[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.91,0:00:01.70,Default,,0000,0000,0000,,Welcome to the presentation
Dialogue: 0,0:00:01.70,0:00:03.77,Default,,0000,0000,0000,,on radians and degrees
Dialogue: 0,0:00:03.77,0:00:06.25,Default,,0000,0000,0000,,So you all are probably already reasonably
Dialogue: 0,0:00:06.25,0:00:07.95,Default,,0000,0000,0000,,familiar with the concept of degrees.
Dialogue: 0,0:00:07.95,0:00:09.57,Default,,0000,0000,0000,,I think in our angles module
Dialogue: 0,0:00:09.57,0:00:11.20,Default,,0000,0000,0000,,I mean, lets see
Dialogue: 0,0:00:11.20,0:00:12.82,Default,,0000,0000,0000,,we actually drill you through a bunch of problems.
Dialogue: 0,0:00:12.82,0:00:18.32,Default,,0000,0000,0000,,You're probably familiar that an angle
Dialogue: 0,0:00:18.32,0:00:23.46,Default,,0000,0000,0000,,a right angle is 90 degrees.
Dialogue: 0,0:00:23.46,0:00:28.65,Default,,0000,0000,0000,,Or half a right angle-- 45 degrees.
Dialogue: 0,0:00:28.65,0:00:31.47,Default,,0000,0000,0000,,And you're also probably familiar with
Dialogue: 0,0:00:31.47,0:00:33.64,Default,,0000,0000,0000,,the concept that in a circle --
Dialogue: 0,0:00:33.64,0:00:36.32,Default,,0000,0000,0000,,and that's my best attempt at a circle --
Dialogue: 0,0:00:36.32,0:00:39.98,Default,,0000,0000,0000,,in a circle, there are 360 degrees.
Dialogue: 0,0:00:39.98,0:00:42.47,Default,,0000,0000,0000,,So today I'm going to introduce you to
Dialogue: 0,0:00:42.47,0:00:45.46,Default,,0000,0000,0000,,another measure or unit for angles
Dialogue: 0,0:00:45.46,0:00:47.27,Default,,0000,0000,0000,,and this is called a radian.
Dialogue: 0,0:00:52.16,0:00:53.45,Default,,0000,0000,0000,,So what is a radian?
Dialogue: 0,0:00:53.45,0:00:55.22,Default,,0000,0000,0000,,So I'll start with the definition
Dialogue: 0,0:00:55.22,0:00:57.10,Default,,0000,0000,0000,,and I think this might give you a little intuition
Dialogue: 0,0:00:57.10,0:00:59.91,Default,,0000,0000,0000,,for why it's even called radian.
Dialogue: 0,0:00:59.91,0:01:01.28,Default,,0000,0000,0000,,Let me use this circle tool
Dialogue: 0,0:01:01.28,0:01:02.85,Default,,0000,0000,0000,,and actually draw a nice circle.
Dialogue: 0,0:01:06.28,0:01:09.24,Default,,0000,0000,0000,,So by definition...
Dialogue: 0,0:01:09.24,0:01:14.27,Default,,0000,0000,0000,,Oh,oh,I'm still using the radian tool, the circle tool.
Dialogue: 0,0:01:14.27,0:01:14.53,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:01:16.43,0:01:19.43,Default,,0000,0000,0000,,So this is --let's say
Dialogue: 0,0:01:19.43,0:01:21.63,Default,,0000,0000,0000,,This is a radius of length r.
Dialogue: 0,0:01:21.63,0:01:25.50,Default,,0000,0000,0000,,A radian is the angle that subtends an arc.
Dialogue: 0,0:01:25.50,0:01:29.25,Default,,0000,0000,0000,,And all subtend means is if this is angle,
Dialogue: 0,0:01:29.25,0:01:30.98,Default,,0000,0000,0000,,and this is the arc,
Dialogue: 0,0:01:30.98,0:01:34.52,Default,,0000,0000,0000,,this angle subtends this arc
Dialogue: 0,0:01:34.52,0:01:36.02,Default,,0000,0000,0000,,and this arc subtends this angle.
