WEBVTT 00:00:00.910 --> 00:00:01.700 Welcome to the presentation 00:00:01.700 --> 00:00:03.770 on radians and degrees 00:00:03.770 --> 00:00:06.250 So you all are probably already reasonably 00:00:06.250 --> 00:00:07.950 familiar with the concept of degrees. 00:00:07.950 --> 00:00:09.573 I think in our angles module 00:00:09.573 --> 00:00:11.196 I mean, lets see 00:00:11.196 --> 00:00:12.820 we actually drill you through a bunch of problems. 00:00:12.820 --> 00:00:18.320 You're probably familiar that an angle 00:00:18.320 --> 00:00:23.460 a right angle is 90 degrees. 00:00:23.460 --> 00:00:28.650 Or half a right angle-- 45 degrees. 00:00:28.650 --> 00:00:31.470 And you're also probably familiar with 00:00:31.470 --> 00:00:33.640 the concept that in a circle -- 00:00:33.640 --> 00:00:36.320 and that's my best attempt at a circle -- 00:00:36.320 --> 00:00:39.980 in a circle, there are 360 degrees. 00:00:39.980 --> 00:00:42.470 So today I'm going to introduce you to 00:00:42.470 --> 00:00:45.460 another measure or unit for angles 00:00:45.460 --> 00:00:47.270 and this is called a radian. 00:00:52.160 --> 00:00:53.450 So what is a radian? 00:00:53.450 --> 00:00:55.220 So I'll start with the definition 00:00:55.220 --> 00:00:57.100 and I think this might give you a little intuition 00:00:57.100 --> 00:00:59.910 for why it's even called radian. 00:00:59.910 --> 00:01:01.280 Let me use this circle tool 00:01:01.280 --> 00:01:02.850 and actually draw a nice circle. 00:01:06.275 --> 00:01:09.240 So by definition... 00:01:09.240 --> 00:01:14.270 Oh,oh,I'm still using the radian tool, the circle tool. 00:01:14.270 --> 00:01:14.530 OK. 00:01:16.430 --> 00:01:19.430 So this is --let's say 00:01:19.430 --> 00:01:21.630 This is a radius of length r. 00:01:21.630 --> 00:01:25.500 A radian is the angle that subtends an arc. 00:01:25.500 --> 00:01:29.250 And all subtend means is if this is angle, 00:01:29.250 --> 00:01:30.980 and this is the arc, 00:01:30.980 --> 00:01:34.520 this angle subtends this arc 00:01:34.520 --> 00:01:36.020 and this arc subtends this angle. 00:01:36.020 --> 00:01:36.830 So a radian 00:01:37.170 --> 00:01:41.050 -- one radian -- is the angle that subtends an arc 00:01:41.050 --> 00:01:44.130 that's the length of the radius. 00:01:44.130 --> 00:01:46.780 So the length of this is also r. 00:01:46.780 --> 00:01:50.440 And this angle is one radian. 00:01:50.440 --> 00:01:51.140 i think that's messy. 00:01:51.140 --> 00:01:52.490 Let me draw a bigger circle. 00:01:55.010 --> 00:01:56.640 Here you go. 00:01:56.640 --> 00:01:57.860 And I'm going to do this because I was wondering 00:01:57.860 --> 00:01:58.780 why they do radians. 00:01:58.780 --> 00:02:00.300 We all know degrees. 00:02:00.300 --> 00:02:01.480 But actually when you think about it 00:02:01.480 --> 00:02:03.100 it actually makes a reasonable amount of sense. 00:02:03.100 --> 00:02:05.870 So let me use the line tool now. 00:02:12.980 --> 00:02:17.340 And let's say that this radius is a length r 00:02:17.340 --> 00:02:21.460 and that this arc right here is also length r. 00:02:21.460 --> 00:02:24.020 Then this angle, what's called theta, 00:02:24.020 --> 00:02:28.210 is equal to one radian. 00:02:28.210 --> 00:02:30.220 And now it makes sense that they call it a radian. 00:02:30.220 --> 00:02:32.440 It's kind of like a radius. 00:02:32.440 --> 00:02:33.560 So let me ask a question: 00:02:33.560 --> 00:02:37.420 how many radians are there in a circle? 00:02:37.420 --> 00:02:39.760 Well, if this is r, 00:02:39.760 --> 00:02:42.050 what is the whole circumference of a circle? 00:02:44.630 --> 00:02:46.540 It's 2 pi r, right? 00:02:46.540 --> 00:02:50.050 You know that from the basic geometry module. 00:02:50.050 --> 00:02:53.100 So if the radian is the angle 00:02:53.100 --> 00:02:55.850 that subtends an arc of r, 00:02:55.850 --> 00:03:00.260 then the angle that subtends an arc of 2 pi r 00:03:00.260 --> 00:03:03.650 is 2 pi radians. 