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Trig ratios of special triangles

Learn to find the sine, cosine, and tangent of 45-45-90 triangles and also 30-60-90 triangles.
Until now, we have used the calculator to evaluate the sine, cosine, and tangent of an angle. However, it is possible to evaluate the trig functions for certain angles without using a calculator.
This is because there are two special triangles whose side ratios we know! These two triangles are the 45-45-90 triangle and the 30-60-90 triangle.

The special triangles

30-60-90 triangles
A 30-60-90 triangle is a right triangle with a 30, degrees degree angle and a 60, degrees degree angle.
45-45-90 triangles
A 45-45-90 triangle is a right triangle with two 45, degrees degree angles.

The trigonometric ratios of 30, degrees

We are now ready to evaluate the trig functions of these special angles. Let's start with 30, degrees.
Study the worked example below to see how this is done.

What is sine, left parenthesis, 30, degrees, right parenthesis?

Here's a worked example:
Step 1: Draw the special triangle that includes the angle of interest.
Step 2: Label the sides of the triangle according to the ratios of that special triangle.
Step 3: Use the definition of the trigonometric ratios to find the value of the indicated expression.
sin(30)=opposite hypotenuse=x2x=1x2x=12\begin{aligned} \sin (30^\circ) &= \dfrac{\text{opposite }}{\text{hypotenuse}} \\\\ &= \dfrac{x}{2x} \\\\ &= \dfrac{1\maroonD{\cancel{x}}}{2\maroonD{\cancel{x}}} \\\\ &=\dfrac{1}{2}\end{aligned}
Note that you can think of x as 1, x so that it is clear that start fraction, x, divided by, 2, x, end fraction, equals, start fraction, 1, x, divided by, 2, x, end fraction, equals, start fraction, 1, divided by, 2, end fraction.
Now let's use this method to find cosine, left parenthesis, 30, degrees, right parenthesis and tangent, left parenthesis, 30, degrees, right parenthesis.

What is cosine, left parenthesis, 30, degrees, right parenthesis?

What is tangent, left parenthesis, 30, degrees, right parenthesis?

The trigonometric ratios of 45, degrees

Let's try this process again with 45, degrees. Here we can start by drawing and labeling the sides of a 45-45-90 triangle.

What is cosine, left parenthesis, 45, degrees, right parenthesis?

What is sine, left parenthesis, 45, degrees, right parenthesis?

What is tangent, left parenthesis, 45, degrees, right parenthesis?

The trigonometric ratios of 60degrees

The process of deriving the trigonometric ratios for the special angles 30, degrees, 45, degrees, and 60, degrees is the same.
While we have not yet explicitly shown how to find the trigonometric ratios of 60, degrees, we have all of the information we need!

What is cosine, left parenthesis, 60, degrees, right parenthesis?

What is sine, left parenthesis, 60, degrees, right parenthesis?

What is tangent, left parenthesis, 60, degrees, right parenthesis?

A summary

We have calculated the trig ratios for 30, degrees, 45, degrees, and 60, degrees. The table below summarizes our results.
cosine, left parenthesis, theta, right parenthesissine, left parenthesis, theta, right parenthesistangent, left parenthesis, theta, right parenthesis
theta, equals, 30, degreesstart color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, square root of, 3, end square root, divided by, 3, end fraction, equals, start fraction, 1, divided by, square root of, 3, end square root, end fraction, end color #1fab54
theta, equals, 45, degreesstart color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ffstart color #aa87ff, start fraction, square root of, 2, end square root, divided by, 2, end fraction, equals, start fraction, 1, divided by, square root of, 2, end square root, end fraction, end color #aa87ffstart color #aa87ff, 1, end color #aa87ff
theta, equals, 60, degreesstart color #1fab54, start fraction, 1, divided by, 2, end fraction, end color #1fab54start color #1fab54, start fraction, square root of, 3, end square root, divided by, 2, end fraction, end color #1fab54start color #1fab54, square root of, 3, end square root, end color #1fab54
These values tend to occur often in advanced trigonometry problems. Because of this, it is helpful to know them.
Some people choose to memorize these values, but memorization is not necessary. In this article, you derived the values yourself, so hopefully you can re-derive them whenever you need them in the future.

Want to join the conversation?

  • winston default style avatar for user Jinho Yoon
    Can these trigonometric rations of special triangles be used for finding the trigonometric ratios for other angles like 50 degrees? For example, if I did sin60 * 5/6, would that equal to sin50?
    (36 votes)
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    • male robot hal style avatar for user Amoy Sasorak
      I am sure before calculators they used these 30-60-90 and 45-45-90 ratios they know to arrive at other answers like 50 degrees. I think you are on the right track, but you have to think on bigger scale. We dealing with degrees here. If 360 has been defined as maximum degrees in a circle (our earth), then 5/6 of 360 would fall on 300th degree. I hope you understand what I mean.
      (2 votes)
  • starky sapling style avatar for user yinger.32319
    When I use a calculator and hit the "Sin" button (or another trig function button) and enter a given angle (ie. SIN(45)), what is the calculator doing to give me the opp/adj ratio?

