1 00:00:00,800 --> 00:00:03,971 Let's just do a ton of more examples, just so we make sure that we're getting 2 00:00:03,971 --> 00:00:07,036 this trig function thing down well. 3 00:00:07,036 --> 00:00:11,447 So let's construct ourselves some right triangles. 4 00:00:11,447 --> 00:00:14,791 Let's construct ourselves some right triangles, and I want to be very clear. 5 00:00:14,791 --> 00:00:18,042 The way I've defined it so far, this will only work in right triangles. 6 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, 7 00:00:23,475 --> 00:00:25,704 we're going to see that we're going to have to construct right triangles, 8 00:00:25,704 --> 00:00:27,867 but let's just focus on the right triangles for now. 9 00:00:27,867 --> 00:00:33,467 So let's say that I have a triangle, where let's say this length down here is seven, 10 00:00:33,467 --> 00:00:39,219 and let's say the length of this side up here, let's say that that is four. 11 00:00:39,219 --> 00:00:42,516 Let's figure out what the hypotenuse over here is going to be. 12 00:00:42,516 --> 00:00:45,720 So we know -let's call the hypotenuse, "h"- 13 00:00:45,720 --> 00:00:52,200 we know that h squared is going to be equal to seven squared plus four squared, 14 00:00:52,200 --> 00:00:55,194 we know that from the Pythagorean theorem, 15 00:00:55,194 --> 00:00:57,469 that the hypotenuse squared is equal to 16 00:00:57,469 --> 00:01:01,974 the square of each of the sum of the squares of the other two sides. 17 00:01:01,974 --> 00:01:04,533 h squared is equal to seven squared plus four squared. 18 00:01:04,533 --> 00:01:09,776 So this is equal to forty-nine plus sixteen, 19 00:01:09,776 --> 00:01:11,800 forty-nine plus sixteen, 20 00:01:11,800 --> 00:01:18,553 forty nine plus ten is fifty-nine, plus six is sixty-five. 21 00:01:18,553 --> 00:01:21,107 It is sixty five. So this h squared, 22 00:01:21,107 --> 00:01:25,705 let me write: h squared -that's different shade of yellow- 23 00:01:25,705 --> 00:01:28,818 so we have h squared is equal to sixty-five. 24 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, 25 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, 26 00:01:37,600 --> 00:01:39,200 square root 27 00:01:39,200 --> 00:01:42,933 square root of sixty five. And we really can't simplify this at all. 28 00:01:42,933 --> 00:01:44,699 This is thirteen. 29 00:01:44,699 --> 00:01:48,971 This is the same thing as thirteen times five, both of those are not perfect squares and 30 00:01:48,971 --> 00:01:51,804 they're both prime so you can't simplify this any more. 31 00:01:51,804 --> 00:01:55,467 So this is equal to the square root of sixty five. 32 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. 33 00:02:02,114 --> 00:02:05,457 Let's call that angle up there theta. 34 00:02:05,457 --> 00:02:06,533 So whenever you do it 35 00:02:06,533 --> 00:02:09,467 you always want to write down - at least for me it works out to write down - 36 00:02:09,467 --> 00:02:11,714 "soh cah toa". 37 00:02:11,714 --> 00:02:13,120 soh... 38 00:02:13,120 --> 00:02:16,464 ...soh cah toa. I have these vague memories 39 00:02:16,464 --> 00:02:18,786 of my trigonometry teacher. 40 00:02:18,786 --> 00:02:21,293 Maybe I've read it in some book. I don't know - you know, some... about 41 00:02:21,293 --> 00:02:23,867 some type of indian princess named "soh cah toa" or whatever, 42 00:02:23,867 --> 00:02:26,123 but it's a very useful mnemonic, 43 00:02:26,123 --> 00:02:27,564 so we can apply "soh cah toa". 44 00:02:27,564 --> 00:02:31,046 Let's find, let's say we want to find the cosine. 45 00:02:31,046 --> 00:02:34,436 We want to find the cosine of our angle. 