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  Intro to the trigonometric ratios: Sal introduces sine, cosine, and tangent, and gives an example of finding them for a given right triangle.

In this video I want to give you the basics of Trigonometry. It sounds like a very complicated topic but you're going to see this is 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. 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. 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 will 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, 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've 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 that this satisfies the Pythagorean theorem. Now with that out of the way let's learn a little bit of Trigonometry. The core functions of trigonometry, we're going to learn a little more about what these functions mean. There is the sine, the sine function. There is the cosine function, and there is the tangent function. And you write sin, or S-I-N, C-O-S, and "tan" for short. And these really just specify, for any angle in this triangle, it will specify the ratios of certain sides. So let me just write something out.

This is really something of a mnemonic here, so something just to help you remember the definitions of these functions, but I'm going to write down something called "soh cah toa", you'll be amazed how far this mnemonic will take you in trigonometry. We have "soh cah toa", and what this tells us is; "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 now, I'll do it in a little 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 the length 4, and the side of length 5. This is theta. So let figure out the sine of theta, the cosine of theta, and what the tangent of theta are. So if we first want to focus on the sine of theta, we just have to remember "soh cah toa", sine is opposit over hypotonuse, so sine of theta is equal to the opposite - so what is the opposite side to the angle? So this is our angle right here, the opposite side, if we just go to the opposite side, not one of the sides that are kind of adjacent to the angle, the opposite side is the 3, if you're just kind of - it's opening on to that 3, so the opposite side is 3. And then what is the hypotenuse? Well, we already know - the hypotenuse here is 5. So it's 3 over 5. The sine of theta is 3/5. And I'm going to show you in a second, that the sine of theta - 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. So 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 we're talking about this angle - this side is opposite to it. When we're talking 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 the adjacent side. And I want to be very clear, this only applies to this angle.

If we're 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 - so the adjacent side of this angle is 4, so the adjacent over the hypotenuse, the adjacent, which is 4, over the hypotenuse, 4 over 5. Now let's do the tangent. Let's do the tangent. The tangent of theta: opposite over adjacent. The opposite side is 3. What is the adjacent side? We've 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 that 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 re-draw it, because my triangle is getting a little bit messy. So I'll re-draw the exact same triangle. The exact same triangle. And, once again, the lengths of this triangle are - we have length 4 there, we have length 3 there, we have length 5 there. In the last example we used this theta. But let's do another angle, 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. It's, I know, a little bit bizarre. Theta is what you normally use, but since I've already used theta, let's use psi. Or actually - let me simplify it, let me call this angle x. Let's call that 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 on to this 4, it opens on to the 4. So in this context, this is now the opposite, this is now the opposite side. 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, that's not the hypotenuse, so this is the adjacent side. That is the adjacent. So it's 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, 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! And 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 angle start to approach 90 degrees, or how could they even get larger than 90 degrees.

And we'll see 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 boundries. 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.

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