[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.98,0:00:06.36,Default,,0000,0000,0000,,A triangle. Has\Na unique shape.
Dialogue: 0,0:00:07.34,0:00:12.76,Default,,0000,0000,0000,,Anyone triangle given 3 pieces\Nof information forms a unique
Dialogue: 0,0:00:12.76,0:00:18.18,Default,,0000,0000,0000,,shape. Having said that, there\Nare two exceptions. Let's think
Dialogue: 0,0:00:18.18,0:00:19.81,Default,,0000,0000,0000,,about a triangle.
Dialogue: 0,0:00:21.85,0:00:28.58,Default,,0000,0000,0000,,There are six pieces\Nof information available. Angles
Dialogue: 0,0:00:28.58,0:00:35.18,Default,,0000,0000,0000,,ABC. And the sides, the clay.\NIt'll be a little see notice
Dialogue: 0,0:00:35.18,0:00:40.06,Default,,0000,0000,0000,,that I've labeled the side\Nlittle B that is opposite the
Dialogue: 0,0:00:40.06,0:00:45.37,Default,,0000,0000,0000,,angle be the side that is\Nlittle. A is the one that's
Dialogue: 0,0:00:45.37,0:00:50.24,Default,,0000,0000,0000,,opposite the angle A and the\Nsame the Anglesey Little sees
Dialogue: 0,0:00:50.24,0:00:52.46,Default,,0000,0000,0000,,the side. That's opposite the
Dialogue: 0,0:00:52.46,0:00:58.50,Default,,0000,0000,0000,,Anglesey. Now. If I\Ntake three of these six
Dialogue: 0,0:00:58.50,0:00:59.96,Default,,0000,0000,0000,,pieces of information.
Dialogue: 0,0:01:01.59,0:01:06.36,Default,,0000,0000,0000,,With two exceptions, I will get\Na unique triangle.
Dialogue: 0,0:01:06.96,0:01:11.92,Default,,0000,0000,0000,,Let's just get rid of the\Nexceptions. The first exception
Dialogue: 0,0:01:11.92,0:01:17.87,Default,,0000,0000,0000,,is the three angles. If I have a\Ntriangle with three given
Dialogue: 0,0:01:17.87,0:01:23.71,Default,,0000,0000,0000,,angles. Let's say that one. Then\NI can draw another triangle.
Dialogue: 0,0:01:24.51,0:01:27.87,Default,,0000,0000,0000,,That is the same shape.
Dialogue: 0,0:01:28.55,0:01:35.92,Default,,0000,0000,0000,,Only bigger.\NSo I haven't fixed on a
Dialogue: 0,0:01:35.92,0:01:37.10,Default,,0000,0000,0000,,unique triangle.
Dialogue: 0,0:01:38.37,0:01:44.60,Default,,0000,0000,0000,,There is another way in which\Nthat can happen where we can get
Dialogue: 0,0:01:44.60,0:01:48.43,Default,,0000,0000,0000,,more than one triangle. Suppose\Nwe are given.
Dialogue: 0,0:01:50.38,0:01:53.59,Default,,0000,0000,0000,,That side and an angle.
Dialogue: 0,0:01:54.09,0:01:59.86,Default,,0000,0000,0000,,Here. Got an angle here? Then\Nthe side could go on and on and
Dialogue: 0,0:01:59.86,0:02:00.88,Default,,0000,0000,0000,,on like that.
Dialogue: 0,0:02:01.72,0:02:04.74,Default,,0000,0000,0000,,Say I was given the length of
Dialogue: 0,0:02:04.74,0:02:10.27,Default,,0000,0000,0000,,this side. And say I was given a\Nland that was that long.
Dialogue: 0,0:02:11.06,0:02:15.79,Default,,0000,0000,0000,,While there's also another\Ntriangle up here that has the
Dialogue: 0,0:02:15.79,0:02:22.28,Default,,0000,0000,0000,,same. Length. There and there,\Nand So what I've got there are
Dialogue: 0,0:02:22.28,0:02:27.06,Default,,0000,0000,0000,,two triangles. I've got a big\None and I've got a little one
Dialogue: 0,0:02:27.06,0:02:31.11,Default,,0000,0000,0000,,and I've got those out of three\Npieces of information. The
Dialogue: 0,0:02:31.11,0:02:35.53,Default,,0000,0000,0000,,length of that side. The length\Nof this one, and that angle.
Dialogue: 0,0:02:36.06,0:02:42.63,Default,,0000,0000,0000,,But those are the only two cases\Nwhere if I take any three of
Dialogue: 0,0:02:42.63,0:02:47.32,Default,,0000,0000,0000,,these pieces of information, I\Nwill get a unique triangle.
Dialogue: 0,0:02:47.32,0:02:49.66,Default,,0000,0000,0000,,These are the two exceptions,
Dialogue: 0,0:02:49.66,0:02:54.52,Default,,0000,0000,0000,,OK? Because that means a\Ntriangle is fixed. Once I've got
Dialogue: 0,0:02:54.52,0:02:57.84,Default,,0000,0000,0000,,these three pieces of\Ninformation, it means that if
Dialogue: 0,0:02:57.84,0:03:01.53,Default,,0000,0000,0000,,you given three piece of\Ninformation, you ought to be
Dialogue: 0,0:03:01.53,0:03:05.96,Default,,0000,0000,0000,,able to calculate the rest and\Nwhat we're going to be looking
Dialogue: 0,0:03:05.96,0:03:08.54,Default,,0000,0000,0000,,at is formally that enable us to
Dialogue: 0,0:03:08.54,0:03:16.28,Default,,0000,0000,0000,,do that. So we will begin with\Na set of formula that are
Dialogue: 0,0:03:16.28,0:03:19.46,Default,,0000,0000,0000,,known as the cosine formula.
Dialogue: 0,0:03:19.47,0:03:26.01,Default,,0000,0000,0000,,First one is cause\Na is equal to
Dialogue: 0,0:03:26.01,0:03:32.56,Default,,0000,0000,0000,,B squared plus C\Nsquared minus a squared
Dialogue: 0,0:03:32.56,0:03:35.83,Default,,0000,0000,0000,,all over 2 BC.
Dialogue: 0,0:03:36.66,0:03:41.92,Default,,0000,0000,0000,,So what we've got here at the\Nright hand side as Givens are
Dialogue: 0,0:03:41.92,0:03:44.36,Default,,0000,0000,0000,,the three sides of the triangle.
Dialogue: 0,0:03:45.26,0:03:48.62,Default,,0000,0000,0000,,So we use these formula.
Dialogue: 0,0:03:49.65,0:03:54.55,Default,,0000,0000,0000,,When we've got three sides of a\Ntriangle given to us and they
Dialogue: 0,0:03:54.55,0:03:56.81,Default,,0000,0000,0000,,enable us to workout the angles.
Dialogue: 0,0:03:57.38,0:04:04.67,Default,,0000,0000,0000,,I use that in the plural there,\Nbut I've only written down one
Dialogue: 0,0:04:04.67,0:04:11.40,Default,,0000,0000,0000,,formula. Let's write down the\Nothers cause B is equal to C
Dialogue: 0,0:04:11.40,0:04:14.77,Default,,0000,0000,0000,,squared plus A squared minus B
Dialogue: 0,0:04:14.77,0:04:18.29,Default,,0000,0000,0000,,squared. All over to
Dialogue: 0,0:04:18.29,0:04:24.57,Default,,0000,0000,0000,,see a. And notice\Nthat I've cycled the letters
Dialogue: 0,0:04:24.57,0:04:26.20,Default,,0000,0000,0000,,through the formula.
Dialogue: 0,0:04:29.14,0:04:36.10,Default,,0000,0000,0000,,Calls of a has minus a squared\Nover 2 BC cause of B has minus B
Dialogue: 0,0:04:36.10,0:04:42.19,Default,,0000,0000,0000,,squared over 2 CA, and so we\Nought to be able to predict what
Dialogue: 0,0:04:42.19,0:04:48.28,Default,,0000,0000,0000,,the cause of C's going to be.\NThat will be A squared plus B
Dialogue: 0,0:04:48.28,0:04:50.02,Default,,0000,0000,0000,,squared minus C squared.
Dialogue: 0,0:04:50.61,0:04:56.34,Default,,0000,0000,0000,,All over 2A B. So whilst\Nthey look complicated.
