[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.33,0:00:05.61,Default,,0000,0000,0000,,There are 6 addition formula.\NI'm not going to prove them.
Dialogue: 0,0:00:06.41,0:00:09.58,Default,,0000,0000,0000,,But I am going to do is start\Nwith one of them.
Dialogue: 0,0:00:10.21,0:00:12.10,Default,,0000,0000,0000,,And derive a second one.
Dialogue: 0,0:00:12.75,0:00:17.44,Default,,0000,0000,0000,,Then I'm going to take another\None is given and derive a second
Dialogue: 0,0:00:17.44,0:00:22.14,Default,,0000,0000,0000,,one from that, and then we're\Ngoing to use those four to help
Dialogue: 0,0:00:22.14,0:00:27.19,Default,,0000,0000,0000,,us derive the final two. So this\Nis the one we're going to start
Dialogue: 0,0:00:27.19,0:00:31.11,Default,,0000,0000,0000,,with. The sign of a
Dialogue: 0,0:00:31.11,0:00:36.36,Default,,0000,0000,0000,,plus B. This is where they get\Ntheir name from addition formula
Dialogue: 0,0:00:36.36,0:00:40.64,Default,,0000,0000,0000,,because here we have a sub and\Nwe're going to find it sign.
Dialogue: 0,0:00:41.26,0:00:48.07,Default,,0000,0000,0000,,This breaks down.\NAssign a Cosby
Dialogue: 0,0:00:48.68,0:00:52.06,Default,,0000,0000,0000,,Plus calls a
Dialogue: 0,0:00:52.06,0:00:58.39,Default,,0000,0000,0000,,sign be. Notice in terms\Nof remembering it, we keep the
Dialogue: 0,0:00:58.39,0:01:03.10,Default,,0000,0000,0000,,A&B in the same order and the\Nsigns and the cosines alternate.
Dialogue: 0,0:01:04.10,0:01:10.64,Default,,0000,0000,0000,,So what about the\Nsign of a minus
Dialogue: 0,0:01:10.64,0:01:17.98,Default,,0000,0000,0000,,B? Well, I just have\Na think about this minus B.
Dialogue: 0,0:01:18.52,0:01:25.82,Default,,0000,0000,0000,,It can be the sign\Nof a plus minus B.
Dialogue: 0,0:01:27.26,0:01:30.86,Default,,0000,0000,0000,,And now. The formula that we
Dialogue: 0,0:01:30.86,0:01:36.10,Default,,0000,0000,0000,,had here. Can be exactly the\Nsame As for this one.
Dialogue: 0,0:01:37.01,0:01:40.40,Default,,0000,0000,0000,,But in this one we can replace
Dialogue: 0,0:01:40.40,0:01:47.52,Default,,0000,0000,0000,,B. By minus be\Nso let's do that sign.
Dialogue: 0,0:01:48.14,0:01:54.50,Default,,0000,0000,0000,,A. Calls\Nof minus B.
Dialogue: 0,0:01:55.13,0:01:58.58,Default,,0000,0000,0000,,Plus calls a.
Dialogue: 0,0:01:59.27,0:02:02.45,Default,,0000,0000,0000,,Sign of minus
Dialogue: 0,0:02:02.45,0:02:09.18,Default,,0000,0000,0000,,B. Well, the\Nsign is OK.
Dialogue: 0,0:02:09.18,0:02:14.39,Default,,0000,0000,0000,,And the cause of minus B is\Njust cause big.
Dialogue: 0,0:02:16.55,0:02:23.16,Default,,0000,0000,0000,,The cause a is OK, but the sign\Nof minus B is minus sign B, so
Dialogue: 0,0:02:23.16,0:02:28.94,Default,,0000,0000,0000,,that's going to change that plus\Nsign into a minus sign and I can
Dialogue: 0,0:02:28.94,0:02:31.83,Default,,0000,0000,0000,,just write sign be at the end.
Dialogue: 0,0:02:32.94,0:02:35.99,Default,,0000,0000,0000,,So that's the second.
Dialogue: 0,0:02:36.94,0:02:40.62,Default,,0000,0000,0000,,Of our addition
Dialogue: 0,0:02:40.62,0:02:46.88,Default,,0000,0000,0000,,formula. If we've got these\Nfor sign, it seemed reasonable
Dialogue: 0,0:02:46.88,0:02:52.18,Default,,0000,0000,0000,,to expect that will have the\Nsame things for cause for
Dialogue: 0,0:02:52.18,0:02:56.52,Default,,0000,0000,0000,,cosine, so let's have a look\Ncause of A+B.
Dialogue: 0,0:02:57.11,0:03:04.09,Default,,0000,0000,0000,,Well, this is cause a\Ncaused B minus sign, a
Dialogue: 0,0:03:04.09,0:03:11.07,Default,,0000,0000,0000,,sign be and that's our\Nstarting one. So let's do
Dialogue: 0,0:03:11.07,0:03:17.35,Default,,0000,0000,0000,,the same as we did\Nhere cause of A-B.