Dialogue: 0,0:01:36.02,0:01:36.83,Default,,0000,0000,0000,,So a radian
Dialogue: 0,0:01:37.17,0:01:41.05,Default,,0000,0000,0000,,-- one radian -- is the angle that subtends an arc
Dialogue: 0,0:01:41.05,0:01:44.13,Default,,0000,0000,0000,,that's the length of the radius.
Dialogue: 0,0:01:44.13,0:01:46.78,Default,,0000,0000,0000,,So the length of this is also r.
Dialogue: 0,0:01:46.78,0:01:50.44,Default,,0000,0000,0000,,And this angle is one radian.
Dialogue: 0,0:01:50.44,0:01:51.14,Default,,0000,0000,0000,,i think that's messy.
Dialogue: 0,0:01:51.14,0:01:52.49,Default,,0000,0000,0000,,Let me draw a bigger circle.
Dialogue: 0,0:01:55.01,0:01:56.64,Default,,0000,0000,0000,,Here you go.
Dialogue: 0,0:01:56.64,0:01:57.86,Default,,0000,0000,0000,,And I'm going to do this because I was wondering
Dialogue: 0,0:01:57.86,0:01:58.78,Default,,0000,0000,0000,,why they do radians.
Dialogue: 0,0:01:58.78,0:02:00.30,Default,,0000,0000,0000,,We all know degrees.
Dialogue: 0,0:02:00.30,0:02:01.48,Default,,0000,0000,0000,,But actually when you think about it
Dialogue: 0,0:02:01.48,0:02:03.10,Default,,0000,0000,0000,,it actually makes a reasonable amount of sense.
Dialogue: 0,0:02:03.10,0:02:05.87,Default,,0000,0000,0000,,So let me use the line tool now.
Dialogue: 0,0:02:12.98,0:02:17.34,Default,,0000,0000,0000,,And let's say that this radius is a length r
Dialogue: 0,0:02:17.34,0:02:21.46,Default,,0000,0000,0000,,and that this arc right here is also length r.
Dialogue: 0,0:02:21.46,0:02:24.02,Default,,0000,0000,0000,,Then this angle, what's called theta,
Dialogue: 0,0:02:24.02,0:02:28.21,Default,,0000,0000,0000,,is equal to one radian.
Dialogue: 0,0:02:28.21,0:02:30.22,Default,,0000,0000,0000,,And now it makes sense that they call it a radian.
Dialogue: 0,0:02:30.22,0:02:32.44,Default,,0000,0000,0000,,It's kind of like a radius.
Dialogue: 0,0:02:32.44,0:02:33.56,Default,,0000,0000,0000,,So let me ask a question:
Dialogue: 0,0:02:33.56,0:02:37.42,Default,,0000,0000,0000,,how many radians are there in a circle?
Dialogue: 0,0:02:37.42,0:02:39.76,Default,,0000,0000,0000,,Well, if this is r,
Dialogue: 0,0:02:39.76,0:02:42.05,Default,,0000,0000,0000,,what is the whole circumference of a circle?
Dialogue: 0,0:02:44.63,0:02:46.54,Default,,0000,0000,0000,,It's 2 pi r, right?
Dialogue: 0,0:02:46.54,0:02:50.05,Default,,0000,0000,0000,,You know that from the basic geometry module.
Dialogue: 0,0:02:50.05,0:02:53.10,Default,,0000,0000,0000,,So if the radian is the angle
Dialogue: 0,0:02:53.10,0:02:55.85,Default,,0000,0000,0000,,that subtends an arc of r,
Dialogue: 0,0:02:55.85,0:03:00.26,Default,,0000,0000,0000,,then the angle that subtends an arc of 2 pi r
Dialogue: 0,0:03:00.26,0:03:03.65,Default,,0000,0000,0000,,is 2 pi radians.
