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Trigonometric ratios in right triangles

Learn how to find the sine, cosine, and tangent of angles in right triangles.
The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). These are defined for acute angle A below:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is labeled opposide. Side A C is labeled adjacent. Side A B is labeled hypotenuse.
In these definitions, the terms opposite, adjacent, and hypotenuse refer to the lengths of the sides.

SOH-CAH-TOA: an easy way to remember trig ratios

The word sohcahtoa helps us remember the definitions of sine, cosine, and tangent. Here's how it works:
Acronym PartVerbal DescriptionMathematical Definition
S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ffstart text, S, end textine is start text, start color #11accd, O, end color #11accd, end textpposite over start text, start color #aa87ff, H, end color #aa87ff, end textypotenusesine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction
C, start color #ed5fa6, A, end color #ed5fa6, start color #aa87ff, H, end color #aa87ffstart text, C, end textosine is start text, start color #ed5fa6, A, end color #ed5fa6, end textdjacent over start text, start color #aa87ff, H, end color #aa87ff, end textypotenusecosine, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, divided by, start text, start color #aa87ff, H, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text, end fraction
T, start color #11accd, O, end color #11accd, start color #ed5fa6, A, end color #ed5fa6start text, T, end textangent is start text, start color #11accd, O, end color #11accd, end textpposite over start text, start color #ed5fa6, A, end color #ed5fa6, end textdjacenttangent, left parenthesis, A, right parenthesis, equals, start fraction, start text, start color #11accd, O, p, p, o, s, i, t, e, end color #11accd, end text, divided by, start text, start color #ed5fa6, A, d, j, a, c, e, n, t, end color #ed5fa6, end text, end fraction
For example, if we want to recall the definition of the sine, we reference S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, since sine starts with the letter S. The start text, start color #11accd, O, end color #11accd, end text and the start text, start color #aa87ff, H, end color #aa87ff, end text help us to remember that sine is start text, start color #11accd, o, p, p, o, s, i, t, e, end color #11accd, end text over start text, start color #aa87ff, h, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text!

Example

Suppose we wanted to find sine, left parenthesis, A, right parenthesis in triangle, A, B, C given below:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is three units. Side A C is four units. Side A B is five units.
Sine is defined as the ratio of the start text, start color #11accd, o, p, p, o, s, i, t, e, end color #11accd, end text to the start text, start color #aa87ff, h, y, p, o, t, e, n, u, s, e, end color #aa87ff, end text left parenthesis, S, start color #11accd, O, end color #11accd, start color #aa87ff, H, end color #aa87ff, right parenthesis. Therefore:
Triangle A B C with angle A C B being ninety degrees. Angle B A C is the angle of reference. Side B C is three units. Side A C is four units. Side A B is five units. Sides A B and B C are highlighted.
sin(A)= opposite  hypotenuse=BCAB=35\begin{aligned}\sin( A)&=\dfrac{\blueD{\text{ opposite }} }{ \purpleC{\text{ hypotenuse}} }\\\\ &=\dfrac{\blueD{BC}}{\purpleC{AB}}\\\\\\ &=\dfrac{\blueD{3}}{\purpleC{5}} \\\\\\ \end{aligned}
Here's another example in which Sal walks through a similar problem:
Khan Academy video wrapper
Trigonometric ratios in right trianglesSee video transcript

Practice

Triangle 1: triangle, D, E, F
Triangle D E F with angle E D F being ninety degrees. Side D F is twelve units. Side E F is thirteen units. Side D F is five units.
cosine, left parenthesis, F, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

sine, left parenthesis, F, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

tangent, left parenthesis, F, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Triangle 2: triangle, G, H, I
Triangle G H I with angle G I H being ninety degrees. Side H I is fifteen units. Side I G is eight units. Side H G is seventeen units.
cosine, left parenthesis, G, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

sine, left parenthesis, G, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

tangent, left parenthesis, G, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Challenge problem
In the triangle below, which of the following is equal to start fraction, a, divided by, c, end fraction?
A right triangle with a ninety-degree angle, a twenty-degree angle, and seventy-degree angle. The side opposide of the twenty-degree angle is a units. The side opposite of seventy-degree angle is b units. The side opposite of the ninety-degree angle is c units.
Choose all answers that apply:
Choose all answers that apply:

Want to join the conversation?

