
  • Instead of ab, we can consider ba.
These new ratios are the reciprocal trig ratios, and we’re about to learn their names.

The cosecant (csc)

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.
sin(A)=oppositehypotenuse=ac
csc(A)=hypotenuseopposite=ca

The secant (sec)

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.
cos(A)=adjacenthypotenuse=bc
sec(A)=hypotenuseadjacent=cb

The cotangent (cot)

The cotangent is the reciprocal of the tangent. It is the ratio of the adjacent side to the opposite side in a right triangle.
tan(A)=oppositeadjacent=ab
cot(A)=adjacentopposite=ba

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(A)=1sin(A)
secantThe secant is the reciprocal of the cosine.sec(A)=1cos(A)
cotangentThe cotangent is the reciprocal of the tangent.cot(A)=1tan(A)

Finding the reciprocal trigonometric ratios

Let's study an example.

In the triangle below, find csc(C), sec(C), and cot(C).

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
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
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

Try it yourself!

Problem 1
csc(X)=
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4

Problem 2
sec(W)=
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4

Problem 3
cot(R)=
  • Your answer should be
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4

Challenge problem
What is the exact value of csc(45)?

Want to join the conversation?

  • aqualine ultimate style avatar for user Nayan Bansal
    What are these new ratios used for?
    (69 votes)
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  • male robot donald style avatar for user Shambhavi
    What is cosh, sinh, and tanh? I saw these functions on the calculator.
    (29 votes)
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  • aqualine ultimate style avatar for user Joaquin Butial
    Why do you need these functions if you already have sine, cosine, and tangent?
    (13 votes)
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    • 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.
      (66 votes)
  • blobby green style avatar for user Wellmaster
    Wouldn't it make more sense for "secant" to be sin, and "cosecant" cos? There is no good reason for it to be the other way around than to absolutely troll us
    (28 votes)
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  • duskpin ultimate style avatar for user rebecca hu
    How would you find the sin, cosine, or tangent of 90 degrees?
    (8 votes)
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    • 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.
      (26 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.
    (18 votes)
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  • 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
    (10 votes)
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    • 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.
      (11 votes)
  • duskpin ultimate style avatar for user C4LOwenZ
    In the sections for secant and cosecant, the article said that secant is the reciprocal of cosine and that cosecant is the reciprocal of sine. Isn't it supposed to be the other way around?
    (4 votes)
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  • 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)
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  • aqualine ultimate style avatar for user Adhya Anil Kumar
    Are there any other ratios we will learn in the future?
    (2 votes)
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    • 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)