Dialogue: 0,0:03:03.65,0:03:06.97,Default,,0000,0000,0000,,So this angle is 2 pi radians.
Dialogue: 0,0:03:12.51,0:03:14.82,Default,,0000,0000,0000,,If you're still confused, think of it this way.
Dialogue: 0,0:03:14.82,0:03:17.22,Default,,0000,0000,0000,,An angle of 2 pi radians going
Dialogue: 0,0:03:17.22,0:03:22.65,Default,,0000,0000,0000,,all the way around subtends an arc of 2 pi radiuses.
Dialogue: 0,0:03:22.65,0:03:23.92,Default,,0000,0000,0000,,Or radii.
Dialogue: 0,0:03:23.92,0:03:26.46,Default,,0000,0000,0000,,I don't know how to say the plural of radius.
Dialogue: 0,0:03:26.46,0:03:27.11,Default,,0000,0000,0000,,Maybe it's radians.
Dialogue: 0,0:03:27.11,0:03:30.13,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:03:30.13,0:03:31.89,Default,,0000,0000,0000,,So why am I going through all of this mess
Dialogue: 0,0:03:31.89,0:03:32.63,Default,,0000,0000,0000,,and confusing you?
Dialogue: 0,0:03:32.63,0:03:34.66,Default,,0000,0000,0000,,I just want to one, give you an intuition
Dialogue: 0,0:03:34.66,0:03:36.05,Default,,0000,0000,0000,,for why it's called a radian
Dialogue: 0,0:03:36.05,0:03:38.13,Default,,0000,0000,0000,,and kind of how it relates to a circle.
Dialogue: 0,0:03:38.13,0:03:41.16,Default,,0000,0000,0000,,And then given that there 2 pi radians in a circle,
Dialogue: 0,0:03:41.16,0:03:42.55,Default,,0000,0000,0000,,we can now figure out
Dialogue: 0,0:03:42.55,0:03:46.98,Default,,0000,0000,0000,,a relationship between radians and degrees.
Dialogue: 0,0:03:46.98,0:03:49.92,Default,,0000,0000,0000,,Let me delete this.
Dialogue: 0,0:03:49.92,0:03:54.19,Default,,0000,0000,0000,,So we said in a circle, there are 2 pi radians.
Dialogue: 0,0:03:54.19,0:03:57.19,Default,,0000,0000,0000,,right, let's say 2 pi raidians
Dialogue: 0,0:03:57.34,0:03:58.97,Default,,0000,0000,0000,,And how many degrees are there in a circle?
Dialogue: 0,0:03:58.97,0:04:00.80,Default,,0000,0000,0000,,If we went around a whole circle how many degrees?
Dialogue: 0,0:04:00.80,0:04:04.36,Default,,0000,0000,0000,,Well that's equal to 360 degrees.
Dialogue: 0,0:04:07.08,0:04:07.52,Default,,0000,0000,0000,,So there.
Dialogue: 0,0:04:07.52,0:04:09.62,Default,,0000,0000,0000,,We have an equation that sets up a conversion between
Dialogue: 0,0:04:09.62,0:04:10.95,Default,,0000,0000,0000,,radians and degrees.
Dialogue: 0,0:04:10.95,0:04:19.39,Default,,0000,0000,0000,,So one radian is equal to 360 over 2 pi degrees.
Dialogue: 0,0:04:19.39,0:04:22.57,Default,,0000,0000,0000,,Right , I just divided both sides by 2 pi.
Dialogue: 0,0:04:22.57,0:04:27.04,Default,,0000,0000,0000,,Which equals 180 over pi degrees.
Dialogue: 0,0:04:29.71,0:04:31.08,Default,,0000,0000,0000,,Similarly, we could have done the other way.