46 00:02:34,436 --> 00:02:37,965 We wanna find the cosine of our angle, you say: "soh cah toa!" 47 00:02:37,965 --> 00:02:40,800 So the "cah". "Cah" tells us what to do with cosine, 48 00:02:40,800 --> 00:02:43,027 the "cah" part tells us 49 00:02:43,027 --> 00:02:46,371 that cosine is adjacent over hypotenuse. 50 00:02:46,371 --> 00:02:51,433 Cosine is equal to adjacent over hypotenuse. 51 00:02:51,433 --> 00:02:55,798 So let's look over here to theta; what side is adjacent? 52 00:02:55,798 --> 00:02:57,702 Well we know that the hypotenuse, 53 00:02:57,702 --> 00:03:00,767 we know that that hypotenuse is this side over here. 54 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 55 00:03:04,761 --> 00:03:07,133 isn't the hypotenuse, is this four. 56 00:03:07,133 --> 00:03:10,473 So the adjacent side over here, that side is, 57 00:03:10,473 --> 00:03:14,374 it's literally right next to the angle, it's one of the sides that kind of forms the angle 58 00:03:14,374 --> 00:03:17,133 it's four over the hypotenuse. 59 00:03:17,133 --> 00:03:21,108 The hypotenuse we already know is square root of sixty-five. 60 00:03:21,108 --> 00:03:25,380 so it's four over the square root of sixty-five. 61 00:03:25,380 --> 00:03:29,142 And sometimes people will want you to rationalize the denominator which means 62 00:03:29,142 --> 00:03:34,111 they don't like to have an irrational number in the denominator, like the square root of sixty five, 63 00:03:34,111 --> 00:03:39,359 and if they - if you wanna rewrite this without a irrational number in the denominator, 64 00:03:39,359 --> 00:03:41,634 you can multiply the numerator and the denominator 65 00:03:41,634 --> 00:03:43,306 by the square root of sixty-five. 66 00:03:43,306 --> 00:03:47,133 This clearly will not change the number, because we're multiplying it by something over itself, 67 00:03:47,133 --> 00:03:49,111 so we're multiplying the number by one. 68 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. 69 00:03:52,780 --> 00:03:54,127 So the numerator becomes 70 00:03:54,127 --> 00:03:57,800 four times the square root of sixty-five, 71 00:03:57,800 --> 00:04:03,461 and the denominator, square root of sixty five times square root of sixty-five, is just going to be sixty-five. 72 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. 73 00:04:07,130 --> 00:04:09,777 Now let's do the other trig functions 74 00:04:09,777 --> 00:04:13,724 or at least the other core trig functions. We'll learn in the future that there's actually a ton of them 75 00:04:13,724 --> 00:04:15,443 but they're all derived from these. 76 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". 77 00:04:19,733 --> 00:04:25,474 The "soh" tells what to do with sine. Sine is opposite over hypotenuse. 78 00:04:25,474 --> 00:04:29,200 Sine is equal to opposite over hypotenuse. 79 00:04:29,200 --> 00:04:31,372 Sine is opposite over hypotenuse. 80 00:04:31,372 --> 00:04:34,390 So for this angle what side is opposite? 81 00:04:34,390 --> 00:04:38,430 We just go opposite it, what it opens into, it's opposite the seven 82 00:04:38,430 --> 00:04:41,200 so the opposite side is the seven. 83 00:04:41,200 --> 00:04:44,468 This is, right here - that is the opposite side 84 00:04:44,468 --> 00:04:47,800 and then the hypotenuse, it's opposite over hypotenuse. 85 00:04:47,800 --> 00:04:51,109 The hypotenuse is the square root of sixty-five. 86 00:04:51,109 --> 00:04:52,966 Square root of sixty-five. 87 00:04:52,966 --> 00:04:55,133 and once again if we wanted to rationalize this, 88 00:04:55,133 --> 00:04:59,933 we could multiply times the square root of sixty-five over the square root of sixty-five 89 00:04:59,933 --> 00:05:04,298 and the the numerator, we will get seven square root of sixty-five 90 00:05:04,298 --> 00:05:07,966 and in the denominator we will get just sixty-five again. 