Dialogue: 0,0:04:56.85,0:05:01.06,Default,,0000,0000,0000,,They're very easy to remember,\Nso we use.
Dialogue: 0,0:05:02.12,0:05:06.11,Default,,0000,0000,0000,,When given three sides.
Dialogue: 0,0:05:06.95,0:05:09.05,Default,,0000,0000,0000,,To find
Dialogue: 0,0:05:09.83,0:05:16.10,Default,,0000,0000,0000,,Angles. Now we\Ncan actually rearrange these, so
Dialogue: 0,0:05:16.10,0:05:21.93,Default,,0000,0000,0000,,let's take this first one. Do a\Nlittle bit of algebra.
Dialogue: 0,0:05:22.61,0:05:29.70,Default,,0000,0000,0000,,And see how we can make use\Nof the result to generate some
Dialogue: 0,0:05:29.70,0:05:35.69,Default,,0000,0000,0000,,more equations. So be squared\Nplus C squared minus a squared
Dialogue: 0,0:05:35.69,0:05:43.32,Default,,0000,0000,0000,,all over 2 BC. First of all,\Nwe can multiply up by the two
Dialogue: 0,0:05:43.32,0:05:50.95,Default,,0000,0000,0000,,BC. So we have two BC cause\Na is equal to B squared plus
Dialogue: 0,0:05:50.95,0:05:53.68,Default,,0000,0000,0000,,C squared minus a squared.
Dialogue: 0,0:05:53.71,0:06:00.77,Default,,0000,0000,0000,,And now can add an A squared\Nto each side and take this lump
Dialogue: 0,0:06:00.77,0:06:07.32,Default,,0000,0000,0000,,away from both sides and that\Nwill give me a squared is equal
Dialogue: 0,0:06:07.32,0:06:12.86,Default,,0000,0000,0000,,to B squared plus C squared\Nminus two BC cause a.
Dialogue: 0,0:06:14.08,0:06:17.69,Default,,0000,0000,0000,,That is a formula for getting
Dialogue: 0,0:06:17.69,0:06:20.42,Default,,0000,0000,0000,,the side. A.
Dialogue: 0,0:06:21.16,0:06:27.52,Default,,0000,0000,0000,,What do the other ones look\Nlike? Well, be squared is going
Dialogue: 0,0:06:27.52,0:06:34.41,Default,,0000,0000,0000,,to be equal to C squared plus\Na squared minus, two CA Cosby,
Dialogue: 0,0:06:34.41,0:06:39.18,Default,,0000,0000,0000,,and notice. We've cycled the\Nletters through the formula
Dialogue: 0,0:06:39.18,0:06:46.07,Default,,0000,0000,0000,,again and see squared is going\Nto be equal to A squared plus
Dialogue: 0,0:06:46.07,0:06:49.25,Default,,0000,0000,0000,,B squared minus two AB cause
Dialogue: 0,0:06:49.25,0:06:55.26,Default,,0000,0000,0000,,see. So the cosine formula\Nactually made up of 6 Formula
Dialogue: 0,0:06:55.26,0:06:59.75,Default,,0000,0000,0000,,One or each of the angles.\NThat's three altogether, and
Dialogue: 0,0:06:59.75,0:07:04.69,Default,,0000,0000,0000,,another one for each of the\Nsides. And again, that's another
Dialogue: 0,0:07:04.69,0:07:10.08,Default,,0000,0000,0000,,three. When would we use these\Nformula? What are we being given
Dialogue: 0,0:07:10.08,0:07:15.02,Default,,0000,0000,0000,,on this side? Well, obviously\Nwe're beginning two of the sides
Dialogue: 0,0:07:15.02,0:07:20.40,Default,,0000,0000,0000,,A&B in this case and an angle.\NLet's just repeat the diagram
Dialogue: 0,0:07:20.40,0:07:25.83,Default,,0000,0000,0000,,again. Well, we've got the\Nangle. A angle B Anglesey, and
Dialogue: 0,0:07:25.83,0:07:30.28,Default,,0000,0000,0000,,the labeling little a. It'll be\Na little C.
Dialogue: 0,0:07:31.51,0:07:36.33,Default,,0000,0000,0000,,And let's have a look at this\Nformula. Here were finding, see
Dialogue: 0,0:07:36.33,0:07:38.34,Default,,0000,0000,0000,,when we've been given a.
Dialogue: 0,0:07:38.87,0:07:44.22,Default,,0000,0000,0000,,When we being given B and when\Nwe've been given the Anglesey.
Dialogue: 0,0:07:44.87,0:07:47.82,Default,,0000,0000,0000,,So we're using these formula.
Dialogue: 0,0:07:48.58,0:07:51.94,Default,,0000,0000,0000,,Defined
Dialogue: 0,0:07:51.94,0:07:59.53,Default,,0000,0000,0000,,aside.\NSee, when we're given
Dialogue: 0,0:07:59.53,0:08:03.92,Default,,0000,0000,0000,,two sides.\NAnd
Dialogue: 0,0:08:03.92,0:08:08.62,Default,,0000,0000,0000,,the\Nangle
Dialogue: 0,0:08:08.62,0:08:10.98,Default,,0000,0000,0000,,between.
Dialogue: 0,0:08:12.21,0:08:15.62,Default,,0000,0000,0000,,Or the angle included.
Dialogue: 0,0:08:16.29,0:08:20.05,Default,,0000,0000,0000,,Between. The
Dialogue: 0,0:08:20.05,0:08:26.98,Default,,0000,0000,0000,,two sides. So\Nthose are the six cosine
Dialogue: 0,0:08:26.98,0:08:34.10,Default,,0000,0000,0000,,formula. You only need to\Nlearn two of them, one
Dialogue: 0,0:08:34.10,0:08:36.24,Default,,0000,0000,0000,,for the angle.
Dialogue: 0,0:08:36.99,0:08:42.44,Default,,0000,0000,0000,,One for the side and then just\Ncycle the letters through to
Dialogue: 0,0:08:42.44,0:08:43.80,Default,,0000,0000,0000,,find the others.
Dialogue: 0,0:08:43.80,0:08:49.31,Default,,0000,0000,0000,,Another formula\Nis the
Dialogue: 0,0:08:49.31,0:08:50.69,Default,,0000,0000,0000,,sign.
Dialogue: 0,0:08:51.68,0:08:58.85,Default,,0000,0000,0000,,Formula. The sign\Nformula looks like this a
Dialogue: 0,0:08:58.85,0:09:05.61,Default,,0000,0000,0000,,over sign a is equal\Nto B over sign B
Dialogue: 0,0:09:05.61,0:09:08.99,Default,,0000,0000,0000,,is equal to see over
Dialogue: 0,0:09:08.99,0:09:12.55,Default,,0000,0000,0000,,sign, see. Is equal
Dialogue: 0,0:09:12.55,0:09:15.40,Default,,0000,0000,0000,,to. Two
Dialogue: 0,0:09:15.40,0:09:21.49,Default,,0000,0000,0000,,are. What Earth is our?\NWhere did that suddenly come
Dialogue: 0,0:09:21.49,0:09:26.01,Default,,0000,0000,0000,,from? Well, that's just a\Ncomplete the formula and what
Dialogue: 0,0:09:26.01,0:09:32.34,Default,,0000,0000,0000,,our is equal to R is the radius\Nof the circum circle, so the
Dialogue: 0,0:09:32.34,0:09:38.21,Default,,0000,0000,0000,,Circum Circle is the circle that\Nwe can draw that will go through
Dialogue: 0,0:09:38.21,0:09:39.57,Default,,0000,0000,0000,,all the points.
Dialogue: 0,0:09:40.50,0:09:47.53,Default,,0000,0000,0000,,Of the triangle, and that's\Nour so where are.