Dialogue: 0,0:04:31.08,0:04:33.12,Default,,0000,0000,0000,,We could have divided both sides by 360
Dialogue: 0,0:04:33.13,0:04:36.04,Default,,0000,0000,0000,,and we could have said 1 degree --
Dialogue: 0,0:04:36.04,0:04:39.23,Default,,0000,0000,0000,,I'm just going to divide both sides by 360
Dialogue: 0,0:04:39.23,0:04:39.97,Default,,0000,0000,0000,,and I'm flipping it.
Dialogue: 0,0:04:39.97,0:04:45.41,Default,,0000,0000,0000,,1 degree is equal to 2 pi over 360 radians.
Dialogue: 0,0:04:48.57,0:04:53.26,Default,,0000,0000,0000,,Which equals pi over 180 radians.
Dialogue: 0,0:04:53.26,0:04:54.40,Default,,0000,0000,0000,,So then we have a conversion:
Dialogue: 0,0:04:54.86,0:04:59.10,Default,,0000,0000,0000,,1 radian equals pi over 180 degrees
Dialogue: 0,0:04:59.10,0:05:02.10,Default,,0000,0000,0000,,(laughs)1 radian equals 180 over pi degrees
Dialogue: 0,0:05:02.10,0:05:05.22,Default,,0000,0000,0000,,and 1 degree equals pi over 180 radians.
Dialogue: 0,0:05:05.22,0:05:06.98,Default,,0000,0000,0000,,And if you ever forget these
Dialogue: 0,0:05:06.98,0:05:08.74,Default,,0000,0000,0000,,it doesn't hurt to memorize this.
Dialogue: 0,0:05:08.74,0:05:12.52,Default,,0000,0000,0000,,But if you ever forget it, I always go back to this.
Dialogue: 0,0:05:12.52,0:05:15.81,Default,,0000,0000,0000,,That 2 pi radians is equal to 360 degrees.
Dialogue: 0,0:05:15.81,0:05:21.23,Default,,0000,0000,0000,,Or another way that actually makes the algebra or the algebric expression
Dialogue: 0,0:05:21.24,0:05:22.78,Default,,0000,0000,0000,,a little simpler is
Dialogue: 0,0:05:22.78,0:05:26.54,Default,,0000,0000,0000,,if you just think of a half circle.
Dialogue: 0,0:05:26.54,0:05:28.09,Default,,0000,0000,0000,,A half circle --
Dialogue: 0,0:05:28.09,0:05:31.55,Default,,0000,0000,0000,,this angle -- is a 180 degrees, right?
Dialogue: 0,0:05:31.55,0:05:34.55,Default,,0000,0000,0000,,180 degrees
Dialogue: 0,0:05:35.21,0:05:36.12,Default,,0000,0000,0000,,That's a degree sign.
Dialogue: 0,0:05:36.12,0:05:37.81,Default,,0000,0000,0000,,I could also write degrees out.
Dialogue: 0,0:05:37.81,0:05:39.68,Default,,0000,0000,0000,,And that's also equal to pi radians,right.
Dialogue: 0,0:05:42.68,0:05:44.49,Default,,0000,0000,0000,,So pi radians equal 180 degrees
Dialogue: 0,0:05:44.49,0:05:46.25,Default,,0000,0000,0000,,and we can get to see the math.
Dialogue: 0,0:05:46.25,0:05:53.32,Default,,0000,0000,0000,,1 radian equals 180 over pi degrees
Dialogue: 0,0:05:53.32,0:06:00.94,Default,,0000,0000,0000,,or 1 degree,that's degree, is equal to pi over 180 radians.
Dialogue: 0,0:06:00.94,0:06:02.49,Default,,0000,0000,0000,,So let's do a couple of problems were
Dialogue: 0,0:06:02.49,0:06:03.54,Default,,0000,0000,0000,,you'll get the intuition for this.
Dialogue: 0,0:06:03.54,0:06:07.42,Default,,0000,0000,0000,,If I asked you 45 degrees --
Dialogue: 0,0:06:07.42,0:06:09.01,Default,,0000,0000,0000,,to convert that into radians.