91 00:05:07,966 --> 00:05:10,474 Now let's do tangent! 92 00:05:10,474 --> 00:05:12,796 Let us do tangent. 93 00:05:12,796 --> 00:05:14,793 So if i ask you the tangent 94 00:05:14,793 --> 00:05:17,394 of - the tangent of theta 95 00:05:17,394 --> 00:05:20,784 once again go back to "soh cah toa". 96 00:05:20,784 --> 00:05:23,106 The toa part tells us what to do with tangent 97 00:05:23,106 --> 00:05:24,800 it tells us... 98 00:05:24,800 --> 00:05:27,053 it tells us that tangent 99 00:05:27,053 --> 00:05:29,867 is equal to opposite over adjacent 100 00:05:29,867 --> 00:05:33,137 is equal to opposite over 101 00:05:33,137 --> 00:05:35,867 opposite over adjacent 102 00:05:35,867 --> 00:05:38,709 So for this angle, what is opposite? We've already figured it out. 103 00:05:38,709 --> 00:05:41,124 it's seven. It opens into the seven. 104 00:05:41,124 --> 00:05:42,533 It is opposite the seven. 105 00:05:42,533 --> 00:05:46,372 So it's seven over what side is adjacent. 106 00:05:46,372 --> 00:05:48,200 well this four is adjacent. 107 00:05:48,200 --> 00:05:51,295 This four is adjacent. So the adjacent side is four. 108 00:05:51,295 --> 00:05:54,638 so it's seven over four, 109 00:05:54,638 --> 00:05:56,133 and we're done. 110 00:05:56,133 --> 00:05:59,375 We figured out all of the trig ratios for theta. let's do another one. 111 00:05:59,375 --> 00:06:02,719 Let's do another one. i'll make it a little bit concrete 'cause right now we've been saying, 112 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. 113 00:06:06,434 --> 00:06:08,431 Let's say... 114 00:06:08,431 --> 00:06:10,799 let's say, let me draw another right triangle, 115 00:06:10,799 --> 00:06:13,772 that's another right triangle here. 116 00:06:13,772 --> 00:06:17,533 Everything we're dealing with, these are going to be right triangles. 117 00:06:17,533 --> 00:06:21,109 let's say the hypotenuse has length four, 118 00:06:21,109 --> 00:06:26,357 let's say that this side over here has length two, 119 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. 120 00:06:31,790 --> 00:06:33,462 We can verify that this works. 121 00:06:33,462 --> 00:06:36,467 If you have this side squared, so you have - let me write it down - 122 00:06:36,467 --> 00:06:38,803 two times the square root of three squared 123 00:06:38,803 --> 00:06:42,471 plus two squared, is equal to what? 124 00:06:42,471 --> 00:06:46,467 this is two. There's going to be four times three. 125 00:06:46,467 --> 00:06:49,763 four times three plus four, 126 00:06:49,763 --> 00:06:53,478 and this is going to be equal to twelve plus four is equal to sixteen 127 00:06:53,478 --> 00:06:57,800 and sixteen is indeed four squared. So this does equal four squared, 128 00:06:57,800 --> 00:07:01,790 it does equal four squared. It satisfies the pythagorean theorem 129 00:07:01,790 --> 00:07:06,133 and if you remember some of your work from thirty sixty ninety triangles 130 00:07:06,133 --> 00:07:07,781 that you might have learned in geometry, 131 00:07:07,781 --> 00:07:11,450 you might recognize that this is a thirty sixty ninety triangle. 132 00:07:11,450 --> 00:07:13,133 This right here is our right angle, 133 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 - 134 00:07:15,867 --> 00:07:20,366 this angle right over here is our thirty degree angle 135 00:07:20,366 --> 00:07:23,385 and then this angle up here, this angle up here is 136 00:07:23,385 --> 00:07:26,125 a sixty degree angle, 137 00:07:26,125 --> 00:07:27,797 and it's a thirty sixteen ninety because 138 00:07:27,797 --> 00:07:31,791 the side opposite the thirty degrees is half the hypotenuse 139 00:07:31,791 --> 00:07:36,800 and then the side opposite the sixty degrees is a squared of three times the other side 140 00:07:36,800 --> 00:07:38,432 that's not the hypotenuse. 