WEBVTT 00:00:00.800 --> 00:00:03.017 Let's just do a ton of more examples, 00:00:03.017 --> 00:00:07.036 just so we make sure that we're getting this trig function thing down well. 00:00:07.036 --> 00:00:11.447 So let's construct ourselves some right triangles. 00:00:11.447 --> 00:00:13.668 Let's construct ourselves some right triangles, 00:00:13.668 --> 00:00:15.186 and I want to be very clear. 00:00:15.186 --> 00:00:18.042 The way I've defined it so far, this will only work in right triangles. 00:00:18.042 --> 00:00:23.475 So if you're trying to find the trig functions of angles that aren't part of right triangles, 00:00:23.475 --> 00:00:25.704 we're going to see that we're going to have to construct right triangles, 00:00:25.704 --> 00:00:27.867 but let's just focus on the right triangles for now. 00:00:27.867 --> 00:00:31.344 So let's say that I have a triangle, 00:00:31.344 --> 00:00:33.897 where let's say this length down here is seven, 00:00:33.897 --> 00:00:37.757 and let's say the length of this side up here, 00:00:37.757 --> 00:00:39.452 let's say that that is four. 00:00:39.452 --> 00:00:42.516 Let's figure out what the hypotenuse over here is going to be. 00:00:42.516 --> 00:00:45.720 So we know -let's call the hypotenuse, "h"- 00:00:45.720 --> 00:00:52.200 we know that h squared is going to be equal to seven squared plus four squared, 00:00:52.200 --> 00:00:55.194 we know that from the Pythagorean theorem, 00:00:55.194 --> 00:00:57.469 that the hypotenuse squared is equal to 00:00:57.469 --> 00:01:01.974 the square of each of the sum of the squares of the other two sides. 00:01:01.974 --> 00:01:04.533 h squared is equal to seven squared plus four squared. 00:01:04.533 --> 00:01:09.776 So this is equal to forty-nine plus sixteen, 00:01:09.776 --> 00:01:11.800 forty-nine plus sixteen, 00:01:11.800 --> 00:01:18.553 forty nine plus ten is fifty-nine, plus six is sixty-five. 00:01:18.553 --> 00:01:21.107 It is sixty five. So this h squared, 00:01:21.107 --> 00:01:25.705 let me write: h squared -that's different shade of yellow- 00:01:25.705 --> 00:01:28.818 so we have h squared is equal to sixty-five. 00:01:28.818 --> 00:01:33.533 Did I do that right? Forty nine plus ten is fifty nine, plus another six is sixty-five, 00:01:33.533 --> 00:01:37.600 or we could say that h is equal to, if we take the square root of both sides, 00:01:37.600 --> 00:01:39.200 square root 00:01:39.200 --> 00:01:42.933 square root of sixty five. And we really can't simplify this at all. 00:01:42.933 --> 00:01:44.699 This is thirteen. 00:01:44.699 --> 00:01:47.463 This is the same thing as thirteen times five, 00:01:47.463 --> 00:01:50.388 both of those are not perfect squares and 00:01:50.388 --> 00:01:51.804 they're both prime so you can't simplify this any more. 00:01:51.804 --> 00:01:55.467 So this is equal to the square root of sixty five. 00:01:55.467 --> 00:02:02.114 Now let's find the trig, let's find the trig functions for this angle up here. 00:02:02.114 --> 00:02:05.457 Let's call that angle up there theta. 00:02:05.457 --> 00:02:06.533 So whenever you do it 00:02:06.533 --> 00:02:09.467 you always want to write down - at least for me it works out to write down - 00:02:09.467 --> 00:02:11.714 "soh cah toa". 00:02:11.714 --> 00:02:13.120 soh... 00:02:13.120 --> 00:02:16.464 ...soh cah toa. I have these vague memories 00:02:16.464 --> 00:02:18.786 of my trigonometry teacher. 00:02:18.786 --> 00:02:21.293 Maybe I've read it in some book. I don't know - you know, some... about 00:02:21.293 --> 00:02:23.867 some type of indian princess named "soh cah toa" or whatever, 00:02:23.867 --> 00:02:26.123 but it's a very useful mnemonic, 00:02:26.123 --> 00:02:27.564 so we can apply "soh cah toa". 