Introduction to the trigonometric ratios
Let's just do a ton of more examples, just so we make sure that we're getting this trig function thing down well. So let's construct ourselves some right triangles. Let's construct ourselves some right triangles, and I want to be very clear. The way I've defined it so far, this will only work in right triangles. So if you're trying to find the trig functions of angles that aren't part of right triangles, we're going to see that we're going to have to construct right triangles, but let's just focus on the right triangles for now. So let's say that I have a triangle, where let's say this length down here is seven, and let's say the length of this side up here, let's say that that is four. Let's figure out what the hypotenuse over here is going to be. So we know -let's call the hypotenuse, "h"- we know that h squared is going to be equal to seven squared plus four squared, we know that from the Pythagorean theorem, that the hypotenuse squared is equal to the square of each of the sum of the squares of the other two sides. h squared is equal to seven squared plus four squared. So this is equal to forty-nine plus sixteen, forty-nine plus sixteen, forty nine plus ten is fifty-nine, plus six is sixty-five. It is sixty five. So this h squared, let me write: h squared -that's different shade of yellow- so we have h squared is equal to sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six is sixty-five, or we could say that h is equal to, if we take the square root of both sides, square root square root of sixty five. And we really can't simplify this at all. This is thirteen. This is the same thing as thirteen times five, both of those are not perfect squares and they're both prime so you can't simplify this any more. So this is equal to the square root of sixty five. Now let's find the trig, let's find the trig functions for this angle up here. Let's call that angle up there theta. So whenever you do it you always want to write down - at least for me it works out to write down - "soh cah toa". soh... ...soh cah toa. I have these vague memories of my trigonometry teacher. Maybe I've read it in some book. I don't know - you know, some... about some type of indian princess named "soh cah toa" or whatever, but it's a very useful mnemonic, so we can apply "soh cah toa". Let's find, let's say we want to find the cosine. We want to find the cosine of our angle. We wanna find the cosine of our angle, you say: "soh cah toa!" So the "cah". "Cah" tells us what to do with cosine, the "cah" part tells us that cosine is adjacent over hypotenuse. Cosine is equal to adjacent over hypotenuse. So let's look over here to theta; what side is adjacent? Well we know that the hypotenuse, we know that that hypotenuse is this side over here. So it can't be that side. The only other side that's kind of adjacent to it that isn't the hypotenuse, is this four. So the adjacent side over here, that side is, it's literally right next to the angle, it's one of the sides that kind of forms the angle it's four over the hypotenuse. The hypotenuse we already know is square root of sixty-five. so it's four over the square root of sixty-five. And sometimes people will want you to rationalize the denominator which means they don't like to have an irrational number in the denominator, like the square root of sixty five, and if they - if you wanna rewrite this without a irrational number in the denominator, you can multiply the numerator and the denominator by the square root of sixty-five. This clearly will not change the number, because we're multiplying it by something over itself, so we're multiplying the number by one. That won't change the number, but at least it gets rid of the irrational number in the denominator. So the numerator becomes four times the square root of sixty-five, and the denominator, square root of 65 times square root of 65, is just going to be 65. We didn't get rid of the irrational number, it's still there, but it's now in the numerator. Now let's do the other trig functions or at least the other core trig functions. We'll learn in the future that there's actually a ton of them but they're all derived from these. so let's think about what the sine of theta is. Once again go to "soh cah toa". The "soh" tells what to do with sine. Sine is opposite over hypotenuse. Sine is equal to opposite over hypotenuse. Sine is opposite over hypotenuse. So for this angle what side is opposite? We just go opposite it, what it opens into, it's opposite the seven so the opposite side is the seven. This is, right here - that is the opposite side and then the hypotenuse, it's opposite over hypotenuse. The hypotenuse is the square root of sixty-five. Square root of sixty-five. and once again if we wanted to rationalize this, we could multiply times the square root of 65 over the square root of 65 and the the numerator, we will get seven square root of 65 and in the denominator we will get just sixty-five again. Now let's do tangent! Let us do tangent. So if i ask you the tangent of - the tangent of theta once again go back to "soh cah toa". The toa part tells us what to do with tangent it tells us... it tells us that tangent is equal to opposite over adjacent is equal to opposite over opposite over adjacent So for this angle, what is opposite? We've already figured it out. it's seven. It opens into the seven. It is opposite the seven. So it's seven over what side is adjacent. well this four is adjacent. This four is adjacent. So the adjacent side is four. so it's seven over four, and we're done. We figured out all of the trig ratios for theta. let's do another one. Let's do another one. i'll make it a little bit concrete 'cause right now we've been saying, "oh, what's tangent