Triangle similarity & the trigonometric ratios

We've got two right triangles here. And let's say we also know that they both have an angle whose measure is equal to theta. So angle A is congruent to angle D. What do we now know about these two triangles? Well for any triangle, if you know two of the angles, you're going to know the third angle, because the sum of the angles of a triangle add up to 180 degrees. So if you have two angles in common, that means you're going to have three angles in common. And if you have three angles in common, you are dealing with similar triangles. Let me make that a little bit clearer. So if this angle is theta, this is 90. They all have to add up to 180 degrees. That means that this angle plus this angle up here have to add up to 90. We've already used up 90 right over here, so angle A and angle B need to be complements. So this angle right over here needs to be 90 minus theta. Well we could use the same logic over here. We already use of 90 degrees over here. So we have a remaining 90 degrees between theta and that angle. So this angle is going to be 90 degrees minus theta. You have three corresponding angles being congruent. You are dealing with similar triangles. Now why is that interesting? Well we know from geometry that the ratio of corresponding sides of similar triangles are always going to be the same. So let's explore the corresponding sides here. Well, the side that jumps out-- when you're dealing with the right triangles-- the most is always the hypotenuse. So this right over here is the hypotenuse. This hypotenuse is going to correspond to this hypotenuse right over here. And then we could write that down. This is the hypotenuse of this triangle. This is the hypotenuse of that triangle. Now this side right over here, side BC, what side does that correspond to? Well if you look at this triangle, you can view it as the side that is opposite this angle theta. So it's opposite. If you go across the triangle, you get there. So let's go opposite angle D. If you go opposite angle A, you get to BC. Opposite angle D, you get to EF. So it corresponds to this side right over here. And then finally, side AC is the one remaining one. We could view it as, well, there's two sides that make up this angle A. One of them is the hypotenuse. We could call this, maybe, the adjacent side to it. And so D corresponds to A, and so this would be the side that corresponds. Now the whole reason I did that is to leverage that, corresponding sides, the ratio between corresponding sides of similar triangles, is always going to be the same. So for example, the ratio between BC and the hypotenuse, BA-- so let me write that down-- BC/BA is going to be equal to EF/ED, the length of segment EF over the length of segment ED. Or we could also write that the length of segment AC over the hypotenuse, over this triangle's hypotenuse, over AB, is equal to DF/DE-- once again, this green side over the orange side. These are similar triangles. They're corresponding to each other. So this is equal to DF/DE. And we could keep going, but I'll just do another one. Or we could say that the ratio of this side right over here-- this blue side to the green side of this triangle-- the length of BC/CA is going to be the same as the ratio between these two corresponding sides, the blue over the green, EF/DF. And we got all of this from the fact that these are similar triangles. So this is true for any right triangle that has an angle theta. Then those two triangles are going to be similar, and all of these ratios are going to be the same. Well, maybe we can give names to these ratios relative to the angle theta. So from angle theta's point of view-- I'll write theta right over here, or we can just remember that-- what is the ratio of these two sides? Well from theta's point of view, that blue side is the opposite side. It's opposite-- so the opposite side of the right triangle. And then the orange side we've already labeled the hypotenuse. So from theta's point of view, this is the opposite side over the hypotenuse. And I keep stating from theta's point of view because that wouldn't be the case for this other angle, for angle B. From angle B's point of view, this is the adjacent side over the hypotenuse. And we'll think about that relationship later on. But let's just all think of it from theta's point of view right over here. So from theta's point of view, what is this? Well theta's right over here. Clearly AB and DE are still the hypotenuses-- hypoteni. I don't know how to say that in plural again. And what is AC, and what are DF? Well, these are adjacent to it. They're one of the two sides that make up this angle that is not the hypotenuse. So this we can view as the ratio, in either of these triangles, between the adjacent side-- so this is relative. Once again, this is opposite angle B, but we're only thinking about angle A right here, or the angle that measures theta, or angle D right over here-- relative to angle A, AC is adjacent. Relative to angle D, DF is adjacent. So this ratio right over here is the adjacent over the hypotenuse. And it's going to be the same for any right triangle that has an angle theta in it. And then finally, this over here, this is going to be the opposite side. Once again, this was the opposite side over here. This ratio for either right triangle is going to be the opposite side over the adjacent side. And I really want to stress the importance-- and we're going to do many, many more examples of this to make this very concrete-- but for any right triangle that has an angle theta, the ratio between its opposite side and its hypotenuse is going to be the same. That comes out of similar triangles. We've just explored that. The ratio between the adjacent side to that angle that is theta and the hypotenuse is going to be the same, for any of these triangles, as long as it has that angle theta in it. And the ratio, relative to the angle theta, between the opposite side and the adjacent side, between the blue side and the green side, is always going to be the same. These are similar triangles. So given that, mathematicians decided to give these things names. Relative to the angle theta, this ratio is always going to be the same, so the opposite over hypotenuse, they call this the sine of the angle theta. Let me do this in a new color-- by definition-- and we're going to extend this definition in the future-- this is sine of theta. This right over here, by definition, is the cosine of theta. And this right over here, by definition, is the tangent of theta. And a mnemonic that will help you remember this-- and these really are just definitions. People realized, wow, by similar triangles, for any angle theta, this ratio is always going to be the same. Because of similar triangles, for any angle theta, this ratio is always going to be the same. This ratio is always going to the same. So let's make these definitions. And to help us remember it, there's the mnemonic soh-cah-toa. So I'll write it like this. soh is sine is opposite over hypotenuse. cah-- cosine is adjacent over hypotenuse. And then finally, tangent is opposite over adjacent-- soh-cah-toa. And in future videos, we'll actually apply these definitions for these trigonometric functions.