Dialogue: 0,0:06:12.44,0:06:18.41,Default,,0000,0000,0000,,Well, we know that 1 degree is pi over 180 radians.
Dialogue: 0,0:06:18.41,0:06:25.26,Default,,0000,0000,0000,,So 45 degrees is equal to
Dialogue: 0,0:06:25.26,0:06:32.91,Default,,0000,0000,0000,,45 times pi over 180 radians.
Dialogue: 0,0:06:32.91,0:06:36.85,Default,,0000,0000,0000,,And let's see, 45 divided by 180.
Dialogue: 0,0:06:36.85,0:06:39.37,Default,,0000,0000,0000,,45 goes into 180 four times
Dialogue: 0,0:06:39.38,0:06:42.36,Default,,0000,0000,0000,,so this equals pi over 4 radians.
Dialogue: 0,0:06:45.65,0:06:49.60,Default,,0000,0000,0000,,45 degrees is equal to pi over 4 radians.
Dialogue: 0,0:06:49.60,0:06:50.81,Default,,0000,0000,0000,,And just keep in mind,
Dialogue: 0,0:06:50.81,0:06:52.61,Default,,0000,0000,0000,,these are just two different units
Dialogue: 0,0:06:52.61,0:06:55.07,Default,,0000,0000,0000,,or two different ways of measuring angles.
Dialogue: 0,0:06:55.07,0:06:56.59,Default,,0000,0000,0000,,And the reason why I do this is this is actually
Dialogue: 0,0:06:56.59,0:06:58.99,Default,,0000,0000,0000,,the mathematical standard for measuring angles,
Dialogue: 0,0:06:58.99,0:07:01.28,Default,,0000,0000,0000,,although most of us are more familiar with degrees
Dialogue: 0,0:07:01.28,0:07:03.03,Default,,0000,0000,0000,,just from everyday life.
Dialogue: 0,0:07:03.03,0:07:04.92,Default,,0000,0000,0000,,Let's do a couple of other examples.
Dialogue: 0,0:07:04.92,0:07:05.84,Default,,0000,0000,0000,,Just always remember:
Dialogue: 0,0:07:05.84,0:07:08.40,Default,,0000,0000,0000,,this 1 radian equals 180 over pi degrees.
Dialogue: 0,0:07:08.40,0:07:10.20,Default,,0000,0000,0000,,1 degree equals pi over 180 radians.
Dialogue: 0,0:07:10.20,0:07:12.63,Default,,0000,0000,0000,,If you ever get confused, just write this out.
Dialogue: 0,0:07:12.63,0:07:15.19,Default,,0000,0000,0000,,this is what I do because I always forget
Dialogue: 0,0:07:15.19,0:07:17.57,Default,,0000,0000,0000,,whether it's pi over 180 or 180 over pi.
Dialogue: 0,0:07:17.57,0:07:21.55,Default,,0000,0000,0000,,I just remember pi radians is equal to 180 degrees.
Dialogue: 0,0:07:21.55,0:07:23.84,Default,,0000,0000,0000,,Let's do another one.
Dialogue: 0,0:07:23.84,0:07:33.06,Default,,0000,0000,0000,,So if I were to say pi over 2 radians equals
Dialogue: 0,0:07:33.06,0:07:33.76,Default,,0000,0000,0000,,how many degrees?
Dialogue: 0,0:07:37.48,0:07:40.13,Default,,0000,0000,0000,,Well I already forgot what I had just written
Dialogue: 0,0:07:40.14,0:07:42.59,Default,,0000,0000,0000,,so I just remind myself that
Dialogue: 0,0:07:42.59,0:07:45.56,Default,,0000,0000,0000,,pi radians is equal to 180 degrees.
Dialogue: 0,0:07:55.72,0:07:56.69,Default,,0000,0000,0000,,Oh, my wife just got home,
Dialogue: 0,0:07:56.69,0:07:57.93,Default,,0000,0000,0000,,so I'm just going to have to leave
Dialogue: 0,0:07:57.93,0:08:00.21,Default,,0000,0000,0000,,the presentation like that
Dialogue: 0,0:08:00.21,0:08:02.67,Default,,0000,0000,0000,,and I will continue it later.
