1 00:00:00,040 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,690 Commons License. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,990 offer high quality educational resources for free. 5 00:00:09,990 --> 00:00:12,830 To make a donation, or to view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:28,873 --> 00:00:30,870 PROFESSOR: Hi. 9 00:00:30,870 --> 00:00:33,840 Today's lesson, well, I settled for the title, 10 00:00:33,840 --> 00:00:36,440 "Circular Functions." But I guess it could have been 11 00:00:36,440 --> 00:00:38,280 called a lot of different things. 12 00:00:38,280 --> 00:00:40,530 It could've been called 'Trigonometry without 13 00:00:40,530 --> 00:00:41,490 Triangles'. 14 00:00:41,490 --> 00:00:44,740 It could have been called 'Trigonometry Revisited'. 15 00:00:44,740 --> 00:00:48,450 And the whole point is that much of what today's lecture 16 00:00:48,450 --> 00:00:52,280 hinges on is a hang-up that bothered me, and which I think 17 00:00:52,280 --> 00:00:55,220 may bother you and is worthwhile discussing. 18 00:00:55,220 --> 00:00:57,570 I remember, when I was in high school, I asked my 19 00:00:57,570 --> 00:00:59,860 trigonometry teacher, why would I have to know 20 00:00:59,860 --> 00:01:00,720 trigonometry? 21 00:01:00,720 --> 00:01:03,280 And his answer was, surveyors use it. 22 00:01:03,280 --> 00:01:06,030 And at that particular time, I didn't know what I was going 23 00:01:06,030 --> 00:01:08,940 to be, but I knew what I wasn't going to be. 24 00:01:08,940 --> 00:01:10,440 I wasn't going to be a surveyor. 25 00:01:10,440 --> 00:01:13,690 And I kind of took the course kind of lightly, and really 26 00:01:13,690 --> 00:01:17,300 got clobbered a year or two later when I got into calculus 27 00:01:17,300 --> 00:01:18,680 and physics courses. 28 00:01:18,680 --> 00:01:22,230 So what I would like to do today is to introduce the 29 00:01:22,230 --> 00:01:26,450 notion of what we call circular functions, and point 30 00:01:26,450 --> 00:01:30,630 out what the connection is between these and the 31 00:01:30,630 --> 00:01:33,680 trigonometric functions that we learned when we studied the 32 00:01:33,680 --> 00:01:36,890 subject that we call trigonometry, and which might 33 00:01:36,890 --> 00:01:39,890 better have been called numerical geometry. 34 00:01:39,890 --> 00:01:41,890 Let me get to the point right away. 35 00:01:41,890 --> 00:01:44,970 Let's imagine that I say circular functions to you. 36 00:01:44,970 --> 00:01:47,280 I think it's rather natural that, as soon as I say that, 37 00:01:47,280 --> 00:01:48,270 you think of a circle. 38 00:01:48,270 --> 00:01:51,480 And because you think of a circle, let me draw a circle 39 00:01:51,480 --> 00:01:55,660 here, and let me assume that the radius of the circle is 1. 40 00:01:55,660 --> 00:01:59,660 In other words, I have the circle here, 'x squared' plus 41 00:01:59,660 --> 00:02:02,750 'y squared' equals 1. 42 00:02:02,750 --> 00:02:04,250 Now, the thing is this. 43 00:02:04,250 --> 00:02:05,620 When I talk about-- 44 00:02:05,620 --> 00:02:08,430 And I'm assuming now that you are familiar with the 45 00:02:08,430 --> 00:02:11,550 trigonometric functions in the traditional sense. 46 00:02:11,550 --> 00:02:15,030 And in fact, the first section of our supplementary notes in 47 00:02:15,030 --> 00:02:19,190 the reading material that goes with the present lecture takes 48 00:02:19,190 --> 00:02:21,050 care of the fact that, if you don't recall some of these 49 00:02:21,050 --> 00:02:24,210 things too well, there's ample opportunity for refreshing 50 00:02:24,210 --> 00:02:26,580 your minds and getting some review in here. 51 00:02:26,580 --> 00:02:28,280 But the idea is something like this. 52 00:02:28,280 --> 00:02:31,610 When we're talking about calculus, we talk about 53 00:02:31,610 --> 00:02:33,710 functions of a real variable. 54 00:02:33,710 --> 00:02:37,240 We are assuming that our functions have the property 55 00:02:37,240 --> 00:02:41,890 that the domain is a set of suitably chosen real numbers, 56 00:02:41,890 --> 00:02:45,120 and the image is a suitably chosen set of real numbers. 57 00:02:45,120 --> 00:02:48,290 We do not think of inputs as being angles and 58 00:02:48,290 --> 00:02:49,530 things of this type. 59 00:02:49,530 --> 00:02:52,730 And so the question is, how can we define, for example-- 60 00:02:52,730 --> 00:02:54,100 let's call it the 'sine machine'. 61 00:02:54,100 --> 00:02:55,510 Let me come down here. 62 00:02:55,510 --> 00:02:56,700 I'll call it the 'sine machine'. 63 00:02:56,700 --> 00:03:00,590 If the input is the number 't', I want the output, say, 64 00:03:00,590 --> 00:03:01,810 to be 'sine t'. 65 00:03:01,810 --> 00:03:07,300 But you see, now I'm talking about a number, not an angle. 66 00:03:07,300 --> 00:03:12,020 Well, one way of doing this thing visually is the old idea 67 00:03:12,020 --> 00:03:13,250 of the number line. 68 00:03:13,250 --> 00:03:17,500 Let us think of a number as being a length, the same as we 69 00:03:17,500 --> 00:03:19,700 do in coordinate geometry. 70 00:03:19,700 --> 00:03:22,680 We knock off lengths along the x-axis and the y-axis. 71 00:03:22,680 --> 00:03:25,850 Let me think of 't' as being a length. 72 00:03:25,850 --> 00:03:32,540 As such, I can take 't' and lay it off along my circle in 73 00:03:32,540 --> 00:03:36,250 such a way that the length originates at 'S' and 74 00:03:36,250 --> 00:03:39,430 terminates, shall we say, at some point 'P' whose 75 00:03:39,430 --> 00:03:41,710 coordinates are 'x' and 'y'. 76 00:03:41,710 --> 00:03:43,050 Now, notice what I'm saying here. 77 00:03:43,050 --> 00:03:47,190 I lay the length off along the circumference. 78 00:03:47,190 --> 00:03:50,360 I'll talk more about that a little bit later. 79 00:03:50,360 --> 00:03:52,090 Now, so far, so good. 80 00:03:52,090 --> 00:03:54,970 No mention of the word "angle" here or anything like this. 81 00:03:54,970 --> 00:03:58,340 Now, wherever t terminates-- and again, conventions here, 82 00:03:58,340 --> 00:04:01,830 if 't' is positive, I lay if off along the circle in the 83 00:04:01,830 --> 00:04:04,580 so-called positive direction, namely, what? 84 00:04:04,580 --> 00:04:05,880 Counter-clockwise. 85 00:04:05,880 --> 00:04:09,710 If 't' is negative, I'll lay it off in the clockwise 86 00:04:09,710 --> 00:04:11,030 direction, et cetera. 87 00:04:11,030 --> 00:04:13,220 The usual trigonometric conventions. 88 00:04:13,220 --> 00:04:18,579 Now what I do is is, at the point 'P', I drop a 89 00:04:18,579 --> 00:04:19,829 perpendicular. 90 00:04:23,050 --> 00:04:30,020 And I define the sine of 't' to be the length, 'PR', and 91 00:04:30,020 --> 00:04:34,150 the cosine of 'P' to be the length, 'OR'. 