00:02:27.564 --> 00:02:31.046 Let's find, let's say we want to find the cosine. 00:02:31.046 --> 00:02:34.436 We want to find the cosine of our angle. 00:02:34.436 --> 00:02:37.965 We wanna find the cosine of our angle, you say: "soh cah toa!" 00:02:37.965 --> 00:02:40.800 So the "cah". "Cah" tells us what to do with cosine, 00:02:40.800 --> 00:02:43.027 the "cah" part tells us 00:02:43.027 --> 00:02:46.371 that cosine is adjacent over hypotenuse. 00:02:46.371 --> 00:02:51.433 Cosine is equal to adjacent over hypotenuse. 00:02:51.433 --> 00:02:55.798 So let's look over here to theta; what side is adjacent? 00:02:55.798 --> 00:02:57.702 Well we know that the hypotenuse, 00:02:57.702 --> 00:03:00.767 we know that that hypotenuse is this side over here. 00:03:00.767 --> 00:03:04.761 So it can't be that side. The only other side that's kind of adjacent to it that 00:03:04.761 --> 00:03:07.133 isn't the hypotenuse, is this four. 00:03:07.133 --> 00:03:10.473 So the adjacent side over here, that side is, 00:03:10.473 --> 00:03:14.374 it's literally right next to the angle, 00:03:14.374 --> 00:03:15.754 it's one of the sides that kind of forms the angle 00:03:15.754 --> 00:03:17.133 it's four over the hypotenuse. 00:03:17.133 --> 00:03:21.108 The hypotenuse we already know is square root of sixty-five. 00:03:21.108 --> 00:03:25.380 so it's four over the square root of sixty-five. 00:03:25.380 --> 00:03:29.142 And sometimes people will want you to rationalize the denominator which means 00:03:29.142 --> 00:03:32.625 they don't like to have an irrational number in the denominator, 00:03:32.625 --> 00:03:35.227 like the square root of sixty five, 00:03:35.227 --> 00:03:39.359 and if they - if you wanna rewrite this without a irrational number in the denominator, 00:03:39.359 --> 00:03:41.634 you can multiply the numerator and the denominator 00:03:41.634 --> 00:03:43.306 by the square root of sixty-five. 00:03:43.306 --> 00:03:45.094 This clearly will not change the number, 00:03:45.094 --> 00:03:48.122 because we're multiplying it by something over itself, 00:03:48.122 --> 00:03:49.111 so we're multiplying the number by one. 00:03:49.111 --> 00:03:52.780 That won't change the number, but at least it gets rid of the irrational number in the denominator. 00:03:52.780 --> 00:03:54.127 So the numerator becomes 00:03:54.127 --> 00:03:57.800 four times the square root of sixty-five, 00:03:57.800 --> 00:04:03.461 and the denominator, square root of 65 times square root of 65, is just going to be 65. 00:04:03.461 --> 00:04:07.130 We didn't get rid of the irrational number, it's still there, but it's now in the numerator. 00:04:07.130 --> 00:04:09.777 Now let's do the other trig functions 00:04:09.777 --> 00:04:12.401 or at least the other core trig functions. 00:04:12.401 --> 00:04:14.399 We'll learn in the future that there's actually a ton of them 00:04:14.399 --> 00:04:15.443 but they're all derived from these. 00:04:15.443 --> 00:04:19.733 so let's think about what the sign of theta is. Once again go to "soh cah toa". 00:04:19.733 --> 00:04:25.474 The "soh" tells what to do with sine. Sine is opposite over hypotenuse. 00:04:25.474 --> 00:04:29.200 Sine is equal to opposite over hypotenuse. 00:04:29.200 --> 00:04:31.372 Sine is opposite over hypotenuse. 00:04:31.372 --> 00:04:34.390 So for this angle what side is opposite? 00:04:34.390 --> 00:04:38.430 We just go opposite it, what it opens into, it's opposite the seven 00:04:38.430 --> 00:04:41.200 so the opposite side is the seven. 00:04:41.200 --> 00:04:44.468 This is, right here - that is the opposite side 00:04:44.468 --> 00:04:47.800 and then the hypotenuse, it's opposite over hypotenuse. 