Dialogue: 0,0:03:18.43,0:03:25.55,Default,,0000,0000,0000,,Will rewrite the minus B\Nas a plus minus B.
Dialogue: 0,0:03:26.37,0:03:32.52,Default,,0000,0000,0000,,And then we can replace the bees\Nin here in this first formula
Dialogue: 0,0:03:32.52,0:03:37.25,Default,,0000,0000,0000,,with the minus be there so will\Nhave cause a.
Dialogue: 0,0:03:38.04,0:03:41.41,Default,,0000,0000,0000,,Cause of minus B.
Dialogue: 0,0:03:42.28,0:03:49.18,Default,,0000,0000,0000,,Minus sign, a sign\Nof minus B.
Dialogue: 0,0:03:50.82,0:03:58.25,Default,,0000,0000,0000,,Now the cause of minus B is\Njust cause be, so we have caused
Dialogue: 0,0:03:58.25,0:03:59.85,Default,,0000,0000,0000,,a caused B.
Dialogue: 0,0:04:01.30,0:04:07.18,Default,,0000,0000,0000,,And then we have minus sign a\NTimes by sign of minus B.
Dialogue: 0,0:04:07.93,0:04:13.42,Default,,0000,0000,0000,,But the sign of minus B is a\Nminus sign be, so we have two
Dialogue: 0,0:04:13.42,0:04:17.45,Default,,0000,0000,0000,,minus signs together, giving us\Na plus sign plus sign a.
Dialogue: 0,0:04:17.95,0:04:19.95,Default,,0000,0000,0000,,Sign be.
Dialogue: 0,0:04:21.27,0:04:26.98,Default,,0000,0000,0000,,So we've now got 4 addition\Nformula. Let me turn over and
Dialogue: 0,0:04:26.98,0:04:29.84,Default,,0000,0000,0000,,write those down as a group.
Dialogue: 0,0:04:31.09,0:04:34.51,Default,,0000,0000,0000,,Sign of A+B?
Dialogue: 0,0:04:35.52,0:04:37.84,Default,,0000,0000,0000,,Is sign a?
Dialogue: 0,0:04:38.46,0:04:44.95,Default,,0000,0000,0000,,Call speedy close calls\Na sign be.
Dialogue: 0,0:04:45.70,0:04:49.94,Default,,0000,0000,0000,,Sign of a minus\NB.
Dialogue: 0,0:04:50.99,0:04:57.77,Default,,0000,0000,0000,,Is sign a\NCosby minus cause
Dialogue: 0,0:04:57.77,0:05:01.16,Default,,0000,0000,0000,,a sign be?
Dialogue: 0,0:05:01.16,0:05:04.80,Default,,0000,0000,0000,,Cause of
Dialogue: 0,0:05:04.80,0:05:10.52,Default,,0000,0000,0000,,A+B? Is cause\Na calls B?
Dialogue: 0,0:05:11.05,0:05:14.95,Default,,0000,0000,0000,,Minus sign a
Dialogue: 0,0:05:14.95,0:05:22.50,Default,,0000,0000,0000,,sign B. The\Ncause of A-B is
Dialogue: 0,0:05:22.50,0:05:30.37,Default,,0000,0000,0000,,cause a caused B\Nplus sign a sign.
Dialogue: 0,0:05:31.06,0:05:33.45,Default,,0000,0000,0000,,So there are our four.
Dialogue: 0,0:05:34.23,0:05:38.62,Default,,0000,0000,0000,,Addition formula. Now we did\Npromise 6 but these are the fall
Dialogue: 0,0:05:38.62,0:05:42.65,Default,,0000,0000,0000,,really basic ones. And really\Nthese are the four that you've
Dialogue: 0,0:05:42.65,0:05:43.75,Default,,0000,0000,0000,,got to learn.
Dialogue: 0,0:05:44.45,0:05:47.59,Default,,0000,0000,0000,,The other two will presumably
Dialogue: 0,0:05:47.59,0:05:51.08,Default,,0000,0000,0000,,be. Tangent tan of
Dialogue: 0,0:05:51.08,0:05:58.16,Default,,0000,0000,0000,,A+B? Well, we\Ncan derive these from the
Dialogue: 0,0:05:58.16,0:06:05.83,Default,,0000,0000,0000,,others. Tan is sign over\Ncause so we have the
Dialogue: 0,0:06:05.83,0:06:12.73,Default,,0000,0000,0000,,sign of A+B divided by\Nthe cost of A+B.
