[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.45,0:00:05.72,Default,,0000,0000,0000,,In this video, we're going\Nto look at a different way
Dialogue: 0,0:00:05.72,0:00:06.88,Default,,0000,0000,0000,,of measuring angles.
Dialogue: 0,0:00:08.23,0:00:12.84,Default,,0000,0000,0000,,Well aware.\NIn a circle.
Dialogue: 0,0:00:15.30,0:00:21.50,Default,,0000,0000,0000,,There are 360 degrees all\Nthe way around.
Dialogue: 0,0:00:22.08,0:00:27.35,Default,,0000,0000,0000,,360 degrees in a circle, 360\Ndegrees all the way around. What
Dialogue: 0,0:00:27.35,0:00:32.62,Default,,0000,0000,0000,,we need is a new way of\Nmeasuring angles and this is
Dialogue: 0,0:00:32.62,0:00:36.57,Default,,0000,0000,0000,,needed because when we look at\Ntrig functions, particularly
Dialogue: 0,0:00:36.57,0:00:40.52,Default,,0000,0000,0000,,their differentiation, we will\Nneed a different measure of
Dialogue: 0,0:00:40.52,0:00:45.79,Default,,0000,0000,0000,,angle from degrees. So let's\Nhave a look at what that measure
Dialogue: 0,0:00:45.79,0:00:51.80,Default,,0000,0000,0000,,is. So\Nwe take
Dialogue: 0,0:00:51.80,0:00:58.48,Default,,0000,0000,0000,,a circle.\NRadius R.
Dialogue: 0,0:01:00.81,0:01:07.18,Default,,0000,0000,0000,,And I want to mark off on the\Ncircumference a length that is
Dialogue: 0,0:01:07.18,0:01:11.59,Default,,0000,0000,0000,,actually equal tool are so that\Ndistance around the
Dialogue: 0,0:01:11.59,0:01:16.98,Default,,0000,0000,0000,,circumference is equal to R, the\Nradius. Now we join up.
Dialogue: 0,0:01:17.70,0:01:24.86,Default,,0000,0000,0000,,There. That's our and this\Nangle here theater we describe
Dialogue: 0,0:01:24.86,0:01:27.54,Default,,0000,0000,0000,,as being one Radian.
Dialogue: 0,0:01:28.05,0:01:31.38,Default,,0000,0000,0000,,So Theta equals 1.
Dialogue: 0,0:01:32.22,0:01:33.03,Default,,0000,0000,0000,,Radian
Dialogue: 0,0:01:34.99,0:01:39.12,Default,,0000,0000,0000,,And if we want a definition of\Nthat, then the definition is.
Dialogue: 0,0:01:40.01,0:01:45.72,Default,,0000,0000,0000,,The angle of one Radian is the\Nangle subtended at the center of
Dialogue: 0,0:01:45.72,0:01:51.86,Default,,0000,0000,0000,,the circle by an arc that is\Nequal in length to the radius of
Dialogue: 0,0:01:51.86,0:01:56.69,Default,,0000,0000,0000,,the circle. So that gives us our\Ndefinition of a radium.
Dialogue: 0,0:01:57.23,0:02:03.08,Default,,0000,0000,0000,,Let me\Njust draw.
Dialogue: 0,0:02:05.02,0:02:09.79,Default,,0000,0000,0000,,3.\NCircles.
Dialogue: 0,0:02:11.36,0:02:18.90,Default,,0000,0000,0000,,Let's put that definition into\Neffect again, where we've got.
Dialogue: 0,0:02:19.78,0:02:24.98,Default,,0000,0000,0000,,A distance equal to R, the\Nradius of the circle around the
Dialogue: 0,0:02:24.98,0:02:29.87,Default,,0000,0000,0000,,circumference. And there's our\Ntheater equals 1.
Dialogue: 0,0:02:30.65,0:02:33.07,Default,,0000,0000,0000,,Radian now let me.
Dialogue: 0,0:02:34.66,0:02:41.10,Default,,0000,0000,0000,,Go round the circumference a\Ndistance of two are.
Dialogue: 0,0:02:43.16,0:02:50.62,Default,,0000,0000,0000,,As the radius again, let's take\Nhalf way around there so that
Dialogue: 0,0:02:50.62,0:02:58.09,Default,,0000,0000,0000,,this is Theta. In here is\None Radian, an theater in here
Dialogue: 0,0:02:58.09,0:03:00.58,Default,,0000,0000,0000,,is also one Radian.
Dialogue: 0,0:03:01.22,0:03:08.38,Default,,0000,0000,0000,,So what we can see there\Nis that this distance to our
Dialogue: 0,0:03:08.38,0:03:15.55,Default,,0000,0000,0000,,is in fact two radians times\Nby the radius, and so that
Dialogue: 0,0:03:15.55,0:03:18.53,Default,,0000,0000,0000,,gives us a general formula.