00:04:47.800 --> 00:04:51.109 The hypotenuse is the square root of sixty-five. 00:04:51.109 --> 00:04:52.966 Square root of sixty-five. 00:04:52.966 --> 00:04:55.133 and once again if we wanted to rationalize this, 00:04:55.133 --> 00:04:59.933 we could multiply times the square root of 65 over the square root of 65 00:04:59.933 --> 00:05:04.298 and the the numerator, we will get seven square root of 65 00:05:04.298 --> 00:05:07.966 and in the denominator we will get just sixty-five again. 00:05:07.966 --> 00:05:10.474 Now let's do tangent! 00:05:10.474 --> 00:05:12.796 Let us do tangent. 00:05:12.796 --> 00:05:14.793 So if i ask you the tangent 00:05:14.793 --> 00:05:17.394 of - the tangent of theta 00:05:17.394 --> 00:05:20.784 once again go back to "soh cah toa". 00:05:20.784 --> 00:05:23.106 The toa part tells us what to do with tangent 00:05:23.106 --> 00:05:24.800 it tells us... 00:05:24.800 --> 00:05:27.053 it tells us that tangent 00:05:27.053 --> 00:05:29.867 is equal to opposite over adjacent 00:05:29.867 --> 00:05:33.137 is equal to opposite over 00:05:33.137 --> 00:05:35.867 opposite over adjacent 00:05:35.867 --> 00:05:38.709 So for this angle, what is opposite? We've already figured it out. 00:05:38.709 --> 00:05:41.124 it's seven. It opens into the seven. 00:05:41.124 --> 00:05:42.533 It is opposite the seven. 00:05:42.533 --> 00:05:46.372 So it's seven over what side is adjacent. 00:05:46.372 --> 00:05:48.200 well this four is adjacent. 00:05:48.200 --> 00:05:51.295 This four is adjacent. So the adjacent side is four. 00:05:51.295 --> 00:05:54.330 so it's seven over four, 00:05:54.330 --> 00:05:56.133 and we're done. 00:05:56.133 --> 00:05:59.375 We figured out all of the trig ratios for theta. let's do another one. 00:05:59.375 --> 00:06:00.416 Let's do another one. 00:06:00.416 --> 00:06:02.719 i'll make it a little bit concrete 'cause right now we've been saying, 00:06:02.719 --> 00:06:06.434 "oh, what's tangent of x, tangent of theta." let's make it a little bit more concrete. 00:06:06.434 --> 00:06:08.431 Let's say... 00:06:08.431 --> 00:06:10.799 let's say, let me draw another right triangle, 00:06:10.799 --> 00:06:13.772 that's another right triangle here. 00:06:13.772 --> 00:06:17.533 Everything we're dealing with, these are going to be right triangles. 00:06:17.533 --> 00:06:21.109 let's say the hypotenuse has length four, 00:06:21.109 --> 00:06:26.357 let's say that this side over here has length two, 00:06:26.357 --> 00:06:31.790 and let's say that this length over here is going to be two times the square root of three. 00:06:31.790 --> 00:06:33.462 We can verify that this works. 00:06:33.462 --> 00:06:36.467 If you have this side squared, so you have - let me write it down - 00:06:36.467 --> 00:06:38.803 two times the square root of three squared 00:06:38.803 --> 00:06:42.471 plus two squared, is equal to what? 00:06:42.471 --> 00:06:46.467 this is two. There's going to be four times three. 00:06:46.467 --> 00:06:49.763 four times three plus four, 00:06:49.763 --> 00:06:53.478 and this is going to be equal to twelve plus four is equal to sixteen 00:06:53.478 --> 00:06:57.800 and sixteen is indeed four squared. So this does equal four squared, 00:06:57.800 --> 00:07:01.790 it does equal four squared. It satisfies the pythagorean theorem 00:07:01.790 --> 00:07:06.133 and if you remember some of your work from 30 60 90 triangles 00:07:06.133 --> 00:07:07.781 that you might have learned in geometry, 00:07:07.781 --> 00:07:11.450 you might recognize that this is a 30 60 90 triangle. 