Triangle similarity & the trigonometric ratios (video) | Khan Academy
Jun 7, Tan really likes Cos, and Cos likes him back but they'e both in a difficult situation because of Tan's relationship with Sin. It's really complicated. In mathematics, the trigonometric functions are functions of an angle. They relate the angles of The most familiar trigonometric functions are the sine, cosine, and tangent. In the This exhibits a deep relationship between the complex sine and cosine functions and their real (sin, cos) and hyperbolic real (sinh, cosh). This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are.
That means that this angle plus this angle up here have to add up to 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. 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. These are similar triangles. They're corresponding to each other. And we could keep going, but I'll just do another one. 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.
- Triangle similarity & the trigonometric ratios
- Sine, Cosine and Tangent
- Intro to the trigonometric ratios
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. 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 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.
Trigonometric functions - Wikipedia
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 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. So if someone says, hey, what's the sine of that?
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. 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.