Dialogue: 0,0:03:19.69,0:03:26.18,Default,,0000,0000,0000,,That if we have any angle\Ntheater subtended at the center
Dialogue: 0,0:03:26.18,0:03:33.26,Default,,0000,0000,0000,,of the circle then the arc\Nlength S around there is given
Dialogue: 0,0:03:33.26,0:03:36.21,Default,,0000,0000,0000,,by S equals R Theater.
Dialogue: 0,0:03:36.93,0:03:40.44,Default,,0000,0000,0000,,And that's the arc
Dialogue: 0,0:03:40.44,0:03:46.67,Default,,0000,0000,0000,,length. And it's\Ntrue if theater
Dialogue: 0,0:03:46.67,0:03:49.85,Default,,0000,0000,0000,,is in radians.
Dialogue: 0,0:03:50.52,0:03:55.67,Default,,0000,0000,0000,,So that gives us a very good way\Nof calculating the arc length
Dialogue: 0,0:03:55.67,0:04:00.82,Default,,0000,0000,0000,,when we know the angle subtended\Nat the center, and we know its
Dialogue: 0,0:04:00.82,0:04:02.00,Default,,0000,0000,0000,,value in radians.
Dialogue: 0,0:04:02.73,0:04:10.28,Default,,0000,0000,0000,,So.\NCan we get an equivalence
Dialogue: 0,0:04:10.28,0:04:12.61,Default,,0000,0000,0000,,between our radiant measure and
Dialogue: 0,0:04:12.61,0:04:16.35,Default,,0000,0000,0000,,our degrees? Well, let's have a\Nlook at a circle again.
Dialogue: 0,0:04:17.29,0:04:21.44,Default,,0000,0000,0000,,The circumference\Nof this circle.
Dialogue: 0,0:04:22.57,0:04:29.49,Default,,0000,0000,0000,,We know by\Nthe formula to
Dialogue: 0,0:04:29.49,0:04:36.47,Default,,0000,0000,0000,,Π R. Now\Nlet's compare that with our
Dialogue: 0,0:04:36.47,0:04:38.44,Default,,0000,0000,0000,,formula arc length.
Dialogue: 0,0:04:39.03,0:04:45.49,Default,,0000,0000,0000,,Our formula for arclength we\Nknow, is our theater.
Dialogue: 0,0:04:46.73,0:04:50.17,Default,,0000,0000,0000,,Well, he is, AH.
Dialogue: 0,0:04:50.98,0:04:53.34,Default,,0000,0000,0000,,And here is our.
Dialogue: 0,0:04:53.97,0:04:58.71,Default,,0000,0000,0000,,And what that means is that for\Nthis circle, the distance all
Dialogue: 0,0:04:58.71,0:05:02.66,Default,,0000,0000,0000,,the way round is 2π, R, and\Ntherefore the angle.
Dialogue: 0,0:05:03.58,0:05:10.34,Default,,0000,0000,0000,,In radians all the way round\Nis this number here 2π.
Dialogue: 0,0:05:10.94,0:05:18.40,Default,,0000,0000,0000,,So what we've got\Nis that 2π radians.
Dialogue: 0,0:05:19.35,0:05:25.31,Default,,0000,0000,0000,,Is exactly the same\Nas 360 degrees.
Dialogue: 0,0:05:26.67,0:05:32.74,Default,,0000,0000,0000,,Now this equivalence enables us\Nto write down a whole series of
Dialogue: 0,0:05:32.74,0:05:40.57,Default,,0000,0000,0000,,Equalities. Between angles, so\Nwe've got 2π radians.
Dialogue: 0,0:05:40.57,0:05:48.25,Default,,0000,0000,0000,,Is the same as 360\Ndegrees, so Pi radians is
Dialogue: 0,0:05:48.25,0:05:55.93,Default,,0000,0000,0000,,the same as 180 degrees.\NJust dividing both sides by
Dialogue: 0,0:05:55.93,0:06:02.84,Default,,0000,0000,0000,,two. If we divide by\Ntwo, again π by two radians.
Dialogue: 0,0:06:03.40,0:06:10.44,Default,,0000,0000,0000,,Notes, so I said that Π by 2 not\Nπ / 2, but pie by two. It's a
Dialogue: 0,0:06:10.44,0:06:15.13,Default,,0000,0000,0000,,shorthand, so Pi by two radians\Nis 90 degrees, and that's an
Dialogue: 0,0:06:15.13,0:06:17.08,Default,,0000,0000,0000,,important one, because that's a
Dialogue: 0,0:06:17.08,0:06:24.13,Default,,0000,0000,0000,,right angle. Pie by 4\N/ 2 again high by
Dialogue: 0,0:06:24.13,0:06:26.83,Default,,0000,0000,0000,,4 radians is 45.