00:07:11.450 --> 00:07:13.133 This right here is our right angle, 00:07:13.133 --> 00:07:15.867 - i should have drawn it from the get go to show that this is a right triangle - 00:07:15.867 --> 00:07:20.366 this angle right over here is our thirty degree angle 00:07:20.366 --> 00:07:23.385 and then this angle up here, this angle up here is 00:07:23.385 --> 00:07:26.125 a sixty degree angle, 00:07:26.125 --> 00:07:27.797 and it's a thirty sixteen ninety because 00:07:27.797 --> 00:07:31.791 the side opposite the thirty degrees is half the hypotenuse 00:07:31.791 --> 00:07:36.800 and then the side opposite the 60 degrees is a squared of 3 times the other side 00:07:36.800 --> 00:07:38.432 that's not the hypotenuse. 00:07:38.432 --> 00:07:40.159 So that said, we're not gonna ... 00:07:40.159 --> 00:07:43.415 this isn't supposed to be a review of 30 60 90 triangles although i just did it. 00:07:43.415 --> 00:07:46.933 Let's actually find the trig ratios for the different angles. 00:07:46.933 --> 00:07:51.295 So if i were to ask you or if anyone were to ask you, what is... 00:07:51.295 --> 00:07:54.639 what is the sine of thirty degrees? 00:07:54.639 --> 00:07:58.447 and remember 30 degrees is one of the angles in this triangle but it would apply 00:07:58.447 --> 00:08:01.698 whenever you have a 30 degree angle and you're dealing with the right triangle. 00:08:01.698 --> 00:08:05.135 We'll have broader definitions in the future but if you say sine of thirty degrees, 00:08:05.135 --> 00:08:09.035 hey, this angle right over here is thirty degrees so i can use this right triangle, 00:08:09.035 --> 00:08:12.133 and we just have to remember "soh cah toa" 00:08:12.133 --> 00:08:17.116 We rewrite it. soh, cah, toa. 00:08:17.116 --> 00:08:22.782 "sine tells us" (correction). soh tells us what to do with sine. sine is opposite over hypotenuse. 00:08:22.782 --> 00:08:26.358 sine of thirty degrees is the opposite side, 00:08:26.358 --> 00:08:30.723 that is the opposite side which is two over the hypotenuse. 00:08:30.723 --> 00:08:32.395 The hypotenuse here is four. 00:08:32.395 --> 00:08:35.646 it is two fourths which is the same thing as one-half. 00:08:35.646 --> 00:08:40.800 sine of thirty degrees you'll see is always going to be equal to one-half. 00:08:40.800 --> 00:08:44.144 now what is the cosine? 00:08:44.144 --> 00:08:46.867 What is the cosine of thirty degrees? 00:08:46.867 --> 00:08:50.135 Once again go back to "soh cah toa". 00:08:50.135 --> 00:08:52.643 The cah tells us what to do with cosine. 00:08:52.643 --> 00:08:56.033 Cosine is adjacent over hypotenuse. 00:08:56.033 --> 00:08:59.051 So for looking at the thirty degree angle it's the adjacent. 00:08:59.051 --> 00:09:01.791 This, right over here is adjacent. it's right next to it. 00:09:01.791 --> 00:09:05.467 it's not the hypotenuse. it's the adjacent over the hypotenuse. 00:09:05.467 --> 00:09:09.129 so it's two square roots of three 00:09:09.129 --> 00:09:13.633 adjacent over...over the hypotenuse, over four. 00:09:13.633 --> 00:09:16.977 or if we simplify that, we divide the numerator and the denominator by two 00:09:16.977 --> 00:09:20.646 it's the square root of three over two. 00:09:20.646 --> 00:09:22.782 Finally, let's do the tangent. 00:09:22.782 --> 00:09:27.800 The tangent of thirty degrees, 00:09:27.800 --> 00:09:30.305 we go back to "soh cah toa". 00:09:30.305 --> 00:09:31.699 soh cah toa 00:09:31.699 --> 00:09:34.800 toa tells us what to do with tangent. It's opposite over adjacent 00:09:34.800 --> 00:09:38.804 you go to the 30 degree angle because that's what we care about, tangent of 30. 00:09:38.804 --> 00:09:42.101 tangent of thirty. Opposite is two, 00:09:42.101 --> 00:09:46.200 opposite is two and the adjacent is two square roots of three. 