Dialogue: 0,0:09:48.04,0:09:51.15,Default,,0000,0000,0000,,Is the
Dialogue: 0,0:09:51.15,0:09:55.66,Default,,0000,0000,0000,,radius.\NOf
Dialogue: 0,0:09:55.66,0:09:57.87,Default,,0000,0000,0000,,the
Dialogue: 0,0:09:57.87,0:10:03.71,Default,,0000,0000,0000,,circum Circle, and\Nthat's the circle that goes
Dialogue: 0,0:10:03.71,0:10:05.97,Default,,0000,0000,0000,,through all the points of the
Dialogue: 0,0:10:05.97,0:10:12.03,Default,,0000,0000,0000,,triangle. Because we can write a\Nover sign a is be over sign BC
Dialogue: 0,0:10:12.03,0:10:17.49,Default,,0000,0000,0000,,over Sciences to our if we just\Nleave off the two are we can
Dialogue: 0,0:10:17.49,0:10:22.95,Default,,0000,0000,0000,,turn that upside down and write\Nit assign a over a equal sign B
Dialogue: 0,0:10:22.95,0:10:29.40,Default,,0000,0000,0000,,over B. Equal sign C over C and\Nwe can use it that way up as
Dialogue: 0,0:10:29.40,0:10:33.96,Default,,0000,0000,0000,,well. And when do we\Nuse this? Well, if we
Dialogue: 0,0:10:33.96,0:10:35.76,Default,,0000,0000,0000,,just look at that bit.
Dialogue: 0,0:10:36.84,0:10:43.62,Default,,0000,0000,0000,,If we need to find one of these\Nfour things, the side a, the
Dialogue: 0,0:10:43.62,0:10:47.00,Default,,0000,0000,0000,,angle a, the side B, or the
Dialogue: 0,0:10:47.00,0:10:52.44,Default,,0000,0000,0000,,angle be. Just need to find one\Nof those four things we've got
Dialogue: 0,0:10:52.44,0:10:56.52,Default,,0000,0000,0000,,to know the three others, so we\Nhave to know two angles.
Dialogue: 0,0:10:57.26,0:11:02.54,Default,,0000,0000,0000,,And the side, and it is the non\Nincluded side. If we look at
Dialogue: 0,0:11:02.54,0:11:08.44,Default,,0000,0000,0000,,this one. And we want to find\None of these four things. Say we
Dialogue: 0,0:11:08.44,0:11:13.83,Default,,0000,0000,0000,,want to find the angle a. We\Nhave to know the two sides and
Dialogue: 0,0:11:13.83,0:11:19.22,Default,,0000,0000,0000,,the angle be and it will be the\Nnon included angle. So we can
Dialogue: 0,0:11:19.22,0:11:23.46,Default,,0000,0000,0000,,use this either to find aside\Ngiven two angles and aside
Dialogue: 0,0:11:23.46,0:11:27.69,Default,,0000,0000,0000,,notices were given two angles,\Nwe actually know all the angles
Dialogue: 0,0:11:27.69,0:11:33.46,Default,,0000,0000,0000,,because the angles of a triangle\Nadd up to 180 or we can use it
Dialogue: 0,0:11:33.46,0:11:39.78,Default,,0000,0000,0000,,given. Two sides and an angle to\Nfind a second angle. Now let's
Dialogue: 0,0:11:39.78,0:11:42.73,Default,,0000,0000,0000,,have a look at some examples.
Dialogue: 0,0:11:43.70,0:11:45.17,Default,,0000,0000,0000,,So we'll take.
Dialogue: 0,0:11:46.15,0:11:53.04,Default,,0000,0000,0000,,An example where we begin\Nwith a is 5.
Dialogue: 0,0:11:53.04,0:11:54.66,Default,,0000,0000,0000,,The is 7.
Dialogue: 0,0:11:55.32,0:12:01.66,Default,,0000,0000,0000,,And see is 10, so we are given\Nall three sides of the triangle
Dialogue: 0,0:12:01.66,0:12:06.64,Default,,0000,0000,0000,,and having been given all three\Nsides of the triangle, what
Dialogue: 0,0:12:06.64,0:12:12.08,Default,,0000,0000,0000,,we've got to do to solve the\Ntriangle is find the three
Dialogue: 0,0:12:12.08,0:12:17.06,Default,,0000,0000,0000,,angles. So that's going to be\Nour cosine formula, so we'll
Dialogue: 0,0:12:17.06,0:12:22.95,Default,,0000,0000,0000,,start with cause a, which is\Ngoing to be B squared plus C
Dialogue: 0,0:12:22.95,0:12:25.22,Default,,0000,0000,0000,,squared minus a squared all
Dialogue: 0,0:12:25.22,0:12:27.22,Default,,0000,0000,0000,,over. 2 BC.
Dialogue: 0,0:12:28.26,0:12:34.96,Default,,0000,0000,0000,,So we can put our\Nnumbers into their B
Dialogue: 0,0:12:34.96,0:12:39.44,Default,,0000,0000,0000,,squared. That's 7\Nsquared plus 10
Dialogue: 0,0:12:39.44,0:12:45.40,Default,,0000,0000,0000,,squared minus 5\Nsquared all over 2 *
Dialogue: 0,0:12:45.40,0:12:47.63,Default,,0000,0000,0000,,7 * 10.
Dialogue: 0,0:12:49.12,0:12:52.46,Default,,0000,0000,0000,,Some settings are
Dialogue: 0,0:12:52.46,0:12:58.33,Default,,0000,0000,0000,,49. 10 squared is\N105 squared is 25.
Dialogue: 0,0:12:58.93,0:13:06.35,Default,,0000,0000,0000,,All over 140 two sons\Nof 14 and times by
Dialogue: 0,0:13:06.35,0:13:13.40,Default,,0000,0000,0000,,10. Arithmetic we've 100\Nadd on 49 takeaway
Dialogue: 0,0:13:13.40,0:13:17.00,Default,,0000,0000,0000,,25. That's 124 over
Dialogue: 0,0:13:17.00,0:13:24.39,Default,,0000,0000,0000,,140. And what we need\Nto do is to work this out and
Dialogue: 0,0:13:24.39,0:13:30.00,Default,,0000,0000,0000,,find out what the angle is.\NThe angle a is the angle
Dialogue: 0,0:13:30.00,0:13:36.55,Default,,0000,0000,0000,,whose cosine is 124 over 140,\Nand for that we need to use a
Dialogue: 0,0:13:36.55,0:13:37.02,Default,,0000,0000,0000,,Calculator.
Dialogue: 0,0:13:38.40,0:13:43.02,Default,,0000,0000,0000,,So let's set a power\NCalculator.
Dialogue: 0,0:13:44.42,0:13:45.62,Default,,0000,0000,0000,,Turn it on.
Dialogue: 0,0:13:46.54,0:13:47.92,Default,,0000,0000,0000,,Choose the right mode.
Dialogue: 0,0:13:48.54,0:13:52.05,Default,,0000,0000,0000,,I mean radians. Normally in\Ndoing these calculations we
Dialogue: 0,0:13:52.05,0:13:55.95,Default,,0000,0000,0000,,would want to have our\NCalculator in degrees, so will
Dialogue: 0,0:13:55.95,0:14:00.24,Default,,0000,0000,0000,,just switch that into degrees.\NNow we can work this out.
Dialogue: 0,0:14:00.87,0:14:04.42,Default,,0000,0000,0000,,We want the angle\Nwhose cosine is.
Dialogue: 0,0:14:05.51,0:14:09.37,Default,,0000,0000,0000,,124 Divided
Dialogue: 0,0:14:09.37,0:14:17.02,Default,,0000,0000,0000,,by 140. And\Nthat is 27.7 degrees
Dialogue: 0,0:14:17.02,0:14:20.94,Default,,0000,0000,0000,,working to one decimal
Dialogue: 0,0:14:20.94,0:14:26.76,Default,,0000,0000,0000,,place. Well, that's one angle of\Nthe triangle. I can go ahead and
Dialogue: 0,0:14:26.76,0:14:30.68,Default,,0000,0000,0000,,use the formula again and find\Nthe second angle of the
Dialogue: 0,0:14:30.68,0:14:34.59,Default,,0000,0000,0000,,triangle, and then I can use\Nthat information to find the
Dialogue: 0,0:14:34.59,0:14:39.22,Default,,0000,0000,0000,,third one by adding the two that\NI know together and taking them
Dialogue: 0,0:14:39.22,0:14:40.65,Default,,0000,0000,0000,,away from 180 degrees.
Dialogue: 0,0:14:41.29,0:14:48.08,Default,,0000,0000,0000,,So now let's take another\Nexample where we need to use a
Dialogue: 0,0:14:48.08,0:14:53.18,Default,,0000,0000,0000,,different set or formally will\Ntake B equals 10.