  • aqualine ultimate style avatar for user Bhavlabhya
    hey I have a question
    what if we have a triangle with no known sides but 2 angles(including one right angle) is given then how will we find the 3rd angle and 3 sides? is it possible?
    (18 votes)
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    • sneak peak green style avatar for user 490139
      If you know two angles of a triangle, it is easy to find the third one. Since the three interior angles of a triangle add up to 180 degrees you can always calculate the third angle like this:

      Let's suppose that you know a triangle has angles 90 and 50 and you want to know the third angle. Let's call the unknown angle x.
      x + 90 + 50 = 180
      x + 140 = 180
      x = 180 - 140
      x = 40

      As for the side lengths of the triangle, you need more information to figure those out. A triangle of side lengths 10, 14, and 9 has the same angles as a triangle with side lengths of 20, 28, and 18.
      (7 votes)
  • blobby green style avatar for user Ira Kulkarni
    How is theta defined in accurate mathematical language?
    (6 votes)
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  • duskpin sapling style avatar for user V
    What is the etymology of sin, cos and tan?
    (9 votes)
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    • duskpin ultimate style avatar for user ianXmiller
      From Wikipedia - Trigonometric Functions - Etymology

      The word sine derives from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin. The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".

      The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.

      The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.
      (27 votes)
  • piceratops sapling style avatar for user akshaysheriff
    IS there ANY way to easily remember the SIN, COS and TAN formulas?? Any tips and tricks?
    (1 vote)
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  • aqualine ultimate style avatar for user Rishika
    How to find the sin, cos and tan of the 90 degree angle? Will we follow the same procedure as we did with the other two angles?
    (5 votes)
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    • leaf green style avatar for user kubleeka
      If we consider the right angle, the side opposite is also the hypotenuse. So sin(90)=h/h=1.
      By pythagorean theorem, we get that sin^2(90)+cos^2(90)=1. So, substituting, 1+cos^2(90)=1
      cos^2(90)=0
      cos(90)=0

      And we see that tan(90)=sin(90)/cos(90)=1/0. So tan(90) is undefined.
      (7 votes)
  • blobby green style avatar for user Brendon Josh Orate
    Based on the first paragraph, "The ratios of the sides of a right triangle are called trigonometric ratios.", if in trigonometry the ratios of the sides of a triangle are called 'trigonometric ratios' then what if the triangle is not a right triangle. Will the ratios of the sides of that triangle have a different label. And based on my question, how will the mnemonic 'soh cah toa' help find the sides of the 'non- right triangle' triangle? Are there more methods to find the sides of a triangle relative to trigonometric functions or formula?
    (3 votes)
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  • leaf green style avatar for user kubleeka
    I've heard that there are other trigonometric functions out there, with names like versine. Who decided that sine, cosine, and tangent would be the ones we learn in school? What happened to the others?
    (3 votes)
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    • blobby green style avatar for user David Calkins
      I would guess that it's because these functions are technically more complex than the ones we learn in school. For example, versine(x) = 1 - cos(x). Applications of these functions seem to be applicable to navigation, especially across a spherical plane. However, with the progression of technology (I assume) these older functions have grown less practical and have fallen away in favor of manipulations of the more familiar 6 trig functions we study today.
      (6 votes)
  • mr pink red style avatar for user ivanov
    why is sin, cos and tan change?
    (2 votes)
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  • male robot hal style avatar for user TheRealJason
    Can you find the sin, cos and tan of an more than 90 degree angle
    (4 votes)
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  • male robot hal style avatar for user Wormy
    Did anyone else notice that Sal said "Sine is opposite over adjacent" instead of "Sine is opposite over hypotenuse" about the time ?
    (4 votes)
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