    Also, is there a way to find a ratio without using a calculator or a special triangle relationship? For example, how would you find what Sin(23.5) equals without a calculator?
    (7 votes)
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  • starky sapling style avatar for user keith
    why does sal specify that tangent of 30 degrees is √3/3 ? isn't it 1/√3
    (2 votes)
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  • blobby green style avatar for user DylanSangyoonYou
    In the second video, when Khan does the princibe root of sqrt2x^2 and c^2, how come the x goes outside and the two is left in?
    (2 votes)
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    • aqualine ultimate style avatar for user Devraj Priyadarshi
      Its simple mathematics, think of this in this way:
      2*x*x = c*c
      we see 'x' and 'c' to be squared i.e.
      2*x^2 = c^2
      If you recall studying 'Real Numbers' or 'Indices', you would know that the root of 'x^2' is 'x' , the root of 'c^2' is 'c' and of '2' is '(root sign)2'.
      So by simply finding root of both side:
      2*x^2 = c^2, will be
      (root)2*x = c
      and so we write it as :
      x*(root)2 = c,
      and so here comes the concept that 2 is left behind, when truly it is not.
      (5 votes)
  • starky ultimate style avatar for user VB :)
    what are the sin, cos and tan of 90 degrees?
    (2 votes)
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    • piceratops tree style avatar for user Austin Castillo
      You can't really have a tangent of 90 degrees, at least when it comes to reference triangles, because that would indicate two 90 degree angles. As Sal mentioned in the videos talking about the Unit Circle, you can't have two 90 degree angles in a Triangle. You could theoretically have < or = 89.9999_` triangle, but theoretically two 90.00* = a line.
      (1 vote)
  • blobby green style avatar for user Edgar Ramos
    I don't get how you're getting these answers, the math does not make sense. How is the Tan(30) not 1/√3 . I don't understand how you go from 1/√3 to √3/3. Same with Sin(45), how does he get √2/2? What am I missing here?
    (2 votes)
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    • aqualine ultimate style avatar for user Timber Lin
      you have to rationalize the denominator.
      to rationalize 1/sqrt(3), you want to have no square root in the denominator, so you multiply numerator and denominator by sqrt(3): 1/sqrt(3)*sqrt(3)/sqrt(3)=sqrt(3)/(sqrt(3)*sqrt(3)).
      sqrt(3)*sqrt(3)=3, so you get sqrt(3)/3.
      1/sqrt(3) is equal to sqrt(3)/3, but they're just written differently because sqrt(3)/3 is considered simpler.
      same goes to the sqrt(2)/2
      (4 votes)
  • blobby green style avatar for user Sofia
    this is related to tan 60 and was a multiple-choice question in one of my papers;

    the angle made by the line x/3 - y/2 = 1 with the positive direction of the x-axis is closest to:

    then it gave some options, the correct answer being 33 degrees.

    how do I get to that answer? I know that the gradient of the line is 2/3 and so the angle made by the line with the positive direction of the x-axis will be:

    angle= tan-1 (inverse of tan) 2/3.

    but how does this become close to 33 degrees?

    I appreciate any help and thanks so much!
    (2 votes)
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  • aqualine tree style avatar for user Alex Hilton
    When looking for the cos(60º), I have the adj/hyp, which is X/2X. Is the reason that the answer is 1/2 because it is X over 2 times X and not X over 2X? (Because it makes more sense if it is X over 2 times X whereas if it was X over 2X then it would just be 1/X.)
    (3 votes)
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  • aqualine ultimate style avatar for user mandevmaan
    I don't get how side opposite of the 30-degree angle is x/2. can someone help me understand this problem.
    (2 votes)
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    • piceratops ultimate style avatar for user Benjamin Tay WX
      When you divide an equilateral triangle into two, you have two 30-60-90 triangles of equal length. The opposite side of the 30 degree angle is the base. When you split the base of an equilateral triangle of side length x into 2, you get x/2. You can also try drawing it out.
      Hope this helps
      (2 votes)
  • orange juice squid orange style avatar for user moleabbu
    does the order of the 30 degrees 60 degres and 90 degrees and a 30-60-90 right triangle have to correspond with certain side lenghts?
    (2 votes)
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    • piceratops ultimate style avatar for user Rob Gillan
      Yes, the sides in a 30-60-90 triangle always have the same ratio. Their "length" will depend on the unit measurement (mm, inches, whatever!). Once you know the length of any side, the method for determining the others will always be the same.
      (1 vote)