Dialogue: 0,0:14:53.75,0:15:00.75,Default,,0000,0000,0000,,C equals 5 and the angle\Na equals 120 degrees. Let's
Dialogue: 0,0:15:00.75,0:15:07.74,Default,,0000,0000,0000,,sketch this first of all so\Nwe can see exactly what
Dialogue: 0,0:15:07.74,0:15:12.83,Default,,0000,0000,0000,,information we've got. So here's\Nthe angle a.
Dialogue: 0,0:15:13.71,0:15:15.69,Default,,0000,0000,0000,,120 degrees.
Dialogue: 0,0:15:16.80,0:15:22.18,Default,,0000,0000,0000,,B and C2 angles that\Nwe don't know.
Dialogue: 0,0:15:22.71,0:15:28.06,Default,,0000,0000,0000,,And here this is the side\Nlittleby which is equal to 10.
Dialogue: 0,0:15:28.06,0:15:33.86,Default,,0000,0000,0000,,The side little see which is\Nequal to five. This is the side
Dialogue: 0,0:15:33.86,0:15:37.43,Default,,0000,0000,0000,,that we want to be able to find
Dialogue: 0,0:15:37.43,0:15:41.03,Default,,0000,0000,0000,,little A. Well, we've got
Dialogue: 0,0:15:41.03,0:15:44.24,Default,,0000,0000,0000,,two sides. And the angle between
Dialogue: 0,0:15:44.24,0:15:51.22,Default,,0000,0000,0000,,them. So that suggests to us\Nthat we want to use a squared
Dialogue: 0,0:15:51.22,0:15:57.04,Default,,0000,0000,0000,,equals B squared plus C squared\Nminus two BC cause a because
Dialogue: 0,0:15:57.04,0:16:03.83,Default,,0000,0000,0000,,this is what we're given. We\Nknow be we know CB&C and we know
Dialogue: 0,0:16:03.83,0:16:07.71,Default,,0000,0000,0000,,A and this will help us to find
Dialogue: 0,0:16:07.71,0:16:11.42,Default,,0000,0000,0000,,a. So let's put the numbers
Dialogue: 0,0:16:11.42,0:16:16.76,Default,,0000,0000,0000,,in. B squared B is 10, so\Nthat's 10 squared.
Dialogue: 0,0:16:18.01,0:16:25.63,Default,,0000,0000,0000,,C squared CS 5 so\Nthat's 5 squared minus two
Dialogue: 0,0:16:25.63,0:16:29.44,Default,,0000,0000,0000,,times B which is 10.
Dialogue: 0,0:16:29.67,0:16:32.36,Default,,0000,0000,0000,,Time see which is 5.
Dialogue: 0,0:16:33.26,0:16:36.74,Default,,0000,0000,0000,,Times the cosine of
Dialogue: 0,0:16:36.74,0:16:43.64,Default,,0000,0000,0000,,120 degrees. So\Nthis is 100.
Dialogue: 0,0:16:44.14,0:16:47.45,Default,,0000,0000,0000,,Plus 25 five squared.
Dialogue: 0,0:16:47.96,0:16:53.24,Default,,0000,0000,0000,,Minus 2 * 10 is 20 *\N5 is 100.
Dialogue: 0,0:16:53.91,0:17:00.41,Default,,0000,0000,0000,,Times and the cosine\Nof 120 is minus
Dialogue: 0,0:17:00.41,0:17:07.97,Default,,0000,0000,0000,,nought .5. So this\Nis 100 + 25, that's
Dialogue: 0,0:17:07.97,0:17:15.37,Default,,0000,0000,0000,,125. 100 times by minus\N1/2 is minus 50, but we've got
Dialogue: 0,0:17:15.37,0:17:22.04,Default,,0000,0000,0000,,this minus sign here, so that's\Nminus minus 50 is plus 50
Dialogue: 0,0:17:22.04,0:17:29.27,Default,,0000,0000,0000,,altogether gives us 175. It's a\Nthat we're after, so we need to
Dialogue: 0,0:17:29.27,0:17:37.05,Default,,0000,0000,0000,,take the square root of 175 and\Nthe square root of that is 13.2
Dialogue: 0,0:17:37.05,0:17:40.39,Default,,0000,0000,0000,,three and will give the answer.
Dialogue: 0,0:17:40.40,0:17:43.28,Default,,0000,0000,0000,,The two places of decimals.
Dialogue: 0,0:17:44.04,0:17:47.27,Default,,0000,0000,0000,,So now let's take\Na third example.
Dialogue: 0,0:17:48.47,0:17:55.55,Default,,0000,0000,0000,,And in this case\Nwill have seen this
Dialogue: 0,0:17:55.55,0:17:58.78,Default,,0000,0000,0000,,8. B is
Dialogue: 0,0:17:58.78,0:18:05.75,Default,,0000,0000,0000,,12. And the\Nangle C is 30 degrees.
Dialogue: 0,0:18:06.83,0:18:13.34,Default,,0000,0000,0000,,OK, first we need a sketch.\NWhat information have we been
Dialogue: 0,0:18:13.34,0:18:16.30,Default,,0000,0000,0000,,given? So label are triangle
Dialogue: 0,0:18:16.30,0:18:20.25,Default,,0000,0000,0000,,ABC. Label the sides with clay.
Dialogue: 0,0:18:20.89,0:18:28.20,Default,,0000,0000,0000,,It'll be little C and put the\Ninformation on so see is 30.
Dialogue: 0,0:18:28.78,0:18:35.78,Default,,0000,0000,0000,,Be. 12\NAnd the side little C is 8.
Dialogue: 0,0:18:36.64,0:18:39.94,Default,,0000,0000,0000,,So we've got two sides\Nand an angle.
Dialogue: 0,0:18:41.23,0:18:45.57,Default,,0000,0000,0000,,We don't have the angle included\Nbetween the two sides, so this
Dialogue: 0,0:18:45.57,0:18:47.02,Default,,0000,0000,0000,,is the sine formula.
Dialogue: 0,0:18:47.58,0:18:49.48,Default,,0000,0000,0000,,So remember that.
Dialogue: 0,0:18:50.33,0:18:57.69,Default,,0000,0000,0000,,A over sign a is B over.\NSign B is C over sign. See
Dialogue: 0,0:18:57.69,0:19:05.06,Default,,0000,0000,0000,,or we can use it the other\Nway up. Sign a over A is
Dialogue: 0,0:19:05.06,0:19:11.90,Default,,0000,0000,0000,,signed B over B is sign C.\NOversee now which bits do we
Dialogue: 0,0:19:11.90,0:19:15.42,Default,,0000,0000,0000,,want? Well, we've got bees 12.
Dialogue: 0,0:19:15.98,0:19:18.00,Default,,0000,0000,0000,,So we've got that one.
Dialogue: 0,0:19:18.82,0:19:23.88,Default,,0000,0000,0000,,We've got little C is 8, that's\Nthat one, and we've got the
Dialogue: 0,0:19:23.88,0:19:28.93,Default,,0000,0000,0000,,Anglesey is 30. That's that one.\NSo it looks as though it's this
Dialogue: 0,0:19:28.93,0:19:34.38,Default,,0000,0000,0000,,box that we're going to be using\Nand the angle we're going to be
Dialogue: 0,0:19:34.38,0:19:39.83,Default,,0000,0000,0000,,finding is B, so let's work with\Nthese because the sign B is on
Dialogue: 0,0:19:39.83,0:19:46.67,Default,,0000,0000,0000,,the top. So let's write that\Ndown separately. Sign B over B
Dialogue: 0,0:19:46.67,0:19:53.66,Default,,0000,0000,0000,,is equal to sign C over C,\Nand let's put some numbers in.
Dialogue: 0,0:19:53.66,0:20:01.20,Default,,0000,0000,0000,,This is signs B over 12 is\Nequal to sign of 30 degrees over
Dialogue: 0,0:20:01.20,0:20:08.92,Default,,0000,0000,0000,,8. And so sign\NB is 12 times
Dialogue: 0,0:20:08.92,0:20:12.73,Default,,0000,0000,0000,,sign 30 degrees over
Dialogue: 0,0:20:12.73,0:20:17.18,Default,,0000,0000,0000,,8. That's fairly complicated and\NI could use a Calculator
Dialogue: 0,0:20:17.18,0:20:21.84,Default,,0000,0000,0000,,straight away, but one of the\Nthings that I do recognize here
Dialogue: 0,0:20:21.84,0:20:24.17,Default,,0000,0000,0000,,is that sign 30 is 1/2.