92 00:04:34,150 --> 00:04:36,500 In other words, I could write that like this. 93 00:04:36,500 --> 00:04:40,540 I could write down that I'm defining 'sine t' to be the 94 00:04:40,540 --> 00:04:45,010 length of 'RP' in that direction, meaning, of course, 95 00:04:45,010 --> 00:04:48,600 that this is just a fancy way of saying that the sine of 't' 96 00:04:48,600 --> 00:04:52,200 will just be the y-coordinate of the point at which the 97 00:04:52,200 --> 00:04:55,260 length 't' terminates on the circle. 98 00:04:55,260 --> 00:05:00,760 And in a similar way, 'cosine t' will be the directed length 99 00:05:00,760 --> 00:05:03,950 from 'O' to 'R', or more conventionally, the 100 00:05:03,950 --> 00:05:05,400 x-coordinate. 101 00:05:05,400 --> 00:05:07,730 Now, notice I can do this with any length. 102 00:05:07,730 --> 00:05:11,940 Whatever length I'm given, I just mark this length off. 103 00:05:11,940 --> 00:05:13,120 It's a finite length. 104 00:05:13,120 --> 00:05:14,760 Eventually, it has to terminate some 105 00:05:14,760 --> 00:05:16,000 place on the circle. 106 00:05:16,000 --> 00:05:19,590 Wherever it terminates, the x-coordinate of the point of 107 00:05:19,590 --> 00:05:23,280 termination is called the cosine of 't', and the 108 00:05:23,280 --> 00:05:25,960 y-coordinate is called the sine of 't'. 109 00:05:25,960 --> 00:05:29,500 And notice that, in this way, both the sine and the cosine 110 00:05:29,500 --> 00:05:35,246 are functions which map real numbers into real numbers. 111 00:05:35,246 --> 00:05:37,070 So that part, I hope, is clear. 112 00:05:39,910 --> 00:05:43,970 Notice again, I can mimic the usual traditional 113 00:05:43,970 --> 00:05:45,130 trigonometry. 114 00:05:45,130 --> 00:05:48,880 I can define the tangent of t to be the number 'sine t', 115 00:05:48,880 --> 00:05:51,770 divided by the number 'cosine t', et cetera. 116 00:05:51,770 --> 00:05:55,450 And I'll leave those details to the reading material. 117 00:05:55,450 --> 00:05:58,870 I can ascertain rather interesting results the same 118 00:05:58,870 --> 00:06:03,520 way as I could in regular traditional trigonometry. 119 00:06:03,520 --> 00:06:06,650 In fact, I can get some certain results very nicely. 120 00:06:06,650 --> 00:06:09,230 I remember, for example-- 121 00:06:09,230 --> 00:06:10,550 Well, I won't even go into these. 122 00:06:10,550 --> 00:06:14,460 But how did you talk about the sine of 0 when one talked 123 00:06:14,460 --> 00:06:16,230 about traditional trigonometry? 124 00:06:16,230 --> 00:06:20,220 How did you embed a 0-degree angle into a triangle, and 125 00:06:20,220 --> 00:06:21,320 things of this type. 126 00:06:21,320 --> 00:06:24,130 Notice that in terms of my tradition here-- 127 00:06:24,130 --> 00:06:26,500 and we'll summarize these results in a minute-- but 128 00:06:26,500 --> 00:06:30,510 notice, for example, that the sine of 0 comes out to be 0 129 00:06:30,510 --> 00:06:34,790 very nicely, because when 't' is 0, the length 0 130 00:06:34,790 --> 00:06:36,630 terminates at 'S'. 131 00:06:36,630 --> 00:06:38,690 'S' is on the x-axis. 132 00:06:38,690 --> 00:06:41,990 That makes, what? 'y' equal to 0. 133 00:06:41,990 --> 00:06:46,900 Notice also that, if the radius of my circle is 1, the 134 00:06:46,900 --> 00:06:49,200 circumference is 2 pi. 135 00:06:49,200 --> 00:06:52,530 So for example, what I usually think of a 90-degree angle 136 00:06:52,530 --> 00:06:55,250 would be the length pi/2. 137 00:06:55,250 --> 00:06:58,370 And without making any fuss over this, again, leaving most 138 00:06:58,370 --> 00:07:01,450 of the details to the reading and to the simplicity of just 139 00:07:01,450 --> 00:07:05,420 plugging these things in, we arrive at these rather 140 00:07:05,420 --> 00:07:07,430 familiar results. 141 00:07:07,430 --> 00:07:12,460 We also get, very quickly, in addition to these results, 142 00:07:12,460 --> 00:07:15,950 things like the fundamental result that we always like 143 00:07:15,950 --> 00:07:17,520 with trigonometric functions. 144 00:07:17,520 --> 00:07:21,890 That's 'sine squared t' plus 'cosine squared t' is 1. 145 00:07:21,890 --> 00:07:23,310 And how do we know that? 146 00:07:23,310 --> 00:07:27,620 Remember that 'cosine t' was just another name for the 147 00:07:27,620 --> 00:07:29,280 x-coordinate at which the point 148 00:07:29,280 --> 00:07:30,670 terminated on the circle. 149 00:07:30,670 --> 00:07:32,610 In other words, notice that 'cosine 150 00:07:32,610 --> 00:07:34,970 squared t' is 'x squared'. 151 00:07:34,970 --> 00:07:37,510 'Sine squared t' is 'y squared'. 152 00:07:37,510 --> 00:07:41,310 The x-coordinate and the y-coordinate are related by 153 00:07:41,310 --> 00:07:43,350 the fact that, what? 154 00:07:43,350 --> 00:07:47,930 The sum of the squares to be on the circle is equal to 1. 155 00:07:47,930 --> 00:07:53,380 We could even graph 'sine t' without any problem at all. 156 00:07:53,380 --> 00:07:58,280 Namely, we observe that when 't' is 0, 'sine t' is 0. 157 00:07:58,280 --> 00:08:04,250 Notice that as we go along the circle, the sine increases up 158 00:08:04,250 --> 00:08:10,510 until we get to pi/2, at which it peaks at 1, then decreases 159 00:08:10,510 --> 00:08:13,330 at pi, back down to 0. 160 00:08:13,330 --> 00:08:15,910 And if that's giving you trouble to follow, let's 161 00:08:15,910 --> 00:08:18,480 simply come back to our diagram to make sure that we 162 00:08:18,480 --> 00:08:20,580 understand this. 163 00:08:20,580 --> 00:08:25,200 In other words, all we're saying is, as 't' gets longer, 164 00:08:25,200 --> 00:08:31,240 its y-coordinate increases from 0 to a maximum of 1, when 165 00:08:31,240 --> 00:08:33,000 the particle was over here. 166 00:08:33,000 --> 00:08:37,340 Then, as 't' goes from to pi/2 to pi, the length of the 167 00:08:37,340 --> 00:08:42,770 y-coordinate decreases until it again becomes 0. 168 00:08:42,770 --> 00:08:46,740 And again, without making much more ado over this, we get the 169 00:08:46,740 --> 00:08:51,490 usual curve that we associate with the sine function even 170 00:08:51,490 --> 00:08:54,860 when we thought of it as a traditional 171 00:08:54,860 --> 00:08:56,240 trigonometric problem. 172 00:08:56,240 --> 00:08:59,470 But the major point that I want you to see right now-- 173 00:08:59,470 --> 00:09:01,800 and we won't worry about why I want to do this-- 174 00:09:01,800 --> 00:09:05,650 I can define the trigonometric functions in such a way that 175 00:09:05,650 --> 00:09:09,150 their domains are real numbers rather than angles. 176 00:09:09,150 --> 00:09:13,700 And in fact, this is the main reason why people invented the 177 00:09:13,700 --> 00:09:15,780 notion of radian measure. 