Dialogue: 0,0:08:02.67,0:08:04.54,Default,,0000,0000,0000,,Actually, let me just finish this problem
Dialogue: 0,0:08:04.54,0:08:07.27,Default,,0000,0000,0000,,and then I'll go attend to my wife.
Dialogue: 0,0:08:07.27,0:08:11.64,Default,,0000,0000,0000,,But we know that pi radians is equal to 180 degrees,
Dialogue: 0,0:08:11.64,0:08:12.14,Default,,0000,0000,0000,,right?
Dialogue: 0,0:08:12.14,0:08:16.31,Default,,0000,0000,0000,,So one radian is equal to 180 over --
Dialogue: 0,0:08:16.31,0:08:21.66,Default,,0000,0000,0000,,that's one radian -- is equal to 180 over pi degrees.
Dialogue: 0,0:08:21.66,0:08:23.47,Default,,0000,0000,0000,,I just figure out the formula again
Dialogue: 0,0:08:23.47,0:08:24.49,Default,,0000,0000,0000,,because I always forget it.
Dialogue: 0,0:08:24.49,0:08:25.50,Default,,0000,0000,0000,,So let's go back here.
Dialogue: 0,0:08:25.50,0:08:33.16,Default,,0000,0000,0000,,So pi over 2 radians is equal to
Dialogue: 0,0:08:33.16,0:08:38.51,Default,,0000,0000,0000,,pi over 2 times 180 over pi degrees.
Dialogue: 0,0:08:38.51,0:08:41.58,Default,,0000,0000,0000,,And that equals 90 degrees.
Dialogue: 0,0:08:47.24,0:08:48.83,Default,,0000,0000,0000,,I'll do one more example.
Dialogue: 0,0:08:54.48,0:08:55.91,Default,,0000,0000,0000,,Let's say 30 degrees.
Dialogue: 0,0:09:00.95,0:09:03.20,Default,,0000,0000,0000,,Once again, I forgot the formula so I just remember
Dialogue: 0,0:09:03.20,0:09:10.96,Default,,0000,0000,0000,,that pi radians is equal to 180 degrees.
Dialogue: 0,0:09:10.96,0:09:19.15,Default,,0000,0000,0000,,So 1 degree is equal to pi over 180 radians.
Dialogue: 0,0:09:19.15,0:09:23.03,Default,,0000,0000,0000,,So 30 degrees is equal to
Dialogue: 0,0:09:23.03,0:09:27.22,Default,,0000,0000,0000,,30 times pi over 180 radians
Dialogue: 0,0:09:27.22,0:09:31.32,Default,,0000,0000,0000,,which equals -- 30 goes into 180 six times.
Dialogue: 0,0:09:31.32,0:09:36.16,Default,,0000,0000,0000,,That equals pi over 6 radians.
Dialogue: 0,0:09:36.16,0:09:37.89,Default,,0000,0000,0000,,Hopefully you have a sense of
Dialogue: 0,0:09:37.89,0:09:40.31,Default,,0000,0000,0000,,how to convert between degrees and radians now
Dialogue: 0,0:09:40.31,0:09:41.77,Default,,0000,0000,0000,,and even why it's called a radian
Dialogue: 0,0:09:41.77,0:09:44.86,Default,,0000,0000,0000,,because it's very closely related to a radius
Dialogue: 0,0:09:44.87,0:09:48.57,Default,,0000,0000,0000,,and you'll feel comfortable when someone asks you to,
Dialogue: 0,0:09:48.57,0:09:52.41,Default,,0000,0000,0000,,I don't know,deal with radians as opposed to degrees.
Dialogue: 0,0:09:52.41,0:09:54.67,Default,,0000,0000,0000,,I'll see you in the next presentation.