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Intro to the trigonometric ratios

Sin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles. Sin is the ratio of the opposite side to the hypotenuse, cos is the ratio of the adjacent side to the hypotenuse, and tan is the ratio of the opposite side to the adjacent side. They are often written as sin(x), cos(x), and tan(x), where x is an angle in radians or degrees. Created by Sal Khan.

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  • mr pants teal style avatar for user estewart
    What do you use for the adjacent side if you're trying to figure out the sin cos and tan for the right angle?
    (209 votes)
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    • leaf green style avatar for user Hello
      Well, in beginning trigonometry, it's convenient to evaluate sin/cos/tan by using soh-cah-toa, but later, as you get into the unit circle and you start taking taking stuff like sin(135) and tan(-45) you don't use the adjacent-opposite-hypotenuse much anymore. If you can think of it intuitively, though, sin(90) means that the opposite side is infinitely long, and the hypotenuse is also infinitely long, so sin(90)=1. cos(90) means adjacent over the hypotenuse, which is infinitely long given that the angle is 90 degrees, so any number over infinity is 0, so cos(90)=0. tan(90)=sin(90)/cos(90)=1/0, so tan(90) doesn't exist.
      (115 votes)
  • female robot grace style avatar for user Hermione
    How can you figure out which is the opposite or the adjacent?
    (105 votes)
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  • blobby green style avatar for user born2bgeeky
    A small question. We name the ratios as "Sine", "Cosine", "Tangent", "Cotagent", "Secant" and "Cosecant". The last four can be drawn of circle. Is "Sine" also a part of circle. I mean can it be drawn on circle like tangent and secant. I know its a useless question, but I was just wondering. Thanks for your time.
    (2 votes)
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    • leaf blue style avatar for user Matthew Daly
      I think that's a great question! This is a pretty cool story (to me at least). The word that the Arabs used for sine was the same as their word for "chord", but when a European translated it into Latin he read it wrong and translated it as sinus, which is the Latin word for chest. So it's a historical accident that secant and tangent have geometric meanings but sine doesn't.
      (118 votes)
  • stelly orange style avatar for user cozy
    Why does Sal (the person talking in the video) use theta or some other greek letter for the angles instead of a normal variable, like x or y, for every angle he shows the sin, cos, and tan for?
    (5 votes)
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    • spunky sam orange style avatar for user Elder Fauth
      The reason is because in the world of math (not khan academy's "world of math"), mathematicians usually use x and y for missing lengths, and use Greek letters for unknown angles, most likely in honor of Elucid, founder of geometry, who was Greek.

      Hope that helps!
      (29 votes)
  • blobby green style avatar for user sifukada
    My question is around how to calculate sin, cos or tan. I've pushed the sin/cos/tan button many times on my calculator with no _idea_ what is actually happening. From doing some of my own research, it seems like a Taylor Series may have to be used? Is this the only way? Did someone once sit down and measure every angle and every side of the triangle to get each ratio into a large table? Would it then be something like a look up table with the calculator simply searching for the closest ratio that matches what is typed into the calculator?
    (12 votes)
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    • hopper cool style avatar for user obiwan kenobi
      sin cos and tan are basically just functions that relate an angle with a ratio of two sides in a right triangle. Sin is equal to the side opposite the angle that you are conducting the functions on over the hypotenuse which is the longest side in the triangle. Cos is adjacent over hypotenuse. And tan is opposite over adjacent, which means tan is sin/cos. this can be proved with some basic algebra.
      (6 votes)
  • male robot hal style avatar for user KhanacademY947
    To help you guys understand SOH CAH TOA even better, I decided to create this comment as a question.

    SOH: [S is Sine, O is Opposite, H is Hypotenuse]. Then, [Sine= Opposite/Hypotenuse].

    CAH: [C is Cosine, A is Adjacent, H is Hypotenuse]. Then, [Cosine= Adjacent/Hypotenuse].

    TOA: [T is Tangent, O is Opposite, A is Adjacent]. Then, [Tangent= Opposite/Adjacent].

    To simplify it to make you guys understand even better, knowing the short form for it, I shall show it down below.

    SOH: S= O/H

    CAH: C= A/H

    TOA: T= O/A

    Now, since I already told you guys about the SOH CAH TOA form, I shall give you guys an example.