Dialogue: 0,0:06:27.35,0:06:34.08,Default,,0000,0000,0000,,Degrees. So these are the\Nimportant angle, so to speak.
Dialogue: 0,0:06:34.08,0:06:39.07,Default,,0000,0000,0000,,Some other quite important\Nangles in trigonometry are 60
Dialogue: 0,0:06:39.07,0:06:46.28,Default,,0000,0000,0000,,degrees, well, 60 degrees is 1/3\Nof 180 degrees. So Pi by three
Dialogue: 0,0:06:46.28,0:06:51.44,Default,,0000,0000,0000,,radians. Is the same as 60\Ndegrees, another angle that's
Dialogue: 0,0:06:51.44,0:06:53.60,Default,,0000,0000,0000,,important is the angle 30
Dialogue: 0,0:06:53.60,0:07:00.49,Default,,0000,0000,0000,,degrees. And 30 degrees is\N1/6 of 180 degrees, so π
Dialogue: 0,0:07:00.49,0:07:02.24,Default,,0000,0000,0000,,by 6 radians.
Dialogue: 0,0:07:03.32,0:07:06.53,Default,,0000,0000,0000,,Is the same as 30 degrees.
Dialogue: 0,0:07:07.13,0:07:12.11,Default,,0000,0000,0000,,OK, we've got all of these\Nequivalences, but the one thing
Dialogue: 0,0:07:12.11,0:07:18.46,Default,,0000,0000,0000,,we haven't got is how big is a\NRadian in terms of degrees. One
Dialogue: 0,0:07:18.46,0:07:20.72,Default,,0000,0000,0000,,Radian equals how many degrees?
Dialogue: 0,0:07:21.43,0:07:27.65,Default,,0000,0000,0000,,Or will start with this\Nrelationship here that Pi
Dialogue: 0,0:07:27.65,0:07:30.41,Default,,0000,0000,0000,,radians is 180 degrees.
Dialogue: 0,0:07:31.09,0:07:38.20,Default,,0000,0000,0000,,So.\NΠ radians
Dialogue: 0,0:07:38.20,0:07:41.27,Default,,0000,0000,0000,,is 180
Dialogue: 0,0:07:41.27,0:07:44.51,Default,,0000,0000,0000,,degrees. So
Dialogue: 0,0:07:44.51,0:07:51.69,Default,,0000,0000,0000,,one Radian. Must\Nbe 180 degrees divided by
Dialogue: 0,0:07:51.69,0:07:58.77,Default,,0000,0000,0000,,pie and if we use\Na Calculator to do that,
Dialogue: 0,0:07:58.77,0:08:04.43,Default,,0000,0000,0000,,we find that the answer\Nis 57.296 degrees.
Dialogue: 0,0:08:05.12,0:08:10.89,Default,,0000,0000,0000,,2 three decimal places now it's\Nactually rare that we need to do
Dialogue: 0,0:08:10.89,0:08:16.22,Default,,0000,0000,0000,,this particular calculation, but\Nwhat it does do is it gives us
Dialogue: 0,0:08:16.22,0:08:19.33,Default,,0000,0000,0000,,some idea of exactly how big a
Dialogue: 0,0:08:19.33,0:08:22.08,Default,,0000,0000,0000,,radiant is. Now.
Dialogue: 0,0:08:23.06,0:08:27.59,Default,,0000,0000,0000,,So far I've written down the\Nword Radian in fall.
Dialogue: 0,0:08:28.30,0:08:33.71,Default,,0000,0000,0000,,But I haven't written down the\Nword degrees in full. Have used
Dialogue: 0,0:08:33.71,0:08:39.12,Default,,0000,0000,0000,,this symbol. This little zero up\Nhere to signify degrees. So what
Dialogue: 0,0:08:39.12,0:08:45.44,Default,,0000,0000,0000,,do we do about radians? How can\Nwe signify Aready? And so if we
Dialogue: 0,0:08:45.44,0:08:50.85,Default,,0000,0000,0000,,write down 1.5 radians, how we\Nable to tell that this is
Dialogue: 0,0:08:50.85,0:08:57.16,Default,,0000,0000,0000,,radians? One way is to put a\Nlittle see up there at the top,
Dialogue: 0,0:08:57.16,0:08:58.52,Default,,0000,0000,0000,,usually in italic.