141 00:07:38,432 --> 00:07:42,852 So that said, we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles although i just did it. 142 00:07:42,852 --> 00:07:46,933 Let's actually find the trig ratios for the different angles. 143 00:07:46,933 --> 00:07:51,295 So if i were to ask you or if anyone were to ask you, what is... 144 00:07:51,295 --> 00:07:54,639 what is the sine of thirty degrees? 145 00:07:54,639 --> 00:07:58,447 and remember thirty degrees is one of the angles in this triangle but it would apply 146 00:07:58,447 --> 00:08:01,698 whenever you have a thirty degree angle and you're dealing with the right triangle. 147 00:08:01,698 --> 00:08:05,135 We'll have broader definitions in the future but if you say sine of thirty degrees, 148 00:08:05,135 --> 00:08:09,035 hey, this angle right over here is thirty degrees so i can use this right triangle, 149 00:08:09,035 --> 00:08:12,133 and we just have to remember "soh cah toa" 150 00:08:12,133 --> 00:08:17,116 We rewrite it. soh, cah, toa. 151 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. 152 00:08:22,782 --> 00:08:26,358 sine of thirty degrees is the opposite side, 153 00:08:26,358 --> 00:08:30,723 that is the opposite side which is two over the hypotenuse. 154 00:08:30,723 --> 00:08:32,395 The hypotenuse here is four. 155 00:08:32,395 --> 00:08:35,646 it is two fourths which is the same thing as one-half. 156 00:08:35,646 --> 00:08:40,800 sine of thirty degrees you'll see is always going to be equal to one-half. 157 00:08:40,800 --> 00:08:44,144 now what is the cosine? 158 00:08:44,144 --> 00:08:46,867 What is the cosine of thirty degrees? 159 00:08:46,867 --> 00:08:50,135 Once again go back to "soh cah toa". 160 00:08:50,135 --> 00:08:52,643 The cah tells us what to do with cosine. 161 00:08:52,643 --> 00:08:56,033 Cosine is adjacent over hypotenuse. 162 00:08:56,033 --> 00:08:59,051 So for looking at the thirty degree angle it's the adjacent. 163 00:08:59,051 --> 00:09:01,791 This, right over here is adjacent. it's right next to it. 164 00:09:01,791 --> 00:09:05,467 it's not the hypotenuse. it's the adjacent over the hypotenuse. 165 00:09:05,467 --> 00:09:09,129 so it's two square roots of three 166 00:09:09,129 --> 00:09:13,633 adjacent over...over the hypotenuse, over four. 167 00:09:13,633 --> 00:09:16,977 or if we simplify that, we divide the numerator and the denominator by two 168 00:09:16,977 --> 00:09:20,646 it's the square root of three over two. 169 00:09:20,646 --> 00:09:22,782 Finally, let's do the tangent. 170 00:09:22,782 --> 00:09:27,800 The tangent of thirty degrees, 171 00:09:27,800 --> 00:09:30,305 we go back to "soh cah toa". 172 00:09:30,305 --> 00:09:31,699 soh cah toa 173 00:09:31,699 --> 00:09:34,800 toa tells us what to do with tangent. It's opposite over adjacent 174 00:09:34,800 --> 00:09:38,804 you go to the thirty degree angle because that's what we care about, tangent of thirty. 175 00:09:38,804 --> 00:09:42,101 tangent of thirty. Opposite is two, 176 00:09:42,101 --> 00:09:46,200 opposite is two and the adjacent is two square roots of three. 