178 00:09:15,780 --> 00:09:18,940 Let me see if I can't make that a little bit clearer, 179 00:09:18,940 --> 00:09:20,780 once and for all. 180 00:09:20,780 --> 00:09:22,270 You see, the question is this. 181 00:09:22,270 --> 00:09:23,990 Let's suppose I'm talking-- 182 00:09:23,990 --> 00:09:26,060 Oh, let me give you some letters over here. 183 00:09:26,060 --> 00:09:28,580 We'll put a 'Q' over here. 184 00:09:28,580 --> 00:09:32,030 Let's talk about angle, 'QOS'. 185 00:09:35,330 --> 00:09:36,560 That's a right angle. 186 00:09:36,560 --> 00:09:39,790 It's 1/4 of a rotation of the circle. 187 00:09:39,790 --> 00:09:43,570 Now, the question that I have in mind is, if something is 188 00:09:43,570 --> 00:09:46,250 1/4 of a rotation, why do you need two different ways of 189 00:09:46,250 --> 00:09:47,310 saying that? 190 00:09:47,310 --> 00:09:52,260 Why do we have to say it's 90 degrees or pi/2 radians, and 191 00:09:52,260 --> 00:09:56,270 bring in a new measure when we already have another way of 192 00:09:56,270 --> 00:09:59,250 measuring circles, angles of circles? 193 00:09:59,250 --> 00:10:01,020 The idea is something like this. 194 00:10:01,020 --> 00:10:05,430 Let's again mimic the idea of taking the length 't' and 195 00:10:05,430 --> 00:10:07,820 laying it off along the circle like this. 196 00:10:07,820 --> 00:10:11,190 Now, here's the idea. 197 00:10:11,190 --> 00:10:15,040 Remember, the radius of this circle is 1. 198 00:10:15,040 --> 00:10:20,660 So notice that 'PR', in other words, this y-coordinate, is 199 00:10:20,660 --> 00:10:23,750 what we call, by definition, the sine of 't'. 200 00:10:23,750 --> 00:10:28,030 In other words, just above we said that 'sine t' was the 201 00:10:28,030 --> 00:10:31,250 length 'R' to 'P' in that direction. 202 00:10:31,250 --> 00:10:34,160 Now, the point is, I said disregard traditional 203 00:10:34,160 --> 00:10:36,860 trigonometry, but we can't really disregard it. 204 00:10:36,860 --> 00:10:38,410 It exists. 205 00:10:38,410 --> 00:10:41,480 For the person who's had traditional trigonometry, how 206 00:10:41,480 --> 00:10:47,580 would he tend to look at this length divided by this length? 207 00:10:47,580 --> 00:10:49,930 He would think of that as being what? 208 00:10:49,930 --> 00:10:52,610 It's side opposite over hypotenuse. 209 00:10:52,610 --> 00:10:54,620 That also suggests sine. 210 00:10:54,620 --> 00:10:55,890 And the sine of what? 211 00:10:55,890 --> 00:10:59,570 Well, the sine of what angle this is. 212 00:10:59,570 --> 00:11:00,940 Now, the thing is this. 213 00:11:00,940 --> 00:11:04,530 Somehow or other, to avoid ambiguity, if we could have 214 00:11:04,530 --> 00:11:07,520 called whatever measure this angle was measured in terms 215 00:11:07,520 --> 00:11:11,530 of, if we could have called that unit 't', then notice 216 00:11:11,530 --> 00:11:15,550 that the sine of the angle 't' would have been numerically 217 00:11:15,550 --> 00:11:18,760 the same as the sine of the number 't'. 218 00:11:18,760 --> 00:11:22,140 And again, if this seems like a hard point to understand, we 219 00:11:22,140 --> 00:11:24,950 explore this in great detail in our notes. 220 00:11:24,950 --> 00:11:26,670 But the idea is this. 221 00:11:26,670 --> 00:11:30,760 You see, somehow or other, if 'sine t' is going to have two 222 00:11:30,760 --> 00:11:33,620 different meanings, we would like to make sure that we pick 223 00:11:33,620 --> 00:11:37,470 the kind of a unit where it makes no difference whether 224 00:11:37,470 --> 00:11:40,130 you're thinking of 't' as being a number or thinking of 225 00:11:40,130 --> 00:11:41,960 't' as being a length. 226 00:11:41,960 --> 00:11:45,370 For example, suppose I now invent the word "radian" to 227 00:11:45,370 --> 00:11:47,310 mean the following. 228 00:11:47,310 --> 00:11:49,910 An angle is said to have 't' radians. 229 00:11:49,910 --> 00:11:53,920 If, when made the central angle of a unit circle, a 230 00:11:53,920 --> 00:11:59,180 circle whose radius is 1, it subtends an arc whose length 231 00:11:59,180 --> 00:12:02,270 is 't' units of length. 232 00:12:02,270 --> 00:12:05,570 See, in other words, I would define the measure called 233 00:12:05,570 --> 00:12:09,690 radians so that an angle of 't' radians intercepts the 234 00:12:09,690 --> 00:12:11,490 length 't' over here. 235 00:12:11,490 --> 00:12:16,210 In that way, 'sine t' is unambiguous whether you're 236 00:12:16,210 --> 00:12:19,005 talking about an angle or a length. 237 00:12:19,005 --> 00:12:22,860 For example, when I say the sine of pi/1 238 00:12:22,860 --> 00:12:26,150 radians, what do I mean? 239 00:12:26,150 --> 00:12:29,280 I mean the angle which is the sine of the angle which 240 00:12:29,280 --> 00:12:34,350 intercepts a length, an arc, pi/2 units long. 241 00:12:34,350 --> 00:12:36,570 Well, see, pi/2 is this length. 242 00:12:36,570 --> 00:12:40,630 I'm now talking about this angle here. 243 00:12:40,630 --> 00:12:43,640 And the sine, therefore, of pi/2 radians, in terms of 244 00:12:43,640 --> 00:12:46,530 classical trigonometry, is 1. 245 00:12:46,530 --> 00:12:51,040 But that's also what the sine of a number pi/2 was. 246 00:12:51,040 --> 00:12:53,890 This explains the convention that one says when one uses 247 00:12:53,890 --> 00:12:56,240 radians, you can leave the label off. 248 00:12:56,240 --> 00:12:59,870 All we're saying is that, if we had used degrees, there 249 00:12:59,870 --> 00:13:01,900 would have been an ambiguity. 250 00:13:01,900 --> 00:13:08,120 Certainly, the sine of 3 degrees is not the same 251 00:13:08,120 --> 00:13:10,490 as the sine of 3. 252 00:13:10,490 --> 00:13:13,970 You see, 3 degrees is a rather small angle. 253 00:13:13,970 --> 00:13:19,550 But 3 is a rather great length when you're talking about the 254 00:13:19,550 --> 00:13:20,740 arc of the unit circle here. 255 00:13:20,740 --> 00:13:26,830 Remember, 1/2 circle is pi units long, so 3 would be just 256 00:13:26,830 --> 00:13:28,000 about this long. 257 00:13:28,000 --> 00:13:32,180 In other words, notice that 3 radians and 3 degrees are 258 00:13:32,180 --> 00:13:33,240 entirely different things. 259 00:13:33,240 --> 00:13:34,280 But the beauty is what? 260 00:13:34,280 --> 00:13:38,130 That if we agreed to use radian measure, then we have 261 00:13:38,130 --> 00:13:41,230 no ambiguity when we talk about the sine. 262 00:13:41,230 --> 00:13:44,670 The sine of the number 't' will equal the sine of the 263 00:13:44,670 --> 00:13:46,170 angle 't' radians. 264 00:13:46,170 --> 00:13:49,490 The cosine of a number 't' will equal the cosine of the 265 00:13:49,490 --> 00:13:51,360 angle 't' radians. 266 00:13:51,360 --> 00:13:53,960 In a certain sense, it was analogous to when we talked 267 00:13:53,960 --> 00:13:57,460 about the derivative 'dy/dx', then wanted to define 268 00:13:57,460 --> 00:14:01,700 differentials 'dy' and 'dx' separately, so that 'dy' 269 00:14:01,700 --> 00:14:05,210 divided by 'dx' would be the same as 'dy/dx', that we 270 00:14:05,210 --> 00:14:11,340 wanted to avoid any ambiguity where the same symbol could be 271 00:14:11,340 --> 00:14:13,400 interpreted in two different ways to give 272 00:14:13,400 --> 00:14:14,840 two different answers. 