    For Example:
    If you are given a triangle where the two significantly shorter sides are given and you wish to obtain the longest side termed the hypotenuse, you recognize that it satisfies the terms necessary to use either the SOH or CAH form; You, therefore, proceed to identify which is adjacent and which is the opposite. The opposite, which is clearly identifiable due to its name, is the side that is directly OPPOSITE the given angle. The adjacent is therefore the side which forms a 90° angle to the opposite.

    Another Example:
    For purposes, the given angle is 45°. So you then proceed to imply due to the SOH form that Sine45= opposite divided by the hypotenuse. For you to obtain the hypotenuse, we transpose it for the hypotenuse to become the subject of the formula. By cross multiplying we obtain the formula:
    Hypotenuse = Opposite divided by Sine45.

    Note: Some of these contents are copied from @machyl69.

    Hope This Helps,
    Thank You!
    (13 votes)
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  • piceratops ultimate style avatar for user Ayush Singhal
    what do you call the hypotenuse if it's not a right angle triangle
    (3 votes)
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  • leaf blue style avatar for user Anthony Jacquez
    How do you use trigonometry on 3d and even 4d shapes and objects?
    (5 votes)
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    • blobby green style avatar for user 01123fibonacci581321
      Trigonometry can be applied to 3d objects. Consider you have a cube, and you know that angle from cube diagonal to diagonal of square is 45° from here you can easily apply these methods. For 4d would be similiar but yet 4d system would need to be defined and you need to derive many of the equations for that abstraction.
      (5 votes)
  • male robot hal style avatar for user berfle
    Is hypotenuse the longest side or what?
    (0 votes)
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  • duskpin ultimate style avatar for user _______
    if the angle changes the sin and cos interchange .why?
    (4 votes)
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    • hopper cool style avatar for user obiwan kenobi
      you need to understand what sin and cos are. They are trigonometric ratios. They relate an acute angle in a right triangle with two sides in the triangle. Lets say you take the sin of one acute angle. that is equal to the opposite side/hypotenuse. Now visualize. The opposite of this angle is the same as the adjacent for the other angle. Therefore, the cos of the other angle is equal to the sin of this angle
      (4 votes)