Dialogue: 0,0:20:24.74,0:20:28.34,Default,,0000,0000,0000,,So I've got 12 times by
Dialogue: 0,0:20:28.34,0:20:35.71,Default,,0000,0000,0000,,1/2. And divided by 812\Ntimes by 1/2 is 6
Dialogue: 0,0:20:35.71,0:20:42.92,Default,,0000,0000,0000,,still to be divided by\N8, which gives me 3/4
Dialogue: 0,0:20:42.92,0:20:50.13,Default,,0000,0000,0000,,or nought .75. So my\Nangle that I want be
Dialogue: 0,0:20:50.13,0:20:55.90,Default,,0000,0000,0000,,is the angle who sign\Nis North .75.
Dialogue: 0,0:20:55.92,0:20:59.14,Default,,0000,0000,0000,,So let's bring up the\NCalculator again.
Dialogue: 0,0:21:00.30,0:21:03.95,Default,,0000,0000,0000,,We want the angle who sign is.
Dialogue: 0,0:21:05.24,0:21:12.93,Default,,0000,0000,0000,,Nought .75\NAnd we see that the angle
Dialogue: 0,0:21:12.93,0:21:19.78,Default,,0000,0000,0000,,that we get is 48.6 degrees\Nworking till 1 decimal place.
Dialogue: 0,0:21:20.98,0:21:22.08,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:21:23.70,0:21:27.24,Default,,0000,0000,0000,,There is a potential\Ncomplication here.
Dialogue: 0,0:21:29.50,0:21:34.71,Default,,0000,0000,0000,,Let's go ahead and just have a\Nlook at the possibilities in
Dialogue: 0,0:21:34.71,0:21:36.01,Default,,0000,0000,0000,,this particular question.
Dialogue: 0,0:21:36.75,0:21:44.70,Default,,0000,0000,0000,,And it's to do with\Nthese angles because B needn't
Dialogue: 0,0:21:44.70,0:21:47.88,Default,,0000,0000,0000,,just be 48.6 degrees.
Dialogue: 0,0:21:48.41,0:21:56.40,Default,,0000,0000,0000,,Remember that C is\N30 degrees, and that's
Dialogue: 0,0:21:56.40,0:22:04.39,Default,,0000,0000,0000,,fixed. B is 48.6\Ndegrees or 180 -
Dialogue: 0,0:22:04.39,0:22:12.39,Default,,0000,0000,0000,,48.6 degrees could be\Neither. Both have a
Dialogue: 0,0:22:12.39,0:22:20.10,Default,,0000,0000,0000,,sign. Of North\N.75 so B. Might
Dialogue: 0,0:22:20.10,0:22:28.08,Default,,0000,0000,0000,,be that, or taking\Nthis away from 180
Dialogue: 0,0:22:28.08,0:22:31.07,Default,,0000,0000,0000,,one 131.4 degrees.
Dialogue: 0,0:22:32.00,0:22:38.54,Default,,0000,0000,0000,,Now the question is, what's the\Nother angle? Is it possible to
Dialogue: 0,0:22:38.54,0:22:45.62,Default,,0000,0000,0000,,have an angle a with these sets\Nof figures? Well, in the first
Dialogue: 0,0:22:45.62,0:22:52.71,Default,,0000,0000,0000,,case we can have C is 30\Ndegrees, B is 48.6 degrees, and
Dialogue: 0,0:22:52.71,0:23:00.34,Default,,0000,0000,0000,,the angle a will be 180 minus\Nthe sum of these two. In other
Dialogue: 0,0:23:00.34,0:23:01.98,Default,,0000,0000,0000,,words, minus 78.6.
Dialogue: 0,0:23:02.22,0:23:09.36,Default,,0000,0000,0000,,And so that will be\N101.4 degrees. So yes, we
Dialogue: 0,0:23:09.36,0:23:12.22,Default,,0000,0000,0000,,can have that particular
Dialogue: 0,0:23:12.22,0:23:19.35,Default,,0000,0000,0000,,combination. What about the\Nother combination? See is 30
Dialogue: 0,0:23:19.35,0:23:25.19,Default,,0000,0000,0000,,degrees be this time would\Nbe 131.4 degrees.
Dialogue: 0,0:23:25.71,0:23:32.38,Default,,0000,0000,0000,,And so a would\Nbe equal to 180
Dialogue: 0,0:23:32.38,0:23:39.05,Default,,0000,0000,0000,,minus the sum of\Nthese 261.4. So this
Dialogue: 0,0:23:39.05,0:23:45.73,Default,,0000,0000,0000,,gives us an angle\Nof 18.6 degrees. It's
Dialogue: 0,0:23:45.73,0:23:51.18,Default,,0000,0000,0000,,still possible. And this is the\Ncase that we came across before,
Dialogue: 0,0:23:51.18,0:23:53.07,Default,,0000,0000,0000,,where we've got one side.
Dialogue: 0,0:23:53.81,0:23:56.19,Default,,0000,0000,0000,,Where we've got an angle?
Dialogue: 0,0:23:56.70,0:24:04.33,Default,,0000,0000,0000,,And where it's possible for the\Nother side to meet twice, once
Dialogue: 0,0:24:04.33,0:24:07.97,Default,,0000,0000,0000,,there. And once there and still.
Dialogue: 0,0:24:08.55,0:24:14.88,Default,,0000,0000,0000,,Produce. A triangle that works.\NWhat this means is that we know
Dialogue: 0,0:24:14.88,0:24:19.85,Default,,0000,0000,0000,,this side. We know that one and\Nthat one 'cause they're the
Dialogue: 0,0:24:19.85,0:24:25.23,Default,,0000,0000,0000,,same, so we would have two sides\Nto find, one for the smaller
Dialogue: 0,0:24:25.23,0:24:28.13,Default,,0000,0000,0000,,angle A and one for the larger
Dialogue: 0,0:24:28.13,0:24:34.42,Default,,0000,0000,0000,,angle a. A difficult one, but\Nwe do have two distinct
Dialogue: 0,0:24:34.42,0:24:37.56,Default,,0000,0000,0000,,triangles from the same set of
Dialogue: 0,0:24:37.56,0:24:42.25,Default,,0000,0000,0000,,information. OK, we've dealt\Nwith the sign formula. We've
Dialogue: 0,0:24:42.25,0:24:47.64,Default,,0000,0000,0000,,dealt with the cosine formula.\NWhat we want to have a look at
Dialogue: 0,0:24:47.64,0:24:53.46,Default,,0000,0000,0000,,now is just the set of formula\Nwhich will give us the area of
Dialogue: 0,0:24:53.46,0:24:56.36,Default,,0000,0000,0000,,a triangle. Let's just draw a\Ntriangle.
Dialogue: 0,0:24:59.24,0:25:06.20,Default,,0000,0000,0000,,Most people are happy with the\Nidea that the area of a triangle
Dialogue: 0,0:25:06.20,0:25:11.54,Default,,0000,0000,0000,,is 1/2 times by the base times\Nby the height.
Dialogue: 0,0:25:12.35,0:25:16.73,Default,,0000,0000,0000,,What does that mean in this\Ntriangle? Well, it's this way
Dialogue: 0,0:25:16.73,0:25:21.90,Default,,0000,0000,0000,,up, so to speak. This is the\Nbottom of the triangle, so this
Dialogue: 0,0:25:21.90,0:25:26.68,Default,,0000,0000,0000,,is, let's say the base. What's\Nthe height? The height is the
Dialogue: 0,0:25:26.68,0:25:31.45,Default,,0000,0000,0000,,distance of the highest point\Nfrom the base, and in this case
Dialogue: 0,0:25:31.45,0:25:35.43,Default,,0000,0000,0000,,we mean the perpendicular\Ndistance so that that line meets
Dialogue: 0,0:25:35.43,0:25:39.81,Default,,0000,0000,0000,,the base at right angles, and\Nthen this is the height.
Dialogue: 0,0:25:40.69,0:25:44.14,Default,,0000,0000,0000,,What if we are given?
