 In this video, I want to introduce the idea of right triangle trigonometry. We've defined the six trigonometric ratios, sine, cosine, tangent, cosecant, secant, and cotangent, with respect to an angle, that is, we use a terminal point on the terminal side of the angle to define the six trigonometric ratios. But we can also define the six trigonometric ratios with respect to an angle of a right triangle, one of the non-right angles. So consider the triangle you see right here, this ABC triangle, where you'll notice C is our right angle. And so we can define sine, cosine, tangent, cosecant, secant, and cotangent with respect to this angle A right here. Now when we talk about the sides of a triangle relative to an angle, we use the following notation. So the segment BC we are going to refer to as the opposite side, that is, it's the side opposite of the angle A. It's often abbreviated OPP. Then the side AC that includes the angle A, this we often refer to as the adjacent side, ADJ for short. And then the final side AB, we call the hypotenuse, because it is a right triangle after all, we'll call that HYP for short. And so when it comes to the trigonometric ratios of a triangle with respect to the angle A, we're going to use the following definition. We define sine to be opposite over hypotenuse, the ratio of opposite divided by the hypotenuse side. Cosine will be defined to be the adjacent side divided by the hypotenuse, or what we say adjacent over hypotenuse. Tangent is defined to be the opposite side divided by the adjacent side. Of course, we're talking about the lengths of these sides here, or so tangents opposite over adjacent. The cosecant ratio is defined to be the hypotenuse divided by the opposite side, or hypotenuse over opposite. Secant is defined to be hypotenuse divided by adjacent. And then lastly, cotangents defined to be adjacent over opposite. And that gives us the six trigonometric ratios for a right triangle. Now, I want to mention that we've defined the six trigonometric ratios for an angle previously. Is there any potential confusion going on here? It's like, am I thinking of this as an angle of a triangle, or the angle of a plane? And it turns out there's not going to be any confusion necessarily going on here, because if we put this triangle in the following orientation, we align the adjacent side on the x-axis so that the point A is the origin there. And then the opposite side runs parallel to the y-axis. We've put this triangle so that the angle A is now in standard position. And so the six trigonometric ratios we defined for this angle A are exactly the ratios you see right here. Because we take as our point, you'll notice the hypotenuse then turns out to be the terminal side of the angle. We have our angle right here. And let's just pick the point B to be our point x and y. Which then the distance of the adjacent side will be the x-coordinate of this point. The opposite side will just be the y-coordinate of this point. And then the distance this point from the origin is just the hypotenuse. And so this usual x, y, and r notation we used previously is exactly the case we have for these right triangles. So the trigonometry of an acute of a right triangle, which the angle necessarily will be acute here, A, that will actually coincide with trigonometric ratios from an angle terminating in the first quadrant here. And so when one studies trigonometry, you often see the definition we saw previously first, or maybe you see this right triangle definition first. It doesn't really matter because the two are going to overlap with each other. I think, honestly, there's sort of a preferred approach to the right triangle approach. Because I think it feels far a little less abstract than what we did previously. Also, there's a very nice mnemonic device. You can help remember this, the right triangle trigonometry. So when you look, because like we've learned previously, because this right triangle trigonometry coincides with the previous trigonometry we introduced, things like the reciprocal identities are still true, right? Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, cotangent is the reciprocal of tangent. We see that. We also get that sine divided by cosine is tangent, those quotient identities still hold. So it turns out because of the reciprocal identities, we just need to focus on sine, cosine, and tangent. If you look at your scientific calculator, it probably only has a sine, cosine, and tangent button. Because if you needed to do secant, just compute cosine and do its reciprocal. So to help you remember this, people often use the following mnemonic device. If you take sine, sine is opposite over hypotenuse. If you just look at the first three letters, you get SOH. If you do cosine, cosine is going to be adjacent over hypotenuse. If you look at just the first three letters there, you're going to get CAH. And then lastly, if you look at tangent, you're going to get tangent is opposite over adjacent. Looking at the first three letters, you get TOA. And so you think of this as a fun little chant. SOH CAH TOA SOH CAH TOA SOH CAH TOA. What does it mean? It means sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. And so, you know, you get into a trigonometric class and people run around with torches chanting SOH CAH TOA SOH CAH TOA. It helps us remember these right triangle calculations. Let's take a look at such a calculation here. Let's suppose we have a right triangle, ABC here, for which we know that the opposite side has length 6, and we know that the hypotenuse has length 10. We actually don't know what the adjacent side looks like here, but we can compute it using the Pythagorean equation, because we know that the adjacent side squared plus the opposite side squared is going to equal the hypotenuse squared. So let's just call the adjacent side X for a moment because we don't know what it is. The opposite side was 6, so we get 6 squared, and then the hypotenuse was 10, so we get 10 squared. So we're going to get X squared plus 36 is equal to 100. So track 36 from both sides, we get X squared equals 64. Take the square root and assuming distance does have to be positive here. We're going to get the square root of 64, which is equal to 8. So it turns out the adjacent side is equal to 8. And then using SOH CAH TOA SOH CAH TOA, we can compute all of these trigonometric ratios. Sine of A is going to be opposite over hypotenuse. We get 6 over 10, or you can simplify that to be three-fifths. Cosine is going to be adjacent over hypotenuse. This is going to be 8 over 10, but we can reduce that fraction to be four-fifths. And then tangent of A, this is opposite over adjacent, so we get 6 over 8, or you can simplify that to be three-fifths. And then the other trigonometric ratios we can compute just either by the definition or we can just use the reciprocal identities. Notice a cosecant of A, that's going to equal five-thirds the reciprocal of sine. We could do secant of A, this will equal five-fourths the reciprocal of cosine. And then cotangent of A, this is going to equal four-thirds the reciprocal of tangent.