Dialogue: 0,0:08:58.55,0:09:04.82,Default,,0000,0000,0000,,Print if it's published in a\Nbook, so 1.5 radians. Another
Dialogue: 0,0:09:04.82,0:09:11.66,Default,,0000,0000,0000,,way is to write 1.5 ramps where\NRams is obviously short for
Dialogue: 0,0:09:11.66,0:09:18.03,Default,,0000,0000,0000,,radians. What we've done does\Nnot affect trig ratios. So for
Dialogue: 0,0:09:18.03,0:09:24.39,Default,,0000,0000,0000,,instance, the cosine of π by\Nthree is exactly the same as the
Dialogue: 0,0:09:24.39,0:09:30.26,Default,,0000,0000,0000,,cosine of 60 degrees because the\Ntwo angles are the same, the
Dialogue: 0,0:09:30.26,0:09:37.10,Default,,0000,0000,0000,,tangent of Π by 4 is exactly the\Nsame as the tangent of 45
Dialogue: 0,0:09:37.10,0:09:44.80,Default,,0000,0000,0000,,degrees. And the sign of three\NPi by two is exactly the same
Dialogue: 0,0:09:44.80,0:09:48.12,Default,,0000,0000,0000,,as the sign of 270 degrees.
Dialogue: 0,0:09:48.89,0:09:53.61,Default,,0000,0000,0000,,One thing to notice here is that\Nbecause I've got pies in.
Dialogue: 0,0:09:54.51,0:09:57.34,Default,,0000,0000,0000,,I haven't written down a symbol.
Dialogue: 0,0:09:58.06,0:10:04.27,Default,,0000,0000,0000,,Because one, we've got a pie in\Nthe angle it's taken as read
Dialogue: 0,0:10:04.27,0:10:06.66,Default,,0000,0000,0000,,that these are measured in
Dialogue: 0,0:10:06.66,0:10:13.48,Default,,0000,0000,0000,,radians, so. We've got our\Nformula S equals R Theater,
Dialogue: 0,0:10:13.48,0:10:20.22,Default,,0000,0000,0000,,where S is the arc\Nlength and theater is in
Dialogue: 0,0:10:20.22,0:10:26.93,Default,,0000,0000,0000,,radians. What's the formula for\NS if Vita is in degrees? Let's
Dialogue: 0,0:10:26.93,0:10:30.51,Default,,0000,0000,0000,,just have a look at our circle.
Dialogue: 0,0:10:31.44,0:10:32.10,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:10:33.91,0:10:40.33,Default,,0000,0000,0000,,Radius R.\NOur arc length S is around the
Dialogue: 0,0:10:40.33,0:10:41.49,Default,,0000,0000,0000,,circle like so.
Dialogue: 0,0:10:43.46,0:10:48.79,Default,,0000,0000,0000,,Radius R and this angle\Nin here let's market as
Dialogue: 0,0:10:48.79,0:10:50.39,Default,,0000,0000,0000,,being theater degrees.
Dialogue: 0,0:10:51.91,0:10:58.89,Default,,0000,0000,0000,,So what's our formula? Well,\NS must be a certain
Dialogue: 0,0:10:58.89,0:11:01.68,Default,,0000,0000,0000,,proportion of the whole
Dialogue: 0,0:11:01.68,0:11:09.48,Default,,0000,0000,0000,,circumference. S over 2π, R\Nand the angle theater will be
Dialogue: 0,0:11:09.48,0:11:16.44,Default,,0000,0000,0000,,exactly the same proportion of\N360, and so we can calculate
Dialogue: 0,0:11:16.44,0:11:23.41,Default,,0000,0000,0000,,the arc length S as 2π\NR that's multiplying both sides
Dialogue: 0,0:11:23.41,0:11:27.20,Default,,0000,0000,0000,,by 2π. R times Theta over
Dialogue: 0,0:11:27.20,0:11:34.18,Default,,0000,0000,0000,,360. So there we've got our\Narc length formula for Theta in
Dialogue: 0,0:11:34.18,0:11:39.29,Default,,0000,0000,0000,,radians. And here's our arc\Nlength formula for theater in
Dialogue: 0,0:11:39.29,0:11:44.40,Default,,0000,0000,0000,,degrees. Notice how much\Nsimplier this one is? What about
Dialogue: 0,0:11:44.40,0:11:49.00,Default,,0000,0000,0000,,another formula? What about the\Narea of this sector?
Dialogue: 0,0:11:49.01,0:11:52.74,Default,,0000,0000,0000,,Well, let's have a look at
Dialogue: 0,0:11:52.74,0:11:57.03,Default,,0000,0000,0000,,that. Is our circle\Nagain?
Dialogue: 0,0:11:58.14,0:12:05.42,Default,,0000,0000,0000,,Radius R.\NAngle
Dialogue: 0,0:12:05.42,0:12:11.80,Default,,0000,0000,0000,,theater\NIt's called the center of
Dialogue: 0,0:12:11.80,0:12:17.04,Default,,0000,0000,0000,,the circle. Oh, and let's\Nlabel the two points A&B.
Dialogue: 0,0:12:17.04,0:12:21.74,Default,,0000,0000,0000,,What's the area of this\Nsector of the circle?