Dialogue: 0,0:25:45.00,0:25:50.72,Default,,0000,0000,0000,,Information about this triangle.\NSo let's label it in the same
Dialogue: 0,0:25:50.72,0:25:53.32,Default,,0000,0000,0000,,way as we did before.
Dialogue: 0,0:25:54.66,0:26:00.73,Default,,0000,0000,0000,,Now this means the base in my\Npicture is the side little A.
Dialogue: 0,0:26:01.37,0:26:04.27,Default,,0000,0000,0000,,That would be the side\Nlittleby and that will
Dialogue: 0,0:26:04.27,0:26:06.20,Default,,0000,0000,0000,,be the side it will see.
Dialogue: 0,0:26:07.49,0:26:12.96,Default,,0000,0000,0000,,Let's look at this right\Nangle triangle.
Dialogue: 0,0:26:16.08,0:26:19.15,Default,,0000,0000,0000,,Here the hypotenuse is, see.
Dialogue: 0,0:26:19.69,0:26:22.24,Default,,0000,0000,0000,,The thing that I've labeled the
Dialogue: 0,0:26:22.24,0:26:26.62,Default,,0000,0000,0000,,height. Is the side that is\Nopposite to the angle be?
Dialogue: 0,0:26:27.79,0:26:35.16,Default,,0000,0000,0000,,Let's assume that I know the\Nangle B and I know the side.
Dialogue: 0,0:26:35.93,0:26:39.65,Default,,0000,0000,0000,,Little C. Then in this right
Dialogue: 0,0:26:39.65,0:26:42.95,Default,,0000,0000,0000,,angle triangle. The
Dialogue: 0,0:26:42.95,0:26:49.67,Default,,0000,0000,0000,,height. Divided by the\Nhypotenuse, C is equal to or.
Dialogue: 0,0:26:49.67,0:26:56.42,Default,,0000,0000,0000,,Remember, height is the opposite\Nside and so that is going to
Dialogue: 0,0:26:56.42,0:27:04.30,Default,,0000,0000,0000,,be sine be. So what I have\Nthere is that the height of this
Dialogue: 0,0:27:04.30,0:27:07.12,Default,,0000,0000,0000,,triangle is C sign be.
Dialogue: 0,0:27:08.40,0:27:15.98,Default,,0000,0000,0000,,The base is little a, so\NI have the area is 1/2.
Dialogue: 0,0:27:17.29,0:27:21.21,Default,,0000,0000,0000,,AC sign be.
Dialogue: 0,0:27:22.14,0:27:25.74,Default,,0000,0000,0000,,At reasonable to ask, since this
Dialogue: 0,0:27:25.74,0:27:30.25,Default,,0000,0000,0000,,formula involves. 3 pieces of\Ninformation. Two sides in the
Dialogue: 0,0:27:30.25,0:27:34.04,Default,,0000,0000,0000,,angle. Can I cycle through\Nagain? Can I cycle these letters
Dialogue: 0,0:27:34.04,0:27:37.48,Default,,0000,0000,0000,,through? Well, let's have a look\Nover at this side.
Dialogue: 0,0:27:38.15,0:27:42.48,Default,,0000,0000,0000,,And again, we see that the\Nheight is the opposite.
Dialogue: 0,0:27:43.20,0:27:50.26,Default,,0000,0000,0000,,To Angle C&B forms the\Nhypotenuse, and so I can
Dialogue: 0,0:27:50.26,0:27:56.61,Default,,0000,0000,0000,,have the area is 1/2\Ntimes the base A.
Dialogue: 0,0:27:57.17,0:28:02.99,Default,,0000,0000,0000,,Times B sign. See because that's\Nwhat the height is in this right
Dialogue: 0,0:28:02.99,0:28:07.47,Default,,0000,0000,0000,,angle triangle. It's not too\Ndifficult to see that the
Dialogue: 0,0:28:07.47,0:28:14.19,Default,,0000,0000,0000,,remaining one is going to be 1/2\NBC sign A and so we have 3
Dialogue: 0,0:28:14.19,0:28:20.02,Default,,0000,0000,0000,,formula that give us the area of\Na triangle. Again we need only
Dialogue: 0,0:28:20.02,0:28:21.81,Default,,0000,0000,0000,,learn one of them.
Dialogue: 0,0:28:22.34,0:28:26.68,Default,,0000,0000,0000,,Because we get the other\Nsimply by cycling through
Dialogue: 0,0:28:26.68,0:28:28.12,Default,,0000,0000,0000,,the various letters.
Dialogue: 0,0:28:29.93,0:28:33.48,Default,,0000,0000,0000,,So the area of that
Dialogue: 0,0:28:33.48,0:28:38.86,Default,,0000,0000,0000,,particular triangle.\NABC
Dialogue: 0,0:28:38.86,0:28:42.31,Default,,0000,0000,0000,,Angles
Dialogue: 0,0:28:42.31,0:28:50.16,Default,,0000,0000,0000,,AB&C.\NArea formula that we had
Dialogue: 0,0:28:50.16,0:28:53.72,Default,,0000,0000,0000,,were a half a B
Dialogue: 0,0:28:53.72,0:29:00.35,Default,,0000,0000,0000,,sign, see. And a\Nhalf BC sign a.
Dialogue: 0,0:29:00.35,0:29:07.13,Default,,0000,0000,0000,,And a half see a sign\NB. Let's check what information
Dialogue: 0,0:29:07.13,0:29:08.97,Default,,0000,0000,0000,,we've got here.
Dialogue: 0,0:29:09.74,0:29:17.74,Default,,0000,0000,0000,,AB sign CAB the angle\Nsee so again it's two
Dialogue: 0,0:29:17.74,0:29:21.74,Default,,0000,0000,0000,,sides and the angle between
Dialogue: 0,0:29:21.74,0:29:27.69,Default,,0000,0000,0000,,two sides.\NAnd the
Dialogue: 0,0:29:27.69,0:29:32.65,Default,,0000,0000,0000,,included.\NAngle.
Dialogue: 0,0:29:33.79,0:29:38.40,Default,,0000,0000,0000,,We don't always get that sort of\Ninformation, though. One of the
Dialogue: 0,0:29:38.40,0:29:44.16,Default,,0000,0000,0000,,things that we do know is we can\Nbe given all three sides of a
Dialogue: 0,0:29:44.16,0:29:47.61,Default,,0000,0000,0000,,triangle. What then? Well, an\Nancient Greek, wouldn't, you
Dialogue: 0,0:29:47.61,0:29:52.22,Default,,0000,0000,0000,,know, by the name of hero?\NAccording to some texts or her
Dialogue: 0,0:29:52.22,0:29:53.76,Default,,0000,0000,0000,,and according to others?
Dialogue: 0,0:29:54.53,0:29:59.57,Default,,0000,0000,0000,,How to formula for calculating\Nthe area of a triangle when you
Dialogue: 0,0:29:59.57,0:30:04.61,Default,,0000,0000,0000,,know all three sides and his\Nformula goes like this. The area
Dialogue: 0,0:30:04.61,0:30:10.49,Default,,0000,0000,0000,,is the square root of. You can\Ntell it's going to be a big
Dialogue: 0,0:30:10.49,0:30:15.95,Default,,0000,0000,0000,,expression 'cause I put a big\Nbar on that square root sign S
Dialogue: 0,0:30:15.95,0:30:20.99,Default,,0000,0000,0000,,times S minus a Times S Minus B\NTimes X minus C.
Dialogue: 0,0:30:21.64,0:30:26.90,Default,,0000,0000,0000,,What's SAB&C are the lengths of\Nthe sides, but what's S?
Dialogue: 0,0:30:27.45,0:30:34.23,Default,,0000,0000,0000,,Where? S is\Nequal to a plus B
Dialogue: 0,0:30:34.23,0:30:39.74,Default,,0000,0000,0000,,Plus C all over 2.\NThe semi perimeter.
Dialogue: 0,0:30:41.52,0:30:46.02,Default,,0000,0000,0000,,Semi perimeter because apples\Npeople see is the perimeter.
Dialogue: 0,0:30:46.02,0:30:52.02,Default,,0000,0000,0000,,It's the distance all the way\Naround. We divide it by two.