273 00:14:14,840 --> 00:14:18,010 By the way, again, there is nothing sacred about our 274 00:14:18,010 --> 00:14:21,000 choice of why we pick circular functions. 275 00:14:21,000 --> 00:14:23,690 We could have picked hyperbolic functions. 276 00:14:23,690 --> 00:14:27,410 Namely, why couldn't we have started, say, with one branch 277 00:14:27,410 --> 00:14:31,510 of the hyperbola, 'x squared' minus 'y squared' equals 1. 278 00:14:31,510 --> 00:14:35,790 Given a length 't', why couldn't we have measured 't' 279 00:14:35,790 --> 00:14:37,340 off along the hyperbola? 280 00:14:37,340 --> 00:14:41,070 Say this way if 't' is positive, the other way if 't' 281 00:14:41,070 --> 00:14:42,060 is negative. 282 00:14:42,060 --> 00:14:43,960 And then what we could have done is drop the 283 00:14:43,960 --> 00:14:47,290 perpendicular again. 284 00:14:47,290 --> 00:14:48,930 And we could have defined what? 285 00:14:48,930 --> 00:14:52,800 The y-coordinate to be the hyperbolic. 286 00:14:52,800 --> 00:14:55,400 Well, we couldn't call it cosine anymore because it 287 00:14:55,400 --> 00:14:57,510 would be confused with the circular functions. 288 00:14:57,510 --> 00:15:00,430 We could have invented a name, as we later will, called the 289 00:15:00,430 --> 00:15:02,810 'hyperbolic cosine'. 290 00:15:02,810 --> 00:15:05,000 I won't go into any more detail on this. 291 00:15:05,000 --> 00:15:07,210 See, this is an abbreviation for hyperbolic 292 00:15:07,210 --> 00:15:10,450 cosine, meaning this-- 293 00:15:10,450 --> 00:15:13,390 I'm sorry, I got this backwards. 294 00:15:13,390 --> 00:15:16,890 Call the x-coordinate the hyperbolic cosine, the 295 00:15:16,890 --> 00:15:20,630 y-coordinate the hyperbolic sine. 296 00:15:20,630 --> 00:15:22,330 You don't have to know anything about advanced 297 00:15:22,330 --> 00:15:23,680 mathematics to see this. 298 00:15:23,680 --> 00:15:27,460 All I'm saying is, I could just as easily have taken any 299 00:15:27,460 --> 00:15:32,130 geometric figure, marked off lengths along it, taken the 300 00:15:32,130 --> 00:15:36,130 x-coordinates and the y-coordinates, and seen what 301 00:15:36,130 --> 00:15:37,860 relationships they obey. 302 00:15:37,860 --> 00:15:41,090 You see, as such, there's nothing sacred about working 303 00:15:41,090 --> 00:15:42,510 on a circle. 304 00:15:42,510 --> 00:15:45,170 Not only that, but even after you agree to work on the 305 00:15:45,170 --> 00:15:47,250 circle, there are many other ways that one 306 00:15:47,250 --> 00:15:47,940 could have done this. 307 00:15:47,940 --> 00:15:50,790 For example, someone might have said, look it, when you 308 00:15:50,790 --> 00:15:54,780 take this length called 't', why did you elect to mark it 309 00:15:54,780 --> 00:15:56,360 off along the circle? 310 00:15:56,360 --> 00:16:01,760 Why couldn't you have taken a radius equal to 't', taken 'S' 311 00:16:01,760 --> 00:16:07,450 as a center, and swung an arc that met the circle, and call 312 00:16:07,450 --> 00:16:10,730 this length 't'? 313 00:16:10,730 --> 00:16:12,960 You see, instead of measuring along the circle, measure 314 00:16:12,960 --> 00:16:14,290 along the straight line. 315 00:16:14,290 --> 00:16:16,510 Again, you could have done this if you wanted to. 316 00:16:16,510 --> 00:16:18,290 Why you would've wanted to do this? 317 00:16:18,290 --> 00:16:20,500 Well, you have the same right to do this 318 00:16:20,500 --> 00:16:21,550 as I had to do mine. 319 00:16:21,550 --> 00:16:23,650 Of course, you have to be a little bit careful. 320 00:16:23,650 --> 00:16:27,350 For example, in this particular configuration, 321 00:16:27,350 --> 00:16:29,420 notice that, if this is how you're going to define your 322 00:16:29,420 --> 00:16:34,230 trigonometric function, your input, your domain, has to be 323 00:16:34,230 --> 00:16:37,070 somewhere between 0 and 2. 324 00:16:37,070 --> 00:16:39,800 In other words, you cannot have a length longer than 2, 325 00:16:39,800 --> 00:16:41,860 because notice that the diameter of the 326 00:16:41,860 --> 00:16:43,690 circle is only 2. 327 00:16:43,690 --> 00:16:46,680 And therefore, if 't' were greater than 2, when you swung 328 00:16:46,680 --> 00:16:49,000 an arc from the point 'S', it wouldn't meet 329 00:16:49,000 --> 00:16:49,900 the circle at all. 330 00:16:49,900 --> 00:16:51,890 Well, that's no great handicap. 331 00:16:51,890 --> 00:16:53,610 It's no great disaster. 332 00:16:53,610 --> 00:16:55,350 You still have the right to make up whatever 333 00:16:55,350 --> 00:16:56,860 functions you want. 334 00:16:56,860 --> 00:17:00,170 I will try to make it clearer why we chose these circular 335 00:17:00,170 --> 00:17:03,190 functions from a physical point of view as we go along. 336 00:17:03,190 --> 00:17:06,790 What I thought I'd like to do now is, having motivated, that 337 00:17:06,790 --> 00:17:09,400 we can invent the trigonometric functions in 338 00:17:09,400 --> 00:17:13,609 terms of numbers definitions along this circle. 339 00:17:13,609 --> 00:17:17,000 And coupled with the fact that, in radian measure, you 340 00:17:17,000 --> 00:17:20,150 can have a very nice identification between what's 341 00:17:20,150 --> 00:17:23,490 happening pictorially and what's happening analytically, 342 00:17:23,490 --> 00:17:26,079 to show, for example, that in terms of our subject called 343 00:17:26,079 --> 00:17:30,610 calculus, that we're pretty much home free once we learn 344 00:17:30,610 --> 00:17:32,280 these basic ideas. 345 00:17:32,280 --> 00:17:35,170 You see, the important point is that, in a manner of 346 00:17:35,170 --> 00:17:38,980 speaking, we have finished differential calculus. 347 00:17:38,980 --> 00:17:41,900 We know what all the recipes are We know what properties 348 00:17:41,900 --> 00:17:42,870 things have. 349 00:17:42,870 --> 00:17:46,180 So all of the rules that we learned will apply to any 350 00:17:46,180 --> 00:17:48,600 particular type of function that we're talking about. 351 00:17:48,600 --> 00:17:51,390 For example, let's suppose we define 'f of 352 00:17:51,390 --> 00:17:54,120 x' to be 'sine x'. 353 00:17:54,120 --> 00:17:56,760 And we want to find the derivative of 'sine x'. 354 00:17:56,760 --> 00:18:00,830 Notice that 'f prime of x' evaluated at any number 'x1' 355 00:18:00,830 --> 00:18:02,780 has already been defined for us. 356 00:18:02,780 --> 00:18:06,770 It's the limit as 'delta x' approaches 0, 'f of 'x1 plus 357 00:18:06,770 --> 00:18:11,100 delta x'', minus 'f of x1' over 'delta x'. 