Dialogue: 0,0:12:21.74,0:12:26.97,Default,,0000,0000,0000,,This slice of the pie, so\Nto speak. What's its
Dialogue: 0,0:12:26.97,0:12:29.06,Default,,0000,0000,0000,,area? Well, the area?
Dialogue: 0,0:12:31.56,0:12:33.66,Default,,0000,0000,0000,,Of the sector.
Dialogue: 0,0:12:35.37,0:12:40.30,Default,,0000,0000,0000,,As a proportion, or a fraction\Nof the area.
Dialogue: 0,0:12:40.83,0:12:42.56,Default,,0000,0000,0000,,Of the circle.
Dialogue: 0,0:12:43.28,0:12:50.55,Default,,0000,0000,0000,,Must be the same fraction,\Nthe same proportion as the
Dialogue: 0,0:12:50.55,0:12:53.46,Default,,0000,0000,0000,,angle is of 2π.
Dialogue: 0,0:12:54.05,0:12:59.75,Default,,0000,0000,0000,,All the way around. So the\Narea of the sector considered
Dialogue: 0,0:12:59.75,0:13:05.96,Default,,0000,0000,0000,,as a fraction of the whole\Ncircle must be in the same
Dialogue: 0,0:13:05.96,0:13:08.04,Default,,0000,0000,0000,,ratio as the angle.
Dialogue: 0,0:13:09.83,0:13:15.22,Default,,0000,0000,0000,,Considered as a fraction of the\Nwhole angle all the way round in
Dialogue: 0,0:13:15.22,0:13:20.40,Default,,0000,0000,0000,,radians. Well, we can work\Nthis out area.
Dialogue: 0,0:13:21.03,0:13:23.31,Default,,0000,0000,0000,,Of the sector.
Dialogue: 0,0:13:23.96,0:13:28.74,Default,,0000,0000,0000,,And we have a formula for the\Narea of the circle. We know
Dialogue: 0,0:13:28.74,0:13:30.22,Default,,0000,0000,0000,,that's π R squared.
Dialogue: 0,0:13:30.77,0:13:36.83,Default,,0000,0000,0000,,Equals feta over 2π, And if we\Nmultiply it by this expression
Dialogue: 0,0:13:36.83,0:13:43.40,Default,,0000,0000,0000,,here, the Pi R-squared the area\Nof the circle. Then we have the
Dialogue: 0,0:13:43.40,0:13:45.42,Default,,0000,0000,0000,,area of the sector.
Dialogue: 0,0:13:46.50,0:13:53.10,Default,,0000,0000,0000,,Is equal to π\NR squared times theater
Dialogue: 0,0:13:53.10,0:13:56.40,Default,,0000,0000,0000,,over 2π and the
Dialogue: 0,0:13:56.40,0:14:00.14,Default,,0000,0000,0000,,pi's cancel. That leaves me with
Dialogue: 0,0:14:00.14,0:14:02.91,Default,,0000,0000,0000,,a half. All
Dialogue: 0,0:14:02.91,0:14:09.21,Default,,0000,0000,0000,,squared Theta. So for\Ntheater in radians this is our
Dialogue: 0,0:14:09.21,0:14:13.67,Default,,0000,0000,0000,,formula. The area of a sector is\N1/2 R-squared Theta.
Dialogue: 0,0:14:14.36,0:14:18.27,Default,,0000,0000,0000,,OK, we've got some formula here.\NWe've got some relationships.
Dialogue: 0,0:14:18.27,0:14:23.35,Default,,0000,0000,0000,,Let's have a look at how we can\Nuse them to do some
Dialogue: 0,0:14:23.35,0:14:26.52,Default,,0000,0000,0000,,calculations. So in our first
Dialogue: 0,0:14:26.52,0:14:29.55,Default,,0000,0000,0000,,example. Let's take a circle.
Dialogue: 0,0:14:31.84,0:14:36.60,Default,,0000,0000,0000,,Of radius. 10\Nunits
Dialogue: 0,0:14:37.73,0:14:40.15,Default,,0000,0000,0000,,And let's take an arc length.
Dialogue: 0,0:14:40.83,0:14:44.12,Default,,0000,0000,0000,,Of 25 units.
Dialogue: 0,0:14:45.39,0:14:51.16,Default,,0000,0000,0000,,Now if I make this a little more\Nrepresentative, 25 more likely
Dialogue: 0,0:14:51.16,0:14:57.42,Default,,0000,0000,0000,,to be around there 10 and 10,\Nand the question is what's this
Dialogue: 0,0:14:57.42,0:15:03.19,Default,,0000,0000,0000,,angle here in terms of radians\Nand also in terms of degrees?