Dialogue: 0,0:30:52.02,0:30:57.52,Default,,0000,0000,0000,,It's the semiperimeter. So this\Nis heroes or herons formula for
Dialogue: 0,0:30:57.52,0:31:04.02,Default,,0000,0000,0000,,finding the area of a triangle.\NSo let's have a look at an
Dialogue: 0,0:31:04.02,0:31:05.52,Default,,0000,0000,0000,,example of each.
Dialogue: 0,0:31:06.24,0:31:09.25,Default,,0000,0000,0000,,So in the first case.
Dialogue: 0,0:31:09.49,0:31:16.50,Default,,0000,0000,0000,,Will take a is 5B is 7 and C\Nis 10 and we're trying to find
Dialogue: 0,0:31:16.50,0:31:22.19,Default,,0000,0000,0000,,the area of a triangle and what\Nwe've been given is the lengths
Dialogue: 0,0:31:22.19,0:31:27.45,Default,,0000,0000,0000,,of the three sides little a\Nlittle bit and little see. So
Dialogue: 0,0:31:27.45,0:31:32.70,Default,,0000,0000,0000,,that means we're going to have\Nto use herons formula. The area
Dialogue: 0,0:31:32.70,0:31:39.71,Default,,0000,0000,0000,,is the square root of S Times S\Nminus a Times S Minus B times S.
Dialogue: 0,0:31:39.76,0:31:42.46,Default,,0000,0000,0000,,Minus see
Dialogue: 0,0:31:42.46,0:31:49.82,Default,,0000,0000,0000,,where. S equals A plus\NB Plus C all over 2, and
Dialogue: 0,0:31:49.82,0:31:53.08,Default,,0000,0000,0000,,that's got to be our first
Dialogue: 0,0:31:53.08,0:32:00.37,Default,,0000,0000,0000,,calculation. So we 5 +\N7 + 10 all over
Dialogue: 0,0:32:00.37,0:32:07.17,Default,,0000,0000,0000,,2, five and Seven is\N12 and 10 is 22
Dialogue: 0,0:32:07.17,0:32:13.97,Default,,0000,0000,0000,,/ 2 gives us 11.\NSo now the area is
Dialogue: 0,0:32:13.97,0:32:20.77,Default,,0000,0000,0000,,equal to the square root\Nof 11 * 11 -
Dialogue: 0,0:32:20.77,0:32:27.82,Default,,0000,0000,0000,,5. Times 11 - 7\N* 11 - 10.
Dialogue: 0,0:32:28.51,0:32:36.22,Default,,0000,0000,0000,,Which is the square root\Nof 11 * 6 *
Dialogue: 0,0:32:36.22,0:32:38.53,Default,,0000,0000,0000,,4 * 1?
Dialogue: 0,0:32:39.05,0:32:46.89,Default,,0000,0000,0000,,Square root of 6 times by 4\Ntimes by one is 24 and what
Dialogue: 0,0:32:46.89,0:32:50.81,Default,,0000,0000,0000,,we need is 24 times by 11.
Dialogue: 0,0:32:50.82,0:32:57.91,Default,,0000,0000,0000,,11 four 44211 is 22 and the four\Ngives us 26 so the area is going
Dialogue: 0,0:32:57.91,0:33:03.67,Default,,0000,0000,0000,,to be the square root of 264 and\Nagain we just need the
Dialogue: 0,0:33:03.67,0:33:06.32,Default,,0000,0000,0000,,Calculator to be able to work
Dialogue: 0,0:33:06.32,0:33:09.09,Default,,0000,0000,0000,,that out. Turn it on.
Dialogue: 0,0:33:10.47,0:33:14.51,Default,,0000,0000,0000,,Get into the right mode and we\Nwant the square root.
Dialogue: 0,0:33:15.23,0:33:18.81,Default,,0000,0000,0000,,Of. 264
Dialogue: 0,0:33:19.95,0:33:27.65,Default,,0000,0000,0000,,And that is 16.223\Nsignificant figures, so the
Dialogue: 0,0:33:27.65,0:33:31.49,Default,,0000,0000,0000,,area is 16.2 square
Dialogue: 0,0:33:31.49,0:33:36.14,Default,,0000,0000,0000,,units. I didn't say what\Nthe units were here for
Dialogue: 0,0:33:36.14,0:33:39.42,Default,,0000,0000,0000,,the lengths of the sides,\Nso these are just units.
Dialogue: 0,0:33:39.42,0:33:41.06,Default,,0000,0000,0000,,Square units for the area.
Dialogue: 0,0:33:42.13,0:33:49.31,Default,,0000,0000,0000,,So now let's have a look at\Nan example using another set of
Dialogue: 0,0:33:49.31,0:33:53.17,Default,,0000,0000,0000,,data. So in this case will take
Dialogue: 0,0:33:53.17,0:33:56.99,Default,,0000,0000,0000,,the 10. And see to
Dialogue: 0,0:33:56.99,0:34:04.05,Default,,0000,0000,0000,,be 5. And the angle A to\Nbe 120 degrees and we want the
Dialogue: 0,0:34:04.05,0:34:09.80,Default,,0000,0000,0000,,area of the triangle. So a quick\Nsketch. Let's just make sure we
Dialogue: 0,0:34:09.80,0:34:16.89,Default,,0000,0000,0000,,know what we've got a BC this is\N120 B we know to be 10 and
Dialogue: 0,0:34:16.89,0:34:23.10,Default,,0000,0000,0000,,Little C we know to be 5. So\Nwe've been given two sides and
Dialogue: 0,0:34:23.10,0:34:27.97,Default,,0000,0000,0000,,the angle between the included\Nangle and so straight away we
Dialogue: 0,0:34:27.97,0:34:35.18,Default,,0000,0000,0000,,know where. All right, to use\Nthe area is 1/2 BC Sign
Dialogue: 0,0:34:35.18,0:34:42.38,Default,,0000,0000,0000,,A. Put the numbers\Ninto the Formula 1/2 *
Dialogue: 0,0:34:42.38,0:34:48.19,Default,,0000,0000,0000,,10 * 5 times sign\Nof 120 degrees.
Dialogue: 0,0:34:48.80,0:34:55.54,Default,,0000,0000,0000,,So 10 times by 5\Nis 50 and a half
Dialogue: 0,0:34:55.54,0:35:01.61,Default,,0000,0000,0000,,is 25 times by the\Nsign of 120 degrees.
Dialogue: 0,0:35:01.61,0:35:05.78,Default,,0000,0000,0000,,And so we need the Calculator\Nagain to help us work this out.
Dialogue: 0,0:35:05.78,0:35:07.71,Default,,0000,0000,0000,,So bring the Calculator up and
Dialogue: 0,0:35:07.71,0:35:14.42,Default,,0000,0000,0000,,turn it on. And get into the\Nright mode and now we need 25.
Dialogue: 0,0:35:14.99,0:35:20.30,Default,,0000,0000,0000,,Times the sign\Nof 120 degrees.
Dialogue: 0,0:35:21.55,0:35:27.42,Default,,0000,0000,0000,,And that gives us an area of\N21.7 working to three
Dialogue: 0,0:35:27.42,0:35:32.76,Default,,0000,0000,0000,,significant figures. And again\Nthis will be 21.7 square units.
Dialogue: 0,0:35:32.76,0:35:38.64,Default,,0000,0000,0000,,I didn't specify what units\Nthese were in. Had they been
Dialogue: 0,0:35:38.64,0:35:42.91,Default,,0000,0000,0000,,centimeters? Then this would be\N21.7 square centimeters.
Dialogue: 0,0:35:43.60,0:35:47.18,Default,,0000,0000,0000,,So now. Let's just sum up what
Dialogue: 0,0:35:47.18,0:35:53.94,Default,,0000,0000,0000,,we've got. We've\Ngot our set
Dialogue: 0,0:35:53.94,0:35:56.48,Default,,0000,0000,0000,,of cosine.
Dialogue: 0,0:35:57.68,0:36:00.26,Default,,0000,0000,0000,,Formally.