358 00:18:11,100 --> 00:18:13,520 This is true for any function 'f'. 359 00:18:13,520 --> 00:18:17,935 In particular, if 'f of x' is 'sine x', all we get is what? 360 00:18:17,935 --> 00:18:20,880 That the derivative is the limit as 'delta x' approaches 361 00:18:20,880 --> 00:18:25,510 0, sine of 'x1 plus delta x' minus sine of 362 00:18:25,510 --> 00:18:27,190 'x1' over 'delta x'. 363 00:18:27,190 --> 00:18:29,910 Now you see, on this particular score, 364 00:18:29,910 --> 00:18:31,450 nobody can fault us. 365 00:18:31,450 --> 00:18:34,160 This is still the basic definition. 366 00:18:34,160 --> 00:18:36,750 All that happens computationally is that, if 367 00:18:36,750 --> 00:18:39,200 we're not familiar with our new functions called the 368 00:18:39,200 --> 00:18:42,740 trigonometric functions, we might not know how to express 369 00:18:42,740 --> 00:18:47,060 sine of 'x1 plus delta x' in a more convenient form. 370 00:18:47,060 --> 00:18:49,080 What do we mean by a more convenient form? 371 00:18:49,080 --> 00:18:51,710 Well, notice again, as is always the case when we take a 372 00:18:51,710 --> 00:18:55,750 derivative, as delta x approaches 0, our numerator 373 00:18:55,750 --> 00:19:01,100 becomes 'sine x1' minus 'sine x1', which is 0/0. 374 00:19:01,100 --> 00:19:05,250 And we're back to our familiar taboo form of 0/0. 375 00:19:05,250 --> 00:19:07,770 Somehow or other, we're going to have to make a refinement 376 00:19:07,770 --> 00:19:10,360 on our numerator that will allow us to get 377 00:19:10,360 --> 00:19:13,000 rid of a 0/0 form. 378 00:19:13,000 --> 00:19:16,600 Well, to make a long story short, if we happen to know 379 00:19:16,600 --> 00:19:18,760 the addition formula for the sine-- 380 00:19:18,760 --> 00:19:22,350 in other words, 'sine 'x1 plus delta x'' is ''sine x1' 381 00:19:22,350 --> 00:19:26,950 'cosine delta x'', plus ''sine delta x' 'cosine x1''-- 382 00:19:26,950 --> 00:19:32,080 then we subtract off 'sine x1' and divide by 'delta x', and 383 00:19:32,080 --> 00:19:34,040 then we factor and collect terms. 384 00:19:34,040 --> 00:19:34,940 We see what? 385 00:19:34,940 --> 00:19:36,820 Without any knowledge of calculus at 386 00:19:36,820 --> 00:19:38,300 all, but just what? 387 00:19:38,300 --> 00:19:40,800 By our definition of derivative, just by our 388 00:19:40,800 --> 00:19:44,150 definition, coupled with properties of the 389 00:19:44,150 --> 00:19:47,510 trigonometric functions, we wind up with the fact that 'f 390 00:19:47,510 --> 00:19:50,440 prime of x1' is this particular limit. 391 00:19:50,440 --> 00:19:53,360 Now certainly, our limit theorems don't change. 392 00:19:53,360 --> 00:19:55,180 The limit of a sum is still going to be 393 00:19:55,180 --> 00:19:57,570 the sum of the limits. 394 00:19:57,570 --> 00:19:59,470 The limit of a product will still be the 395 00:19:59,470 --> 00:20:00,680 product of the limits. 396 00:20:00,680 --> 00:20:04,680 So all in all, what we have to sort of do is figure out what 397 00:20:04,680 --> 00:20:05,940 these limits will be. 398 00:20:05,940 --> 00:20:09,880 Certainly, as 'delta x' approaches 0, this will stay 399 00:20:09,880 --> 00:20:11,870 'cosine x1'. 400 00:20:11,870 --> 00:20:16,570 Certainly this will stay 'sine x1', because 'x1' is a fixed 401 00:20:16,570 --> 00:20:19,040 number that doesn't depend on 'delta x'. 402 00:20:19,040 --> 00:20:22,320 But notice, rather interestingly, that both of my 403 00:20:22,320 --> 00:20:26,980 expressions in parentheses happen to take on that 0/0 404 00:20:26,980 --> 00:20:28,580 form if we're not careful. 405 00:20:28,580 --> 00:20:34,250 Namely, if you replace 'delta x' by 0, sine 0 is 0, 0/0 is 406 00:20:34,250 --> 00:20:37,840 0, and we run into trouble here if we replace 'delta x' 407 00:20:37,840 --> 00:20:40,130 by 0, which of course we can't do. 408 00:20:40,130 --> 00:20:43,480 This is the same definition of limit as we had before. 409 00:20:43,480 --> 00:20:46,450 'Delta x' gets arbitrarily close to 0, but never is 410 00:20:46,450 --> 00:20:47,810 allowed to get there. 411 00:20:47,810 --> 00:20:51,340 Well, you see, if nothing else, this motivates why we 412 00:20:51,340 --> 00:20:54,330 would like to learn this particular type of limit. 413 00:20:54,330 --> 00:20:56,680 In other words, what we would like to know is, how do you-- 414 00:20:56,680 --> 00:20:59,400 the 'delta x' symbol here isn't that important. 415 00:20:59,400 --> 00:21:01,370 'Delta x' just stands for any number. 416 00:21:01,370 --> 00:21:03,760 Notice that what we would like to know is, if you take the 417 00:21:03,760 --> 00:21:07,490 sine of something over that same something, and take the 418 00:21:07,490 --> 00:21:10,570 limit as that same something goes to 0, we would like to 419 00:21:10,570 --> 00:21:12,320 know what that becomes. 420 00:21:12,320 --> 00:21:14,980 In a similar way, we would like to know how to handle 421 00:21:14,980 --> 00:21:17,480 this quotient here, because notice that when 'delta x' is 422 00:21:17,480 --> 00:21:20,040 0, cosine 0 is 1. 423 00:21:20,040 --> 00:21:22,460 This is 1 minus 1 over 0. 424 00:21:22,460 --> 00:21:25,200 It's another 0/0 form. 425 00:21:25,200 --> 00:21:28,556 So the problem that we're confronted with is that, what 426 00:21:28,556 --> 00:21:31,140 we would like to do is to figure out how to handle the 427 00:21:31,140 --> 00:21:35,360 limit of 'sine t' over 't' as 't' approaches 0. 428 00:21:35,360 --> 00:21:38,440 Now, what's 't' here? 't' is a number. 429 00:21:38,440 --> 00:21:38,900 Remember that. 430 00:21:38,900 --> 00:21:40,630 This is the big pitch I've been making. 431 00:21:40,630 --> 00:21:42,170 We're thinking of 't' as a number. 432 00:21:42,170 --> 00:21:45,650 If, on the other hand, you feel more comfortable thinking 433 00:21:45,650 --> 00:21:47,800 in terms of traditional trigonometry-- 434 00:21:47,800 --> 00:21:50,480 and let's face it, the more background you've had in 435 00:21:50,480 --> 00:21:54,780 traditional trigonometry, the more comfortable you're going 436 00:21:54,780 --> 00:21:56,480 to feel using it. 437 00:21:56,480 --> 00:21:59,620 Let's simply agree to do this, that if it bothers you to 438 00:21:59,620 --> 00:22:02,700 think of this as a length divided by a length, et 439 00:22:02,700 --> 00:22:05,500 cetera, and that this is a length or a number, let's 440 00:22:05,500 --> 00:22:10,460 agree that we will go back to angles but use radian measure. 441 00:22:10,460 --> 00:22:11,510 Why? 442 00:22:11,510 --> 00:22:15,970 Because if the angle is measured in radians, the sine 443 00:22:15,970 --> 00:22:21,220 of the angle 't' radians is the same as the number, the 444 00:22:21,220 --> 00:22:24,050 sine, of the number 't'. 445 00:22:24,050 --> 00:22:27,220 Well again, here's how this problem is tackled. 