00:09:46.200 --> 00:09:48.045 It's right next to it. It's adjacent to it. 00:09:48.045 --> 00:09:49.439 adjacent means next to. 00:09:49.439 --> 00:09:52.039 so two square roots of three 00:09:52.039 --> 00:09:54.454 so this is equal to... the twos cancel out 00:09:54.454 --> 00:09:56.776 one over the square root of three 00:09:56.776 --> 00:10:00.723 or we could multiply the numerator and the denominator by the square root of three. 00:10:00.723 --> 00:10:05.367 So we have square root of three over square root of three 00:10:05.367 --> 00:10:08.804 and so this is going to be equal to the numerator square root of three and then 00:10:08.804 --> 00:10:12.473 the denominator right over here is just going to be three. 00:10:12.473 --> 00:10:15.800 So that we've rationalized a square root of three over three. 00:10:15.800 --> 00:10:17.442 Fair enough. 00:10:17.442 --> 00:10:20.693 Now lets use the same triangle to figure out the trig ratios for the sixty degrees, 00:10:20.693 --> 00:10:22.457 since we've already drawn it. 00:10:22.457 --> 00:10:28.328 so what is... what is the sine of the sixty degrees? 00:10:28.328 --> 00:10:30.166 and i think you're hopefully getting the hang of it now. 00:10:30.166 --> 00:10:34.253 Sine is opposite over adjacent. soh from the "soh cah toa". 00:10:34.253 --> 00:10:36.668 for the sixty degree angle what side is opposite? 00:10:36.668 --> 00:10:39.315 what opens out into the two square roots of three, 00:10:39.315 --> 00:10:42.566 so the opposite side is two square roots of three, 00:10:42.566 --> 00:10:45.306 and from the sixty degree angle the adj-oh sorry 00:10:45.306 --> 00:10:47.999 its the opposite over hypotenuse, don't want to confuse you. 00:10:47.999 --> 00:10:50.507 so it is opposite over hypotenuse 00:10:50.507 --> 00:10:54.315 so it's two square roots of three over four. four is the hypotenuse. 00:10:54.315 --> 00:10:59.981 so it is equal to, this simplifies to square root of three over two. 00:10:59.981 --> 00:11:05.507 What is the cosine of sixty degrees? cosine of sixty degrees. 00:11:05.507 --> 00:11:10.244 so remember "soh cah toa". cosine is adjacent over hypotenuse. 00:11:10.244 --> 00:11:13.667 adjacent is the two sides, right next to the sixty degree angle. 00:11:13.667 --> 00:11:17.907 So it's two over the hypotenuse which is four. 00:11:17.907 --> 00:11:20.972 So this is equal to one-half 00:11:20.972 --> 00:11:24.176 and then finally, what is the tangent? 00:11:24.176 --> 00:11:27.984 what is the tangent of sixty degrees? 00:11:27.984 --> 00:11:32.349 Well tangent, "soh cah toa". Tangent is opposite over adjacent 00:11:32.349 --> 00:11:34.671 opposite the sixty degrees 00:11:34.671 --> 00:11:36.400 is two square roots of three 00:11:36.400 --> 00:11:38.000 two square roots of three 00:11:38.000 --> 00:11:39.919 and adjacent to that 00:11:39.919 --> 00:11:42.733 adjacent to that is two. 00:11:42.733 --> 00:11:44.800 Adjacent to sixty degrees is two. 00:11:44.800 --> 00:11:48.650 So its opposite over adjacent, two square roots of three over two 00:11:48.650 --> 00:11:52.644 which is just equal to the square root of three. 00:11:52.644 --> 00:11:54.641 And I just wanted to -look how these are related- 00:11:54.641 --> 00:11:57.984 the sine of thirty degrees is the same as the cosine of sixty degrees. 00:11:57.984 --> 00:12:01.333 The cosine of 30 degrees is the same thing as the sine of 60 degrees 00:12:01.333 --> 00:12:03.966 and then these guys are the inverse of each other 00:12:03.966 --> 00:12:05.635 and i think if you think a little bit about this triangle 00:12:05.635 --> 00:12:07.105 it will start to make sense why. 00:12:07.105 --> 00:12:08.461 we'll keep extending this and 00:12:08.461 --> 99:59:59.999 give you a lot more practice in the next few videos.