177 00:09:46,200 --> 00:09:48,045 It's right next to it it's adjacent to it. 178 00:09:48,045 --> 00:09:49,439 adjacent means next to. 179 00:09:49,439 --> 00:09:52,039 so two square roots of three 180 00:09:52,039 --> 00:09:54,454 so this is equal to... the twos cancel out 181 00:09:54,454 --> 00:09:56,776 one over the square root of three 182 00:09:56,776 --> 00:10:00,723 or we could multiply the numerator and the denominator by the square root of three. 183 00:10:00,723 --> 00:10:05,367 So we have square root of three over square root of three 184 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 185 00:10:08,804 --> 00:10:12,473 the denominator right over here is just going to be three. 186 00:10:12,473 --> 00:10:15,800 So that we've rationalized a square root of three over three. 187 00:10:15,800 --> 00:10:17,442 Fair enough. 188 00:10:17,442 --> 00:10:20,693 Now lets use the same triangle to figure out the trig ratios for the sixty degrees, 189 00:10:20,693 --> 00:10:22,457 since we've already drawn it. 190 00:10:22,457 --> 00:10:28,328 so what is... what is the sine of the sixty degrees? 191 00:10:28,328 --> 00:10:30,166 and i think you're hopefully getting the hang of it now. 192 00:10:30,166 --> 00:10:34,253 Sine is opposite over adjacent. soh from the "soh cah toa". 193 00:10:34,253 --> 00:10:36,668 for the sixty degree angle what side is opposite? 194 00:10:36,668 --> 00:10:39,315 what opens out into the two square roots of three, 195 00:10:39,315 --> 00:10:42,566 so the opposite side is two square roots of three, 196 00:10:42,566 --> 00:10:45,306 and from the sixty degree angle the adj-oh sorry 197 00:10:45,306 --> 00:10:47,999 its the opposite over hypotenuse, don't want to confuse you. 198 00:10:47,999 --> 00:10:50,507 so it is opposite over hypotenuse 199 00:10:50,507 --> 00:10:54,315 so it's two square roots of three over four. four is the hypotenuse. 200 00:10:54,315 --> 00:10:59,981 so it is equal to, this simplifies to square root of three over two. 201 00:10:59,981 --> 00:11:05,507 What is the cosine of sixty degrees? cosine of sixty degrees. 202 00:11:05,507 --> 00:11:10,244 so remember "soh cah toa". cosine is adjacent over hypotenuse. 203 00:11:10,244 --> 00:11:13,667 adjacent is the two sides, right next to the sixty degree angle. 204 00:11:13,667 --> 00:11:17,907 So it's two over the hypotenuse which is four. 205 00:11:17,907 --> 00:11:20,972 So this is equal to one-half 206 00:11:20,972 --> 00:11:24,176 and then finally, what is the tangent? 207 00:11:24,176 --> 00:11:27,984 what is the tangent of sixty degrees? 208 00:11:27,984 --> 00:11:32,349 Well tangent, "soh cah toa". Tangent is opposite over adjacent 209 00:11:32,349 --> 00:11:34,671 opposite the sixty degrees 210 00:11:34,671 --> 00:11:36,400 is two square roots of three 211 00:11:36,400 --> 00:11:38,000 two square roots of three 212 00:11:38,000 --> 00:11:39,919 and adjacent to that 213 00:11:39,919 --> 00:11:42,733 adjacent to that is two. 214 00:11:42,733 --> 00:11:44,800 Adjacent to sixty degrees is two. 215 00:11:44,800 --> 00:11:48,650 So its opposite over adjacent, two square roots of three over two 216 00:11:48,650 --> 00:11:52,644 which is just equal to the square root of three. 217 00:11:52,644 --> 00:11:54,641 And I just wanted to -look how these are related- 218 00:11:54,641 --> 00:11:57,984 the sine of thirty degrees is the same as the cosine of sixty degrees. 219 00:11:57,984 --> 00:12:01,333 The cosine of thirty degrees is the same thing as the sine of sixty degrees 220 00:12:01,333 --> 00:12:04,997 and then these guys are the inverse of each other and i think if you think a little bit about this triangle 221 00:12:04,997 --> 00:12:07,597 it will start to make sense why. 222 00:12:07,597 --> 99:59:59,999 we'll keep extending this and give you a lot more practice in the next few videos.