446 00:22:27,220 --> 00:22:30,250 What we do is we mark off the angle of 't' radians. 447 00:22:30,250 --> 00:22:32,260 Remember that we have the unit circle. 448 00:22:32,260 --> 00:22:37,510 And what we very cleverly do is we catch our wedge, our 449 00:22:37,510 --> 00:22:40,840 circular wedge, between two right triangles. 450 00:22:40,840 --> 00:22:44,400 Again, without making a big issue over this, notice that 451 00:22:44,400 --> 00:22:49,060 this length is 'sine t', this length is 'cosine t', so the 452 00:22:49,060 --> 00:22:52,710 area of the small triangle is 'sine t' times 453 00:22:52,710 --> 00:22:55,650 'cosine t' over 2. 454 00:22:55,650 --> 00:22:59,500 See, 'sine t' times 'cosine t' over 2. 455 00:22:59,500 --> 00:23:02,630 Now, on the other hand, since that's caught in our wedge, 456 00:23:02,630 --> 00:23:04,650 what is the area of our wedge? 457 00:23:04,650 --> 00:23:08,480 Well, since the area of the entire circle is pi-- see, pi 458 00:23:08,480 --> 00:23:10,180 'R squared' and 'R' is 1-- 459 00:23:10,180 --> 00:23:12,870 since the area of the entire circle is pi-- 460 00:23:12,870 --> 00:23:14,490 and we're taking what? 461 00:23:14,490 --> 00:23:17,290 't' of the 2pi. 462 00:23:17,290 --> 00:23:19,980 So there are two pi radians in a circle. 463 00:23:19,980 --> 00:23:24,230 So the sector of the circle that we have, it's 't/2pi' of 464 00:23:24,230 --> 00:23:25,390 the entire circle. 465 00:23:25,390 --> 00:23:28,490 And by the way, this is done more rigorously and carried 466 00:23:28,490 --> 00:23:30,030 out in detail in the notes. 467 00:23:30,030 --> 00:23:33,080 Let me point out that, if we insisted on working with 468 00:23:33,080 --> 00:23:36,450 degrees, instead of 't/2pi', we just 469 00:23:36,450 --> 00:23:40,310 would have had 't/360'. 470 00:23:40,310 --> 00:23:43,170 Because, you see, if we're dealing with degrees, the 471 00:23:43,170 --> 00:23:47,110 entire angle and measure of the circle is 360 degrees, and 472 00:23:47,110 --> 00:23:49,060 we would have had 't/360'. 473 00:23:49,060 --> 00:23:50,740 But here we've used the fact that we're 474 00:23:50,740 --> 00:23:53,630 dealing with radians. 475 00:23:53,630 --> 00:23:56,670 And finally, the bigger triangle, which includes the 476 00:23:56,670 --> 00:24:01,460 wedge, has, as its base, 1, so that's the radius. 477 00:24:01,460 --> 00:24:05,410 And since the tangent is side opposite over side adjacent, 478 00:24:05,410 --> 00:24:07,520 this length is 'tangent t'. 479 00:24:07,520 --> 00:24:09,980 And so what we have is what? 480 00:24:09,980 --> 00:24:15,200 That ''sine t' 'cosine t/2' must be less than this, which 481 00:24:15,200 --> 00:24:20,220 in turn must be less than this, multiplying through by 2 482 00:24:20,220 --> 00:24:25,380 and dividing through by 'sine t'. 483 00:24:25,380 --> 00:24:27,780 And by the way, this hinges on the fact that 't' is positive. 484 00:24:27,780 --> 00:24:30,740 Again, in our notes, we treat the case where 't' is negative 485 00:24:30,740 --> 00:24:32,390 to arrive at the same result. 486 00:24:32,390 --> 00:24:37,000 Remembering that 'tan t' is 'sine t' over 'cosine t', we 487 00:24:37,000 --> 00:24:39,000 wind up with this result. 488 00:24:39,000 --> 00:24:44,240 And now, observing that as 't' approaches 0, 489 00:24:44,240 --> 00:24:46,000 this approaches 1. 490 00:24:46,000 --> 00:24:48,270 This also approaches 1. 491 00:24:48,270 --> 00:24:51,230 And 't' over 'sine t' is caught between these two. 492 00:24:51,230 --> 00:24:54,510 We get that the limit of 't' over 'sine t' as 't' 493 00:24:54,510 --> 00:24:56,690 approaches 0 is 1. 494 00:24:56,690 --> 00:25:00,340 Now of course, since this limit is 1, the limit of the 495 00:25:00,340 --> 00:25:03,930 reciprocal of this will be the reciprocal of this. 496 00:25:03,930 --> 00:25:07,290 But what's very nice about the number 1 is that it's equal to 497 00:25:07,290 --> 00:25:10,500 its own reciprocal. 498 00:25:10,500 --> 00:25:14,050 In other words, what we've now shown is that the limit of 499 00:25:14,050 --> 00:25:18,400 'sine t' over 't' as 't' approaches 0 is 1. 500 00:25:18,400 --> 00:25:20,710 That, as I said before, is done in the text. 501 00:25:20,710 --> 00:25:22,090 We do it in our notes. 502 00:25:22,090 --> 00:25:25,750 But the thing that I hope this motivates is why we want to do 503 00:25:25,750 --> 00:25:27,230 this in the first place. 504 00:25:27,230 --> 00:25:30,640 Notice that this was a limit that we had to compute if we 505 00:25:30,640 --> 00:25:34,630 wanted to compute the derivative of the sine. 506 00:25:34,630 --> 00:25:38,260 Now, the next thing was, how do we handle '1 - cosine t' 507 00:25:38,260 --> 00:25:40,790 over 't' as 't' approaches 0? 508 00:25:40,790 --> 00:25:43,930 Again, leaving the details to you to sketch in as you see 509 00:25:43,930 --> 00:25:46,680 fit, let me point out simply what the mathematics 510 00:25:46,680 --> 00:25:48,100 involved here is. 511 00:25:48,100 --> 00:25:51,820 You see, what we can handle is 'sine t' over 't'. 512 00:25:51,820 --> 00:25:54,610 That means that what we would like to do is, whenever we're 513 00:25:54,610 --> 00:25:57,920 given an alien form, we would somehow or other like to 514 00:25:57,920 --> 00:26:00,900 figure some way of factoring a sine t over 515 00:26:00,900 --> 00:26:02,580 t out of this thing. 516 00:26:02,580 --> 00:26:06,970 When you look at '1 - cosine t', the identity, 'sine 517 00:26:06,970 --> 00:26:10,040 squared' equals '1 - 'cosine squared t'', 518 00:26:10,040 --> 00:26:12,240 should suggest itself. 519 00:26:12,240 --> 00:26:16,130 Now, how do you get from '1 - cosine t' to '1 - 'cosine 520 00:26:16,130 --> 00:26:17,170 squared t''? 521 00:26:17,170 --> 00:26:20,800 You have to multiply by '1 + cosine t'. 522 00:26:20,800 --> 00:26:24,050 And if you multiply by '1 + cosine t' upstairs, you must 523 00:26:24,050 --> 00:26:27,780 multiply by '1 + cosine t' downstairs. 524 00:26:27,780 --> 00:26:30,970 By the way, the only time you can't multiply by something is 525 00:26:30,970 --> 00:26:33,310 when the thing is 0. 526 00:26:33,310 --> 00:26:35,140 You can't put that into the denominator. 527 00:26:35,140 --> 00:26:39,410 Notice that 'cosine t' is not 0 in a neighborhood 528 00:26:39,410 --> 00:26:40,790 of 't' equals 0. 529 00:26:40,790 --> 00:26:45,170 See, 'cosine t' behaves like 1 when 't' is near 0, so this is 530 00:26:45,170 --> 00:26:47,870 a permissible step in this particular problem. 531 00:26:47,870 --> 00:26:52,880 The point is, we now factor '1 - 'cosine squared t'' as 'sine 532 00:26:52,880 --> 00:26:54,920 t' times 'sine t'. 533 00:26:54,920 --> 00:26:56,370 See, that's 'sine squared t'. 534 00:26:56,370 --> 00:27:01,740 We break up our 't' times '1 + cosine t' this way. 535 00:27:01,740 --> 00:27:04,880 Now we know that the limit of a product is the product of 536 00:27:04,880 --> 00:27:05,710 the limits. 