Dialogue: 0,0:36:01.71,0:36:09.63,Default,,0000,0000,0000,,One representative is cause a is\NB squared plus C squared minus
Dialogue: 0,0:36:09.63,0:36:17.55,Default,,0000,0000,0000,,a squared all over 2 BC,\Nand there are another two like
Dialogue: 0,0:36:17.55,0:36:22.91,Default,,0000,0000,0000,,that. For the angle B and\Nfor the angle, see and we
Dialogue: 0,0:36:22.91,0:36:26.63,Default,,0000,0000,0000,,know how to generate them by\Ncycling the lettuce through.
Dialogue: 0,0:36:28.45,0:36:35.48,Default,,0000,0000,0000,,We also know that A squared\Nis equal to B squared plus
Dialogue: 0,0:36:35.48,0:36:38.100,Default,,0000,0000,0000,,C squared minus two BC cause
Dialogue: 0,0:36:38.100,0:36:44.15,Default,,0000,0000,0000,,a. And we know that there are\Nanother two like this.
Dialogue: 0,0:36:44.78,0:36:49.48,Default,,0000,0000,0000,,We know how to get them. We\Nsimply cycle the letters through
Dialogue: 0,0:36:49.48,0:36:55.75,Default,,0000,0000,0000,,the formula. This one\Nwe find angles.
Dialogue: 0,0:36:57.47,0:37:01.44,Default,,0000,0000,0000,,Using three
Dialogue: 0,0:37:01.44,0:37:08.86,Default,,0000,0000,0000,,sides. This\None we find aside.
Dialogue: 0,0:37:09.40,0:37:12.28,Default,,0000,0000,0000,,Using two
Dialogue: 0,0:37:12.28,0:37:15.79,Default,,0000,0000,0000,,sides. And
Dialogue: 0,0:37:15.79,0:37:18.61,Default,,0000,0000,0000,,the included.
Dialogue: 0,0:37:19.19,0:37:26.08,Default,,0000,0000,0000,,Angle. We\Nnext half hour sign
Dialogue: 0,0:37:26.08,0:37:26.98,Default,,0000,0000,0000,,formula.
Dialogue: 0,0:37:29.33,0:37:36.80,Default,,0000,0000,0000,,A over sign a is equal to B\Nover sign B is equal to see over
Dialogue: 0,0:37:36.80,0:37:43.34,Default,,0000,0000,0000,,sign C is equal to two R and\Nremember that are was the radius
Dialogue: 0,0:37:43.34,0:37:45.21,Default,,0000,0000,0000,,of the circum circle.
Dialogue: 0,0:37:45.79,0:37:51.32,Default,,0000,0000,0000,,The circle that went through all\Nthree points of our triangle and
Dialogue: 0,0:37:51.32,0:37:58.24,Default,,0000,0000,0000,,we can turn this the other way\Nup and we can say sign a over
Dialogue: 0,0:37:58.24,0:38:01.92,Default,,0000,0000,0000,,a sign B over B is sign, see
Dialogue: 0,0:38:01.92,0:38:07.21,Default,,0000,0000,0000,,oversee. And we use this\Nwhen were given.
Dialogue: 0,0:38:07.21,0:38:14.01,Default,,0000,0000,0000,,Two sides plus an angle, but it\Nmust be the non included angle
Dialogue: 0,0:38:14.01,0:38:17.15,Default,,0000,0000,0000,,or when we're given two angles.
Dialogue: 0,0:38:18.03,0:38:20.19,Default,,0000,0000,0000,,And aside.
Dialogue: 0,0:38:22.51,0:38:26.86,Default,,0000,0000,0000,,So those are our cosine.\NFormally, those are our sign
Dialogue: 0,0:38:26.86,0:38:29.83,Default,,0000,0000,0000,,formula. The area formerly we've
Dialogue: 0,0:38:29.83,0:38:32.69,Default,,0000,0000,0000,,just had. Remember herons
Dialogue: 0,0:38:32.69,0:38:38.61,Default,,0000,0000,0000,,formula? And the other formula\Nfor the area half a B sign. See,
Dialogue: 0,0:38:38.61,0:38:42.98,Default,,0000,0000,0000,,that's just one representative\Nand the others. We cycle the let
Dialogue: 0,0:38:42.98,0:38:48.05,Default,,0000,0000,0000,,us through. And finally, let's\Nsee if we can connect two of
Dialogue: 0,0:38:48.05,0:38:51.47,Default,,0000,0000,0000,,these sets of formally, that\Nwe've just had.
Dialogue: 0,0:38:52.22,0:38:55.29,Default,,0000,0000,0000,,Just draw a
Dialogue: 0,0:38:55.29,0:39:01.76,Default,,0000,0000,0000,,triangle. So that we can\Nrecall the notation. The capital
Dialogue: 0,0:39:01.76,0:39:07.42,Default,,0000,0000,0000,,letters for the angles, little\Nletters on the sides opposite to
Dialogue: 0,0:39:07.42,0:39:12.06,Default,,0000,0000,0000,,the angles for the lengths of\Nthe sides now.
Dialogue: 0,0:39:12.80,0:39:20.21,Default,,0000,0000,0000,,Area formally told us that\Nthe area was equal to
Dialogue: 0,0:39:20.21,0:39:23.26,Default,,0000,0000,0000,,1/2. AB
Dialogue: 0,0:39:23.26,0:39:28.80,Default,,0000,0000,0000,,sign C.\N1/2
Dialogue: 0,0:39:28.80,0:39:35.32,Default,,0000,0000,0000,,BC. Find\Na under half see
Dialogue: 0,0:39:35.32,0:39:37.77,Default,,0000,0000,0000,,a sign be.
Dialogue: 0,0:39:39.18,0:39:43.94,Default,,0000,0000,0000,,Well, let's take two of these\Nand actually put them equal to
Dialogue: 0,0:39:43.94,0:39:48.31,Default,,0000,0000,0000,,each other. After all, they're\Nboth expressions for the area of
Dialogue: 0,0:39:48.31,0:39:53.47,Default,,0000,0000,0000,,this triangle, and so they are\Nin fact equal, so it write them
Dialogue: 0,0:39:53.47,0:39:55.46,Default,,0000,0000,0000,,down a half a bee.
Dialogue: 0,0:39:56.06,0:40:02.96,Default,,0000,0000,0000,,Sign of say is equal\Nto 1/2 BC sign of
Dialogue: 0,0:40:02.96,0:40:07.68,Default,,0000,0000,0000,,A. Will immediately we see,\Nwe've got a common factor on
Dialogue: 0,0:40:07.68,0:40:08.95,Default,,0000,0000,0000,,each side of 1/2.
Dialogue: 0,0:40:09.53,0:40:15.59,Default,,0000,0000,0000,,And a common factor on each side\Nof B so we can cancel those out
Dialogue: 0,0:40:15.59,0:40:21.65,Default,,0000,0000,0000,,on each side. That will leave us\Non this side here with a sign C.
Dialogue: 0,0:40:21.72,0:40:28.57,Default,,0000,0000,0000,,And on this side\Nit leaves us with
Dialogue: 0,0:40:28.57,0:40:31.14,Default,,0000,0000,0000,,C Sign A.
Dialogue: 0,0:40:31.14,0:40:38.05,Default,,0000,0000,0000,,Now if I divide both\Nsides by sign A and
Dialogue: 0,0:40:38.05,0:40:44.96,Default,,0000,0000,0000,,divide both sides by sign,\Nsee, then I have a
Dialogue: 0,0:40:44.96,0:40:47.03,Default,,0000,0000,0000,,over sign a.
Dialogue: 0,0:40:47.65,0:40:51.58,Default,,0000,0000,0000,,Is equal to see over sign, see.
Dialogue: 0,0:40:52.45,0:40:55.44,Default,,0000,0000,0000,,So working with the formula for
Dialogue: 0,0:40:55.44,0:41:01.55,Default,,0000,0000,0000,,the area. We have derived a\Npart of the sine formula.
Dialogue: 0,0:41:02.42,0:41:08.36,Default,,0000,0000,0000,,If I take another two of these,\Nsay these two together, I'll get
Dialogue: 0,0:41:08.36,0:41:14.30,Default,,0000,0000,0000,,another bit of the sign formula.\NSo we see that these two are
Dialogue: 0,0:41:14.30,0:41:19.33,Default,,0000,0000,0000,,related. The area formula and\Nthe sign formula, and we can
Dialogue: 0,0:41:19.33,0:41:23.90,Default,,0000,0000,0000,,derive the sign formally from\Nthe area formula very simply.