Dialogue: 0,0:15:03.79,0:15:10.55,Default,,0000,0000,0000,,So. What's our formula\Nfor Arclength? That's arc length
Dialogue: 0,0:15:10.55,0:15:17.03,Default,,0000,0000,0000,,equals are times by theater. If\Ntheater is in radians.
Dialogue: 0,0:15:17.67,0:15:24.67,Default,,0000,0000,0000,,So as we know\Nto be 25.
Dialogue: 0,0:15:24.95,0:15:32.14,Default,,0000,0000,0000,,We know to be 10 times\Nby theater, so theater is 25
Dialogue: 0,0:15:32.14,0:15:39.33,Default,,0000,0000,0000,,over 10 which is 2.5 and\NI can put a little see
Dialogue: 0,0:15:39.33,0:15:45.92,Default,,0000,0000,0000,,up there to stand for\Nradians. Or I could write it
Dialogue: 0,0:15:45.92,0:15:47.71,Default,,0000,0000,0000,,as 2.5 rats.
Dialogue: 0,0:15:48.96,0:15:55.86,Default,,0000,0000,0000,,So that's our angle in terms\Nof radians. What will our angle
Dialogue: 0,0:15:55.86,0:16:02.76,Default,,0000,0000,0000,,be in terms of degrees? Well,\None way might be to convert
Dialogue: 0,0:16:02.76,0:16:08.51,Default,,0000,0000,0000,,this answer here. These two\Npoint 5 radians into degrees.
Dialogue: 0,0:16:08.51,0:16:11.96,Default,,0000,0000,0000,,Let's remember that we had Pi
Dialogue: 0,0:16:11.96,0:16:14.83,Default,,0000,0000,0000,,radians. Was
Dialogue: 0,0:16:14.83,0:16:21.60,Default,,0000,0000,0000,,180 degrees.\NSo one Radian
Dialogue: 0,0:16:21.60,0:16:28.41,Default,,0000,0000,0000,,was 180 /\Nπ degrees, so
Dialogue: 0,0:16:28.41,0:16:30.69,Default,,0000,0000,0000,,2.5 radians.
Dialogue: 0,0:16:31.74,0:16:39.47,Default,,0000,0000,0000,,Must be 180\Nover π times
Dialogue: 0,0:16:39.47,0:16:46.96,Default,,0000,0000,0000,,by 2.5. And\Nwe can use a Calculator to work
Dialogue: 0,0:16:46.96,0:16:50.05,Default,,0000,0000,0000,,this out. And the answer is
Dialogue: 0,0:16:50.05,0:16:55.88,Default,,0000,0000,0000,,143.2 degrees. Now let's have a\Nlook at another example.
Dialogue: 0,0:16:56.95,0:17:00.99,Default,,0000,0000,0000,,Start from the same point. That\Nis, we've got a circle.
Dialogue: 0,0:17:02.94,0:17:08.30,Default,,0000,0000,0000,,We've got circle of radius 10\Ncentimeters, and this time let's
Dialogue: 0,0:17:08.30,0:17:13.65,Default,,0000,0000,0000,,say we're going to take an arc\Nlength of 15 centimeters.
Dialogue: 0,0:17:14.29,0:17:21.61,Default,,0000,0000,0000,,The question is, what is the\Nangle in radians? Here at the
Dialogue: 0,0:17:21.61,0:17:28.32,Default,,0000,0000,0000,,center, but also, what's the\Narea of this sector oab? So
Dialogue: 0,0:17:28.32,0:17:35.64,Default,,0000,0000,0000,,let's begin with our formula for\Narc length S equals R Theater.
Dialogue: 0,0:17:36.35,0:17:43.55,Default,,0000,0000,0000,,Our arc length is 15 hour,\Nradius is 10 and the angle
Dialogue: 0,0:17:43.55,0:17:50.75,Default,,0000,0000,0000,,is theater. So we divide both\Nsides by 10 to give us
Dialogue: 0,0:17:50.75,0:17:56.15,Default,,0000,0000,0000,,theater is 15 over 10, which\Nis 1.5 radians.
Dialogue: 0,0:17:56.84,0:18:01.10,Default,,0000,0000,0000,,Now area of the sector. We have\Na formula for that.
Dialogue: 0,0:18:02.25,0:18:08.87,Default,,0000,0000,0000,,Area of sector. We\Nknow that that is
Dialogue: 0,0:18:08.87,0:18:11.36,Default,,0000,0000,0000,,1/2 R-squared Theta.
Dialogue: 0,0:18:12.23,0:18:19.65,Default,,0000,0000,0000,,So we can take our values.\NWe know that R is 10,
Dialogue: 0,0:18:19.65,0:18:26.44,Default,,0000,0000,0000,,so that's 10 squared for the\NR-squared, times by angle we've
Dialogue: 0,0:18:26.44,0:18:28.30,Default,,0000,0000,0000,,just calculated 1.5.