537 00:27:05,710 --> 00:27:08,050 This we already know goes to 1. 538 00:27:08,050 --> 00:27:12,050 And as 't' approaches 0, from our previous limit work on the 539 00:27:12,050 --> 00:27:15,140 like, notice here, the limit of a quotient is the quotient 540 00:27:15,140 --> 00:27:19,360 of the limits, the numerator goes to 0, the denominator 541 00:27:19,360 --> 00:27:23,900 goes to 2, because as t approaches 0, cosine 0 is 1. 542 00:27:23,900 --> 00:27:28,370 At any rate, that's 0/2, which is 0. 543 00:27:28,370 --> 00:27:31,570 And so this limit is 0. 544 00:27:31,570 --> 00:27:35,410 Now, at the risk of giving you a slight headache as I take 545 00:27:35,410 --> 00:27:39,000 the board down here, let me just review what 546 00:27:39,000 --> 00:27:40,130 it was that we did. 547 00:27:40,130 --> 00:27:42,410 You see, notice that, without any knowledge of these limits 548 00:27:42,410 --> 00:27:45,720 at all, we were able to show that whatever the derivative 549 00:27:45,720 --> 00:27:49,110 of 'sine x' was, it was this particular thing here. 550 00:27:49,110 --> 00:27:53,230 Now what we've done is we've shown that this is 1, and 551 00:27:53,230 --> 00:27:55,630 we've shown that this is 0. 552 00:27:55,630 --> 00:27:59,090 And using our limit theorems, what we now see is what? 553 00:27:59,090 --> 00:28:06,650 That if 'f of x' is 'sine x', 'f prime of x' is 'cosine x'. 554 00:28:10,370 --> 00:28:15,780 Let me just write that down over here, that if 'y' equals 555 00:28:15,780 --> 00:28:23,620 'sine x', 'dy/dx' is 'cosine x'. 556 00:28:23,620 --> 00:28:27,560 And again, notice how much of the calculus involved here was 557 00:28:27,560 --> 00:28:28,350 nothing new. 558 00:28:28,350 --> 00:28:31,880 It goes back to the so-called baby chapter that nobody 559 00:28:31,880 --> 00:28:36,690 likes, where we go back to epsilons, deltas, you see 560 00:28:36,690 --> 00:28:39,430 derivatives by 'delta x', et cetera. 561 00:28:39,430 --> 00:28:42,370 See, those recipes always remain the same. 562 00:28:42,370 --> 00:28:45,670 What happens is, as you invent new functions, you need a 563 00:28:45,670 --> 00:28:48,920 different degree of computational sophistication 564 00:28:48,920 --> 00:28:50,740 to find the desired limits. 565 00:28:50,740 --> 00:28:53,880 By the way, once you get over these hurdles, everything 566 00:28:53,880 --> 00:28:56,520 again starts to go smoothly as before. 567 00:28:56,520 --> 00:28:58,460 For example, our chain rule. 568 00:28:58,460 --> 00:29:02,030 Suppose we have now that 'y' equals 'sine u', where 'u' is 569 00:29:02,030 --> 00:29:04,220 some differentiable function of 'x'. 570 00:29:04,220 --> 00:29:06,920 And we now want to find the 'dy/dx'. 571 00:29:06,920 --> 00:29:09,160 Well, you see, the point is that we know that the 572 00:29:09,160 --> 00:29:12,750 derivative of 'sine u' with respect to 'u' would be 573 00:29:12,750 --> 00:29:15,510 'cosine u'. 574 00:29:15,510 --> 00:29:16,850 What we want is the derivative of 'sine u' 575 00:29:16,850 --> 00:29:17,990 with respect to 'x'. 576 00:29:17,990 --> 00:29:21,480 And we motivate the chain rule the same way as we did before. 577 00:29:21,480 --> 00:29:23,920 It happens to be that we're dealing with the specific 578 00:29:23,920 --> 00:29:26,035 value called sine, but it could've 579 00:29:26,035 --> 00:29:27,000 been any old function. 580 00:29:27,000 --> 00:29:30,580 How would you differentiate 'f of u' with respect to 'x' if 581 00:29:30,580 --> 00:29:33,550 you know how to differentiate 'f of u' with respect to 'u'? 582 00:29:33,550 --> 00:29:35,430 And the answer is, you would just differentiate with 583 00:29:35,430 --> 00:29:39,060 respect to 'u', and multiply that by a derivative of 'u' 584 00:29:39,060 --> 00:29:40,490 with respect to 'x'. 585 00:29:40,490 --> 00:29:42,350 In other words, we get the result what? 586 00:29:42,350 --> 00:29:46,990 That since 'dy/du' is 'cosine u', we get that the derivative 587 00:29:46,990 --> 00:29:49,570 of 'sine u' with respect to 'x' is 588 00:29:49,570 --> 00:29:52,000 'cosine u' times 'du/dx'. 589 00:29:52,000 --> 00:29:55,690 And by the way, one rather nice application of this is 590 00:29:55,690 --> 00:29:58,740 that it gives us a very quick way of getting the derivative 591 00:29:58,740 --> 00:30:00,510 of 'cosine x'. 592 00:30:00,510 --> 00:30:04,870 After all, our basic identity is that 'cosine x' is sine 593 00:30:04,870 --> 00:30:06,900 pi/2 minus 'x'. 594 00:30:06,900 --> 00:30:09,910 Again, a number or an angle, either way. 595 00:30:09,910 --> 00:30:12,870 As long as the measurement is in radians, it makes no 596 00:30:12,870 --> 00:30:15,360 difference whether you think of this as being an angle or 597 00:30:15,360 --> 00:30:16,320 being a number. 598 00:30:16,320 --> 00:30:18,000 The answer will be the same. 599 00:30:18,000 --> 00:30:19,320 The idea is this. 600 00:30:19,320 --> 00:30:21,410 To take the derivative of 'cosine x' with respect to 601 00:30:21,410 --> 00:30:25,430 'x', all I have to differentiate is sine pi/2 602 00:30:25,430 --> 00:30:27,420 minus 'x' with respect to 'x'. 603 00:30:27,420 --> 00:30:28,850 But I know how to do that. 604 00:30:28,850 --> 00:30:35,290 Namely, the derivative of sine pi/2 minus 'x' is cosine pi/2 605 00:30:35,290 --> 00:30:39,680 minus 'x,' and by the chain rule, times the derivative of 606 00:30:39,680 --> 00:30:41,230 this with respect to 'x'. 607 00:30:41,230 --> 00:30:43,370 Well, pi/2 is a constant. 608 00:30:43,370 --> 00:30:46,580 The derivative of 'minus x' is minus 1. 609 00:30:46,580 --> 00:30:51,230 And then, remembering that the cosine of pi/2 minus 'x' is 610 00:30:51,230 --> 00:30:55,110 'sine x', I now have the result that the derivative of 611 00:30:55,110 --> 00:30:58,460 the cosine is minus the sine. 612 00:30:58,460 --> 00:31:01,070 And again, I can do all sorts of things this way. 613 00:31:01,070 --> 00:31:04,350 If I want the derivative of a tangent, I could write tangent 614 00:31:04,350 --> 00:31:06,040 as sine over cosine. 615 00:31:06,040 --> 00:31:07,450 Use the quotient rule. 616 00:31:07,450 --> 00:31:10,330 You see, as soon as I make one breakthrough, all of the 617 00:31:10,330 --> 00:31:13,260 previous body of calculus comes to my 618 00:31:13,260 --> 00:31:15,660 rescue, so to speak. 619 00:31:15,660 --> 00:31:18,990 By the way, what I'd like to do now is point out why, from 620 00:31:18,990 --> 00:31:23,440 a physical point of view, we like circular functions to be 621 00:31:23,440 --> 00:31:26,570 independent of angles and the like. 622 00:31:26,570 --> 00:31:29,430 With the results that we've derived so far, it's rather 623 00:31:29,430 --> 00:31:31,650 easy to derive one more result. 