00:03:03.650 --> 00:03:06.970 So this angle is 2 pi radians. 00:03:12.510 --> 00:03:14.820 If you're still confused, think of it this way. 00:03:14.820 --> 00:03:17.220 An angle of 2 pi radians going 00:03:17.220 --> 00:03:22.650 all the way around subtends an arc of 2 pi radiuses. 00:03:22.650 --> 00:03:23.920 Or radii. 00:03:23.920 --> 00:03:26.460 I don't know how to say the plural of radius. 00:03:26.460 --> 00:03:27.110 Maybe it's radians. 00:03:27.110 --> 00:03:30.130 I don't know. 00:03:30.130 --> 00:03:31.890 So why am I going through all of this mess 00:03:31.890 --> 00:03:32.630 and confusing you? 00:03:32.630 --> 00:03:34.660 I just want to one, give you an intuition 00:03:34.660 --> 00:03:36.050 for why it's called a radian 00:03:36.050 --> 00:03:38.130 and kind of how it relates to a circle. 00:03:38.130 --> 00:03:41.160 And then given that there 2 pi radians in a circle, 00:03:41.160 --> 00:03:42.550 we can now figure out 00:03:42.550 --> 00:03:46.980 a relationship between radians and degrees. 00:03:46.980 --> 00:03:49.920 Let me delete this. 00:03:49.920 --> 00:03:54.190 So we said in a circle, there are 2 pi radians. 00:03:54.190 --> 00:03:57.190 right, let's say 2 pi raidians 00:03:57.340 --> 00:03:58.970 And how many degrees are there in a circle? 00:03:58.970 --> 00:04:00.800 If we went around a whole circle how many degrees? 00:04:00.800 --> 00:04:04.360 Well that's equal to 360 degrees. 00:04:07.080 --> 00:04:07.520 So there. 00:04:07.520 --> 00:04:09.620 We have an equation that sets up a conversion between 00:04:09.620 --> 00:04:10.950 radians and degrees. 00:04:10.950 --> 00:04:19.390 So one radian is equal to 360 over 2 pi degrees. 00:04:19.390 --> 00:04:22.570 Right , I just divided both sides by 2 pi. 00:04:22.570 --> 00:04:27.040 Which equals 180 over pi degrees. 00:04:29.710 --> 00:04:31.080 Similarly, we could have done the other way. 00:04:31.080 --> 00:04:33.120 We could have divided both sides by 360 00:04:33.130 --> 00:04:36.040 and we could have said 1 degree -- 00:04:36.040 --> 00:04:39.230 I'm just going to divide both sides by 360 00:04:39.230 --> 00:04:39.970 and I'm flipping it. 00:04:39.970 --> 00:04:45.410 1 degree is equal to 2 pi over 360 radians. 00:04:48.570 --> 00:04:53.260 Which equals pi over 180 radians. 00:04:53.260 --> 00:04:54.400 So then we have a conversion: 00:04:54.860 --> 00:04:59.100 1 radian equals pi over 180 degrees 00:04:59.100 --> 00:05:02.100 (laughs)1 radian equals 180 over pi degrees 00:05:02.100 --> 00:05:05.220 and 1 degree equals pi over 180 radians. 00:05:05.220 --> 00:05:06.980 And if you ever forget these 00:05:06.980 --> 00:05:08.740 it doesn't hurt to memorize this. 00:05:08.740 --> 00:05:12.520 But if you ever forget it, I always go back to this. 00:05:12.520 --> 00:05:15.810 That 2 pi radians is equal to 360 degrees. 00:05:15.810 --> 00:05:21.230 Or another way that actually makes the algebra or the algebric expression 00:05:21.240 --> 00:05:22.780 a little simpler is 00:05:22.780 --> 00:05:26.540 if you just think of a half circle. 00:05:26.540 --> 00:05:28.090 A half circle -- 00:05:28.090 --> 00:05:31.550 this angle -- is a 180 degrees, right? 00:05:31.550 --> 00:05:34.550 180 degrees 00:05:35.210 --> 00:05:36.120 That's a degree sign. 00:05:36.120 --> 00:05:37.810 I could also write degrees out. 00:05:37.810 --> 00:05:39.680 And that's also equal to pi radians,right. 00:05:42.680 --> 00:05:44.490 So pi radians equal 180 degrees 00:05:44.490 --> 00:05:46.250 and we can get to see the math. 00:05:46.250 --> 00:05:53.320 1 radian equals 180 over pi degrees 00:05:53.320 --> 00:06:00.940 or 1 degree,that's degree, is equal to pi over 180 radians. 00:06:00.940 --> 00:06:02.490 So let's do a couple of problems were 00:06:02.490 --> 00:06:03.540 you'll get the intuition for this. 00:06:03.540 --> 00:06:07.420 If I asked you 45 degrees -- 00:06:07.420 --> 00:06:09.010 to convert that into radians. 