Dialogue: 0,0:18:28.90,0:18:36.05,Default,,0000,0000,0000,,So that's 1/2 times by 100\Ntimes by 1.5 now 100 times
Dialogue: 0,0:18:36.05,0:18:43.20,Default,,0000,0000,0000,,by 1.5 is 150 and we\Nwant half of it, so that
Dialogue: 0,0:18:43.20,0:18:49.76,Default,,0000,0000,0000,,is 75 centimeters squared. So\Nthat's the area of this sector.
Dialogue: 0,0:18:51.21,0:18:56.88,Default,,0000,0000,0000,,Further questions we could ask\Nis what's the area of the
Dialogue: 0,0:18:56.88,0:19:03.06,Default,,0000,0000,0000,,segment the minor segment? If\Nyou take a line across a circle,
Dialogue: 0,0:19:03.06,0:19:09.24,Default,,0000,0000,0000,,it divides a circle into two\Nparts, a big bit, the major
Dialogue: 0,0:19:09.24,0:19:14.90,Default,,0000,0000,0000,,segment and a smaller piece.\NThey minus segment, and we can
Dialogue: 0,0:19:14.90,0:19:17.99,Default,,0000,0000,0000,,ask what's the area of that
Dialogue: 0,0:19:17.99,0:19:23.23,Default,,0000,0000,0000,,minor segment? Let's just draw\Nanother diagram and look at that
Dialogue: 0,0:19:23.23,0:19:28.39,Default,,0000,0000,0000,,again, 'cause it was quite a lot\Nof vocabulary in that quite a
Dialogue: 0,0:19:28.39,0:19:33.56,Default,,0000,0000,0000,,lot of words. So we're going to\Ntake a line across the circle.
Dialogue: 0,0:19:33.56,0:19:38.72,Default,,0000,0000,0000,,We're keeping this as 50. We've\Ngot the center of the circle and
Dialogue: 0,0:19:38.72,0:19:44.27,Default,,0000,0000,0000,,we know what the angle is, so\Nthere's our as are we know that
Dialogue: 0,0:19:44.27,0:19:45.46,Default,,0000,0000,0000,,to be 10.
Dialogue: 0,0:19:46.17,0:19:50.43,Default,,0000,0000,0000,,And we've calculated what this\Nangle is. So here's the line
Dialogue: 0,0:19:50.43,0:19:54.68,Default,,0000,0000,0000,,that we drew across the circle,\Nand it divides circling two
Dialogue: 0,0:19:54.68,0:19:59.33,Default,,0000,0000,0000,,parts of big part like that\Nwhich we call the major segment
Dialogue: 0,0:19:59.33,0:20:03.97,Default,,0000,0000,0000,,and a small part like that which\Nwe call the minor segment.
Dialogue: 0,0:20:04.51,0:20:11.73,Default,,0000,0000,0000,,The question is, what's the area\Nof this minor segment? How can
Dialogue: 0,0:20:11.73,0:20:18.96,Default,,0000,0000,0000,,we work it out? Well, we\Nknow the area of the sector
Dialogue: 0,0:20:18.96,0:20:21.37,Default,,0000,0000,0000,,of the circle area.
Dialogue: 0,0:20:21.53,0:20:28.53,Default,,0000,0000,0000,,All sector 'cause we calculated\Nthat and that was 75
Dialogue: 0,0:20:28.53,0:20:35.91,Default,,0000,0000,0000,,centimeters squared. What we\Nneed to be able to do is
Dialogue: 0,0:20:35.91,0:20:38.71,Default,,0000,0000,0000,,workout the area of this
Dialogue: 0,0:20:38.71,0:20:46.23,Default,,0000,0000,0000,,triangle. Area of the triangle\NAOB. The area of the
Dialogue: 0,0:20:46.23,0:20:49.32,Default,,0000,0000,0000,,triangle will be 1/2.
Dialogue: 0,0:20:49.86,0:20:53.85,Default,,0000,0000,0000,,All squared sine theater.
Dialogue: 0,0:20:55.42,0:21:02.49,Default,,0000,0000,0000,,So that's 1/2 times by 10\Nsquared times by the sign of 1.5
Dialogue: 0,0:21:02.49,0:21:09.02,Default,,0000,0000,0000,,radians. And again we can work\Nthis out on a Calculator. We
Dialogue: 0,0:21:09.02,0:21:16.64,Default,,0000,0000,0000,,have to be very careful that are\NCalculator a set up to work in
Dialogue: 0,0:21:16.64,0:21:23.71,Default,,0000,0000,0000,,radians, not in degrees, to work\Nin radians. And so here we have
Dialogue: 0,0:21:23.71,0:21:25.88,Default,,0000,0000,0000,,1/2 times by 100.