624 00:31:31,650 --> 00:31:34,390 Namely, let's assume that a particle is 625 00:31:34,390 --> 00:31:35,980 moving along the x-axis-- 626 00:31:35,980 --> 00:31:38,030 I'm going to start with the answer, sort of, and work 627 00:31:38,030 --> 00:31:38,900 backwards-- 628 00:31:38,900 --> 00:31:42,410 according to the rule, 'x' equals 'sine kt', where 't' is 629 00:31:42,410 --> 00:31:44,810 time and 'k' is a constant. 630 00:31:44,810 --> 00:31:47,680 Then its speed, 'dx/dt', is what? 631 00:31:47,680 --> 00:31:51,150 It's the derivative of 'sine kt', which is 'cosine kt', 632 00:31:51,150 --> 00:31:54,090 times the derivative of what's inside with respect to 't'. 633 00:31:54,090 --> 00:31:56,960 In other words, it's 'k cosine kt'. 634 00:31:56,960 --> 00:32:00,100 The second derivative of 'x' with respect to 't', namely, 635 00:32:00,100 --> 00:32:01,650 the acceleration is what? 636 00:32:01,650 --> 00:32:04,080 How do you differentiate the cosine? 637 00:32:04,080 --> 00:32:07,850 The derivative of the cosine is minus the sine. 638 00:32:07,850 --> 00:32:10,490 By the chain rule, I must multiply by the derivative of 639 00:32:10,490 --> 00:32:13,830 'kt' with respect to 't', which gives me another factor 640 00:32:13,830 --> 00:32:15,450 of 't' over here. 641 00:32:15,450 --> 00:32:24,780 Remembering that 'x' equals 'sine kt', I arrive at this 642 00:32:24,780 --> 00:32:27,370 particular so-called differential equation. 643 00:32:27,370 --> 00:32:28,510 And what does this say? 644 00:32:28,510 --> 00:32:32,620 It says that 'd2x/ dt squared', the acceleration, is 645 00:32:32,620 --> 00:32:35,090 proportional to the displacement, the distance 646 00:32:35,090 --> 00:32:37,580 traveled, but in the opposite direction. 647 00:32:37,580 --> 00:32:40,900 You see, 'k squared' can't be negative, so 'minus 'k 648 00:32:40,900 --> 00:32:42,940 squared'' can't be positive. 649 00:32:42,940 --> 00:32:43,810 This says what? 650 00:32:43,810 --> 00:32:46,510 The acceleration is proportional to the 651 00:32:46,510 --> 00:32:49,610 displacement, but in the opposite direction. 652 00:32:49,610 --> 00:32:54,800 Does that problem require any knowledge of angles to solve? 653 00:32:54,800 --> 00:32:56,470 Notice that this is a perfectly 654 00:32:56,470 --> 00:32:57,950 good physical problem. 655 00:32:57,950 --> 00:33:00,940 It's known as simple harmonic motion. 656 00:33:00,940 --> 00:33:04,310 And all I'm trying to have you see is that, by inventing the 657 00:33:04,310 --> 00:33:08,960 circular functions in the proper way, not only can we do 658 00:33:08,960 --> 00:33:12,660 their calculus, but even more importantly, if we reverse 659 00:33:12,660 --> 00:33:17,220 these steps, for example, we can show that, to solve the 660 00:33:17,220 --> 00:33:20,920 physical problem of simple harmonic motion, we have to 661 00:33:20,920 --> 00:33:24,890 know the so-called circular trigonometric functions. 662 00:33:24,890 --> 00:33:29,030 And this is a far cry, you see, from using trigonometry 663 00:33:29,030 --> 00:33:32,385 in the sense that the surveyor uses trigonometry. 664 00:33:32,385 --> 00:33:34,740 You see, this ties up with my initial hang-up that I was 665 00:33:34,740 --> 00:33:37,110 telling you about at the beginning of the program. 666 00:33:37,110 --> 00:33:40,270 By the way, in closing, I should also make reference to 667 00:33:40,270 --> 00:33:42,640 something that we pointed out in our last lecture, namely, 668 00:33:42,640 --> 00:33:44,780 inverse differentiation. 669 00:33:44,780 --> 00:33:48,200 Keep in mind, also, that as you read the calculus of the 670 00:33:48,200 --> 00:33:51,440 trigonometric functions, that the fact that we know that the 671 00:33:51,440 --> 00:33:55,230 derivative of sine u with respect to 'u' was 'cosine u' 672 00:33:55,230 --> 00:33:58,560 gives us, with a switch in emphasis, the result that the 673 00:33:58,560 --> 00:34:03,290 integral 'cosine u', 'du' is 'sine u' plus a constant. 674 00:34:03,290 --> 00:34:06,060 And in a similar way, since the derivative of cosine is 675 00:34:06,060 --> 00:34:09,070 minus the sine, the integral of 'sine u' with respect to 676 00:34:09,070 --> 00:34:12,590 'u' is 'minus cosine u' plus a constant. 677 00:34:12,590 --> 00:34:13,280 Be careful. 678 00:34:13,280 --> 00:34:16,420 Notice how the sines can screw you up. 679 00:34:16,420 --> 00:34:18,310 Namely, they're in the opposite sense when you're 680 00:34:18,310 --> 00:34:20,929 integrating as when you were differentiating. 681 00:34:20,929 --> 00:34:24,239 But again, these are the details which I expect you can 682 00:34:24,239 --> 00:34:26,909 have come out in the wash rather nicely. 683 00:34:26,909 --> 00:34:29,510 We can continue on this way, from knowing how to 684 00:34:29,510 --> 00:34:32,389 differentiate 'sine x' to the nth power. 685 00:34:32,389 --> 00:34:38,380 Namely, it's 'n - 1' 'x', times the derivative of 'sine 686 00:34:38,380 --> 00:34:40,900 x', which is 'cosine x'. 687 00:34:40,900 --> 00:34:42,510 We don't want this in here. 688 00:34:42,510 --> 00:34:44,050 That's a differential form. 689 00:34:44,050 --> 00:34:46,790 Without going into any detail here, notice that a 690 00:34:46,790 --> 00:34:50,690 modification of this shows us that, if we differentiate 691 00:34:50,690 --> 00:34:53,630 this, we wind up with this. 692 00:34:53,630 --> 00:34:57,730 We could now take the time, if this were the proper place, to 693 00:34:57,730 --> 00:35:01,430 develop all sorts of derivative formulas and 694 00:35:01,430 --> 00:35:02,650 integral formulas. 695 00:35:02,650 --> 00:35:06,000 As you study your study guide, you will notice that the 696 00:35:06,000 --> 00:35:11,000 lesson after this is concerned with the calculus of the 697 00:35:11,000 --> 00:35:12,650 circular functions. 698 00:35:12,650 --> 00:35:16,190 My feeling is is that, with this as background, a very 699 00:35:16,190 --> 00:35:20,390 good review of the previous part of the course will be to 700 00:35:20,390 --> 00:35:25,350 see how much of this you can apply on your own to these new 701 00:35:25,350 --> 00:35:28,420 functions called the circular functions. 702 00:35:28,420 --> 00:35:31,810 Next time, we will talk, as you may be able to guess, 703 00:35:31,810 --> 00:35:34,200 about the inverse circular functions 704 00:35:34,200 --> 00:35:35,800 and why they're important. 705 00:35:35,800 --> 00:35:37,340 But until next time, goodbye. 706 00:35:40,240 --> 00:35:43,440 Funding for the publication of this video was provided by the 707 00:35:43,440 --> 00:35:47,490 Gabriella and Paul Rosenbaum Foundation. 708 00:35:47,490 --> 00:35:51,660 Help OCW continue to provide free and open access to MIT 709 00:35:51,660 --> 00:35:55,860 courses by making a donation at ocw.mit.edu/donate.