of x, tangent of theta." let's make it a little bit more concrete. Let's say... let's say, let me draw another right triangle, that's another right triangle here. Everything we're dealing with, these are going to be right triangles. let's say the hypotenuse has length four, let's say that this side over here has length two, and let's say that this length over here is going to be two times the square root of three. We can verify that this works. If you have this side squared, so you have - let me write it down - two times the square root of three squared plus two squared, is equal to what? this is two. There's going to be four times three. four times three plus four, and this is going to be equal to twelve plus four is equal to sixteen and sixteen is indeed four squared. So this does equal four squared, it does equal four squared. It satisfies the pythagorean theorem and if you remember some of your work from 30 60 90 triangles that you might have learned in geometry, you might recognize that this is a 30 60 90 triangle. This right here is our right angle, - i should have drawn it from the get go to show that this is a right triangle - this angle right over here is our thirty degree angle and then this angle up here, this angle up here is a sixty degree angle, and it's a thirty sixteen ninety because the side opposite the thirty degrees is half the hypotenuse and then the side opposite the 60 degrees is a squared of 3 times the other side that's not the hypotenuse. So that said, we're not gonna ... this isn't supposed to be a review of 30 60 90 triangles although i just did it. Let's actually find the trig ratios for the different angles. So if i were to ask you or if anyone were to ask you, what is... what is the sine of thirty degrees? and remember 30 degrees is one of the angles in this triangle but it would apply whenever you have a 30 degree angle and you're dealing with the right triangle. We'll have broader definitions in the future but if you say sine of thirty degrees, hey, this angle right over here is thirty degrees so i can use this right triangle, and we just have to remember "soh cah toa" We rewrite it. soh, cah, toa. "sine tells us" (correction). soh tells us what to do with sine. sine is opposite over hypotenuse. sine of thirty degrees is the opposite side, that is the opposite side which is two over the hypotenuse. The hypotenuse here is four. it is two fourths which is the same thing as one-half. sine of thirty degrees you'll see is always going to be equal to one-half. now what is the cosine? What is the cosine of thirty degrees? Once again go back to "soh cah toa". The cah tells us what to do with cosine. Cosine is adjacent over hypotenuse. So for looking at the thirty degree angle it's the adjacent. This, right over here is adjacent. it's right next to it. it's not the hypotenuse. it's the adjacent over the hypotenuse. so it's two square roots of three adjacent over...over the hypotenuse, over four. or if we simplify that, we divide the numerator and the denominator by two it's the square root of three over two. Finally, let's do the tangent. The tangent of thirty degrees, we go back to "soh cah toa". soh cah toa toa tells us what to do with tangent. It's opposite over adjacent you go to the 30 degree angle because that's what we care about, tangent of 30. tangent of thirty. Opposite is two, opposite is two and the adjacent is two square roots of three. It's right next to it. It's adjacent to it. adjacent means next to. so two square roots of three so this is equal to... the twos cancel out one over the square root of three or we could multiply the numerator and the denominator by the square root of three. So we have square root of three over square root of three and so this is going to be equal to the numerator square root of three and then the denominator right over here is just going to be three. So that we've rationalized a square root of three over three. Fair enough. Now lets use the same triangle to figure out the trig ratios for the sixty degrees, since we've already drawn it. so what is... what is the sine of the sixty degrees? and i think you're hopefully getting the hang of it now. Sine is opposite over adjacent. soh from the "soh cah toa". for the sixty degree angle what side is opposite? what opens out into the two square roots of three, so the opposite side is two square roots of three, and from the sixty degree angle the adj-oh sorry its the opposite over hypotenuse, don't want to confuse you. so it is opposite over hypotenuse so it's two square roots of three over four. four is the hypotenuse. so it is equal to, this simplifies to square root of three over two. What is the cosine of sixty degrees? cosine of sixty degrees. so remember "soh cah toa". cosine is adjacent over hypotenuse. adjacent is the two sides, right next to the sixty degree angle. So it's two over the hypotenuse which is four. So this is equal to one-half and then finally, what is the tangent? what is the tangent of sixty degrees? Well tangent, "soh cah toa". Tangent is opposite over adjacent opposite the sixty degrees is two square roots of three two square roots of three and adjacent to that adjacent to that is two. Adjacent to sixty degrees is two. So its opposite over adjacent, two square roots of three over two which is just equal to the square root of three. And I just wanted to -look how these are related- the sine of thirty degrees is the same as the cosine of sixty degrees. The cosine of 30 degrees is the same thing as the sine of 60 degrees and then these guys are the inverse of each other and i think if you think a little bit about this triangle it will start to make sense why. we'll keep extending this and give you a lot more practice in the next few videos.