Dialogue: 0,0:21:25.93,0:21:33.21,Default,,0000,0000,0000,,Times by and the sign of this\Nangle is nought. .997 and then
Dialogue: 0,0:21:33.21,0:21:40.49,Default,,0000,0000,0000,,lots of decimal places. And if\Nwe work that out on a Calculator
Dialogue: 0,0:21:40.49,0:21:47.77,Default,,0000,0000,0000,,you get 49.8747 and so our final\Nanswer, which we want is the
Dialogue: 0,0:21:47.77,0:21:50.01,Default,,0000,0000,0000,,shaded area here area.
Dialogue: 0,0:21:50.02,0:21:53.29,Default,,0000,0000,0000,,All the minor.
Dialogue: 0,0:21:54.08,0:21:59.87,Default,,0000,0000,0000,,Segment. And that's\Ngot to be the difference between
Dialogue: 0,0:21:59.87,0:22:01.65,Default,,0000,0000,0000,,the area of the sector.
Dialogue: 0,0:22:02.57,0:22:06.14,Default,,0000,0000,0000,,And the area of the triangle so
Dialogue: 0,0:22:06.14,0:22:12.94,Default,,0000,0000,0000,,that 75. Take away\N49.8747 and to a
Dialogue: 0,0:22:12.94,0:22:19.50,Default,,0000,0000,0000,,suitable degree of accuracy\N2 decimal places. That's
Dialogue: 0,0:22:19.50,0:22:21.95,Default,,0000,0000,0000,,25.13 centimeters squared.
Dialogue: 0,0:22:23.54,0:22:29.02,Default,,0000,0000,0000,,Let's have a look at one final\Nexample. We have converted
Dialogue: 0,0:22:29.02,0:22:34.99,Default,,0000,0000,0000,,radians into degrees, but one of\Nthe things that we haven't done
Dialogue: 0,0:22:34.99,0:22:39.97,Default,,0000,0000,0000,,is to convert an angle that's in\Ndegrees into radians.
Dialogue: 0,0:22:40.11,0:22:47.63,Default,,0000,0000,0000,,So let's take an angle\N120 degrees. What is this
Dialogue: 0,0:22:47.63,0:22:55.15,Default,,0000,0000,0000,,angle in radians? Let's start\Noff with our relationship that
Dialogue: 0,0:22:55.15,0:22:58.91,Default,,0000,0000,0000,,Pi radians is 180 degrees.
Dialogue: 0,0:22:59.43,0:23:06.71,Default,,0000,0000,0000,,Now we were able to say\Nthat one Radian was 180 over
Dialogue: 0,0:23:06.71,0:23:13.100,Default,,0000,0000,0000,,π, dividing both sides by pie.\NSo if we want 1 degree,
Dialogue: 0,0:23:13.100,0:23:21.28,Default,,0000,0000,0000,,then we can divide both sides\Nby 180. So π over 180
Dialogue: 0,0:23:21.28,0:23:28.57,Default,,0000,0000,0000,,radians is equal to 1 degree.\NWe want 120 degrees, so π
Dialogue: 0,0:23:28.57,0:23:35.86,Default,,0000,0000,0000,,over 180. Times 120 is equal\Nto 120 degrees and again we
Dialogue: 0,0:23:35.86,0:23:43.11,Default,,0000,0000,0000,,need a Calculator to work this\Nout. And when we work it
Dialogue: 0,0:23:43.11,0:23:50.36,Default,,0000,0000,0000,,out we get 2.09 radians and\Nthat's to two decimal places. Is
Dialogue: 0,0:23:50.36,0:23:53.38,Default,,0000,0000,0000,,the same as 120 degrees.
Dialogue: 0,0:23:54.86,0:24:00.15,Default,,0000,0000,0000,,Important things to take away\Nfrom this video are this
Dialogue: 0,0:24:00.15,0:24:06.25,Default,,0000,0000,0000,,relationship. Π radians is 180\Ndegrees and the two formally
Dialogue: 0,0:24:06.25,0:24:07.83,Default,,0000,0000,0000,,that we had.
Dialogue: 0,0:24:08.36,0:24:15.03,Default,,0000,0000,0000,,One for the arc length,\Nwhich was S equals R
Dialogue: 0,0:24:15.03,0:24:21.70,Default,,0000,0000,0000,,theater and one for the\Narea of the sector which
Dialogue: 0,0:24:21.70,0:24:28.37,Default,,0000,0000,0000,,was area of the sector\Nis equal to 1/2 R-squared
Dialogue: 0,0:24:28.37,0:24:35.04,Default,,0000,0000,0000,,Theater and both of these\Nto apply when theater is
Dialogue: 0,0:24:35.04,0:24:36.37,Default,,0000,0000,0000,,in radiance.