Video transcript

In this video, I want to give you the basics of trigonometry. And it sounds like a very complicated topic, but you're going to see that it's really just the study of the ratios of sides of triangles. The "trig" part of trigonometry literally means triangle. And the "metry" part literally means measure. So let me just give you some examples here. And I think it'll make everything pretty clear. So let me draw some right triangles. Let me just draw one right triangle. So this is a right triangle. And when I say it's a right triangle, it's because one of the angles here is 90 degrees. This right here is a right angle. It is equal to 90 degrees. And we'll talk about other ways to show the magnitude of angles in future videos. So we have a 90-degree angle. It's a right triangle. And let me put some lengths to the sides here. So this side over here is maybe 3. This height right over there is 3. Maybe the base of the triangle right over here is 4. And then the hypotenuse of the triangle over here is 5. You only have a hypotenuse when you have a right triangle. It is the side opposite the right angle. And it is the longest side of a right triangle. So that right there is the hypotenuse. You probably learned that already from geometry. And you can verify that this right triangle, the sides work out. We know from the Pythagorean theorem that 3 squared plus 4 squared has got to be equal to the length of the longest side, the length of the hypotenuse squared, is equal to 5 squared. So you can verify that this works out. This satisfies the Pythagorean theorem. Now, with that out of the way, let's learn a little bit of trigonometry. So the core functions of trigonometry-- we're going to learn a little bit more about what these functions mean. There is the sine function. There is the cosine function. And there is the tangent function. And you write S-I-N, C-O-S, and tan for short. And these really just specify-- for any angle in this triangle, it'll specify the ratios of certain sides. So let me just write something out. And this is a little bit of a mnemonic here, so something just to help you remember the definitions of these functions. But I'm going to write down something. It's called soh cah toa. And you'll be amazed how far this mnemonic will take you in trigonometry. So we have soh cah toa. And what this tells us-- soh tells us that sine is equal to opposite over hypotenuse. It's telling us-- and this won't make a lot of sense just yet. I'll do it a little bit more detail in a second. And then cosine is equal to adjacent over hypotenuse. And then you finally have tangent. Tangent is equal to opposite over adjacent. So you're probably saying, hey, Sal. What is all this opposite, hypotenuse, adjacent? What are we talking about? Well, let's take an angle here. Let's say that this angle right over here is theta, between the side of length 4 and the side of length 5. This angle right here is theta. So let's figure out what the sine of theta, the cosine of theta, and what the tangent of theta are. So if we want to first focus on the sine of theta, we just have to remember soh cah toa. Sine is opposite over hypotenuse. So sine of theta is equal to the opposite. So what's the opposite side to the angle? So this is our angle right here. The opposite side, so not one of the sides that are kind of adjacent to the angle. The opposite side is the 3. It's opening onto that 3. So the opposite side is 3. And then what's the hypotenuse? Well, we already know. The hypotenuse here is 5. So it's 3 over 5. The sine of theta is 3/5. So if someone says, hey, what's the sine of that? It's 3/5. And I'm going to show you in a second that if this angle is a certain angle, it's always going to be 3/5. The ratio of the opposite to the hypotenuse is always going to be the same, even if the actual triangle were a larger triangle or a smaller one. So I'll show you that in a second. But let's go through all of the trig functions. Let's think about what the cosine of theta is. Cosine is adjacent over hypotenuse. So remember. Let me label them. We already figured out that the 3 was the opposite side. This is the opposite side. And only when we're talking about this angle. When you talk about this angle, this side is opposite to it. When you talk about this angle, this 4 side is adjacent to it. It's one of the sides that kind of make up, that kind of form the vertex here. So this right here is an adjacent side. And I want to be very clear. This only applies to this angle. If we were talking about that angle, then this green side would be opposite and this yellow side would be adjacent. But we're just focusing on this angle right over here. So cosine of this angle-- we care about adjacent. Well, the adjacent side to this angle is 4. So it is adjacent over the hypotenuse. It's the adjacent, which is 4, over the hypotenuse-- 4/5. Now let's do the tangent. The tangent of theta, opposite over adjacent. The opposite side is 3. What is the adjacent side? We already figured that out. The adjacent side is 4. So knowing the sides of this right triangle, we were able to figure out the major trig ratios. And we'll see there are other trig ratios, but they can all be derived from these three basic trig functions. Now, let's think about another angle in this triangle. And I'll redraw it just because my triangle is getting a little bit messy. So let's redraw the exact same triangle. And once again, the lengths of this triangle are we have length 4 there, we have length 3 there, and we have length 5 there. The last example, we used this theta. But let's do another angle up here. And let's call this angle-- I don't know. I'll think of something, a random Greek letter. So let's say it's psi. I know it's a little bit bizarre. Theta is what you normally use. But since I already used theta, let's use psi. Actually, instead of psi, let me just simplify it. Let me call this angle x. So let's figure out the trig functions for that angle x. So we have sine of x is going to be equal to what. Well, sine is opposite over hypotenuse. So what side is opposite to x? Well, it opens onto this 4. So in this context, this is now the opposite. Remember, 4 was adjacent to this theta, but it's opposite to x. So it's going to be 4 over-- now, what's the hypotenuse? Well, the hypotenuse is going to be the same regardless of which angle you pick. So the hypotenuse is now going to be 5. So it's 4/5. Now let's do another one. What is the cosine of x? So cosine is adjacent over hypotenuse. What side is adjacent to x? That's not the hypotenuse. You have the hypotenuse here. Well, the 3 side-- it's one of the sides that forms the vertex that x is at, and it's not the hypotenuse. So this is the adjacent side. That is adjacent. So it's equal to 3 over the hypotenuse. The hypotenuse is 5. And then finally, the tangent. We want to figure out the tangent of x. Tangent is opposite over adjacent. Soh cah toa-- tangent is opposite over adjacent. The opposite side is 4. I want to do it in that blue color. The opposite side is 4, and the adjacent side is 3. And we're done. In the next video, I'll do a ton of more examples of this just so that we really get a feel for it. But I'll leave you thinking of what happens when these angles start to approach 90 degrees, or how could they even get larger than 90 degrees. And what we're going to see is that this definition, the soh cah toa definition, takes us a long way for angles that are between 0 and 90 degrees, or that are less than 90 degrees. But they kind of start to mess up really at the boundaries. And we're going to introduce a new definition, that's kind of derived from the soh cah toa definition, for finding the sine, cosine, and tangent of really any angle.