00:06:12.440 --> 00:06:18.410 Well, we know that 1 degree is pi over 180 radians. 00:06:18.410 --> 00:06:25.260 So 45 degrees is equal to 00:06:25.260 --> 00:06:32.910 45 times pi over 180 radians. 00:06:32.910 --> 00:06:36.850 And let's see, 45 divided by 180. 00:06:36.850 --> 00:06:39.370 45 goes into 180 four times 00:06:39.380 --> 00:06:42.360 so this equals pi over 4 radians. 00:06:45.650 --> 00:06:49.600 45 degrees is equal to pi over 4 radians. 00:06:49.600 --> 00:06:50.810 And just keep in mind, 00:06:50.810 --> 00:06:52.610 these are just two different units 00:06:52.610 --> 00:06:55.070 or two different ways of measuring angles. 00:06:55.070 --> 00:06:56.590 And the reason why I do this is this is actually 00:06:56.590 --> 00:06:58.990 the mathematical standard for measuring angles, 00:06:58.990 --> 00:07:01.280 although most of us are more familiar with degrees 00:07:01.280 --> 00:07:03.030 just from everyday life. 00:07:03.030 --> 00:07:04.920 Let's do a couple of other examples. 00:07:04.920 --> 00:07:05.840 Just always remember: 00:07:05.840 --> 00:07:08.400 this 1 radian equals 180 over pi degrees. 00:07:08.400 --> 00:07:10.200 1 degree equals pi over 180 radians. 00:07:10.200 --> 00:07:12.630 If you ever get confused, just write this out. 00:07:12.630 --> 00:07:15.190 this is what I do because I always forget 00:07:15.190 --> 00:07:17.570 whether it's pi over 180 or 180 over pi. 00:07:17.570 --> 00:07:21.550 I just remember pi radians is equal to 180 degrees. 00:07:21.550 --> 00:07:23.840 Let's do another one. 00:07:23.840 --> 00:07:33.060 So if I were to say pi over 2 radians equals 00:07:33.060 --> 00:07:33.760 how many degrees? 00:07:37.480 --> 00:07:40.130 Well I already forgot what I had just written 00:07:40.140 --> 00:07:42.590 so I just remind myself that 00:07:42.590 --> 00:07:45.560 pi radians is equal to 180 degrees. 00:07:55.720 --> 00:07:56.690 Oh, my wife just got home, 00:07:56.690 --> 00:07:57.930 so I'm just going to have to leave 00:07:57.930 --> 00:08:00.210 the presentation like that 00:08:00.210 --> 00:08:02.670 and I will continue it later. 00:08:02.670 --> 00:08:04.540 Actually, let me just finish this problem 00:08:04.540 --> 00:08:07.270 and then I'll go attend to my wife. 00:08:07.270 --> 00:08:11.640 But we know that pi radians is equal to 180 degrees, 00:08:11.640 --> 00:08:12.140 right? 00:08:12.140 --> 00:08:16.310 So one radian is equal to 180 over -- 00:08:16.310 --> 00:08:21.660 that's one radian -- is equal to 180 over pi degrees. 00:08:21.660 --> 00:08:23.470 I just figure out the formula again 00:08:23.470 --> 00:08:24.490 because I always forget it. 00:08:24.490 --> 00:08:25.500 So let's go back here. 00:08:25.500 --> 00:08:33.160 So pi over 2 radians is equal to 00:08:33.160 --> 00:08:38.510 pi over 2 times 180 over pi degrees. 00:08:38.510 --> 00:08:41.580 And that equals 90 degrees. 00:08:47.240 --> 00:08:48.830 I'll do one more example. 00:08:54.480 --> 00:08:55.910 Let's say 30 degrees. 00:09:00.950 --> 00:09:03.200 Once again, I forgot the formula so I just remember 00:09:03.200 --> 00:09:10.960 that pi radians is equal to 180 degrees. 00:09:10.960 --> 00:09:19.150 So 1 degree is equal to pi over 180 radians. 00:09:19.150 --> 00:09:23.030 So 30 degrees is equal to 00:09:23.030 --> 00:09:27.220 30 times pi over 180 radians 00:09:27.220 --> 00:09:31.320 which equals -- 30 goes into 180 six times. 00:09:31.320 --> 00:09:36.160 That equals pi over 6 radians. 00:09:36.160 --> 00:09:37.890 Hopefully you have a sense of 00:09:37.890 --> 00:09:40.310 how to convert between degrees and radians now 00:09:40.310 --> 00:09:41.770 and even why it's called a radian 00:09:41.770 --> 00:09:44.860 because it's very closely related to a radius 00:09:44.870 --> 00:09:48.570 and you'll feel comfortable when someone asks you to, 00:09:48.570 --> 00:09:52.410 I don't know,deal with radians as opposed to degrees. 00:09:52.410 --> 00:09:54.670 I'll see you in the next presentation.