Dialogue: 0,0:06:13.77,0:06:18.86,Default,,0000,0000,0000,,So we can now make the\Nreplacement for sign of A+B by
Dialogue: 0,0:06:18.86,0:06:20.13,Default,,0000,0000,0000,,its expansion here.
Dialogue: 0,0:06:20.67,0:06:26.04,Default,,0000,0000,0000,,And cause of A+B by\Nits expansion here?
Dialogue: 0,0:06:26.04,0:06:29.22,Default,,0000,0000,0000,,Sign a caused
Dialogue: 0,0:06:29.22,0:06:33.03,Default,,0000,0000,0000,,B. Close calls a
Dialogue: 0,0:06:33.03,0:06:38.19,Default,,0000,0000,0000,,sign be. All\Nover.
Dialogue: 0,0:06:40.14,0:06:43.56,Default,,0000,0000,0000,,Calls a calls B.
Dialogue: 0,0:06:44.08,0:06:47.47,Default,,0000,0000,0000,,Minus. Sign a
Dialogue: 0,0:06:47.47,0:06:53.66,Default,,0000,0000,0000,,sign be. Now this looks\Nvery unwieldy and it would be
Dialogue: 0,0:06:53.66,0:06:59.01,Default,,0000,0000,0000,,nice if in the same way that\Nthis is in terms of signs and
Dialogue: 0,0:06:59.01,0:07:03.97,Default,,0000,0000,0000,,causes and this ones in terms of\Nsigns and causes. I could have
Dialogue: 0,0:07:03.97,0:07:08.17,Default,,0000,0000,0000,,tan of A+B somehow in terms of\Ntangent and possibly cotangent,
Dialogue: 0,0:07:08.17,0:07:12.76,Default,,0000,0000,0000,,but certainly I want to have\Nsome tangents in there. So what
Dialogue: 0,0:07:12.76,0:07:17.34,Default,,0000,0000,0000,,can I do? Well, look at this\Nterm here cause a Cosby.
Dialogue: 0,0:07:18.01,0:07:24.07,Default,,0000,0000,0000,,Supposing I divided everything\Nby that term cause a Cosby.
Dialogue: 0,0:07:24.60,0:07:29.14,Default,,0000,0000,0000,,Well, that would give me one\Nhere and I'd have this over
Dialogue: 0,0:07:29.14,0:07:34.05,Default,,0000,0000,0000,,cause A cause be of course I'd\Nhave sign over cause sign over
Dialogue: 0,0:07:34.05,0:07:38.96,Default,,0000,0000,0000,,cause for each of A&B so I have\Ntan a tan be there.
Dialogue: 0,0:07:39.99,0:07:47.15,Default,,0000,0000,0000,,Would I get anything nice on the\Ntop? Well, let's write it down
Dialogue: 0,0:07:47.15,0:07:54.48,Default,,0000,0000,0000,,in full. So we're\Ngoing to divide
Dialogue: 0,0:07:54.48,0:07:57.91,Default,,0000,0000,0000,,everything by calls
Dialogue: 0,0:07:57.91,0:08:05.12,Default,,0000,0000,0000,,a Cosby. And\NI have to divide
Dialogue: 0,0:08:05.12,0:08:08.98,Default,,0000,0000,0000,,everything by this because
Dialogue: 0,0:08:08.98,0:08:15.13,Default,,0000,0000,0000,,I've got. A pair of\Nequal signs here a balance. So
Dialogue: 0,0:08:15.13,0:08:21.13,Default,,0000,0000,0000,,what I do to one side I must do\Nthe other. I must do everything
Dialogue: 0,0:08:21.13,0:08:23.93,Default,,0000,0000,0000,,to all of the terms to preserve
Dialogue: 0,0:08:23.93,0:08:30.88,Default,,0000,0000,0000,,the equality. Now that looks\Nabsolutely awful. Absolutely
Dialogue: 0,0:08:30.88,0:08:37.03,Default,,0000,0000,0000,,massive algebra. So\Nhow can we
Dialogue: 0,0:08:37.03,0:08:43.17,Default,,0000,0000,0000,,make it any\Nsimpler? Well, let's
Dialogue: 0,0:08:43.17,0:08:49.32,Default,,0000,0000,0000,,just go back\Nto the denominator
Dialogue: 0,0:08:49.32,0:08:52.39,Default,,0000,0000,0000,,here. This bottom
Dialogue: 0,0:08:52.39,0:08:59.61,Default,,0000,0000,0000,,term. Cause a over cause\Na cancels down Cosby over. Cosby
Dialogue: 0,0:08:59.61,0:09:02.98,Default,,0000,0000,0000,,cancels down. This is just one.
Dialogue: 0,0:09:03.82,0:09:09.99,Default,,0000,0000,0000,,Sinai over 'cause I is Tanay and\Nsign be over. Cosby is tan be.
Dialogue: 0,0:09:10.53,0:09:13.30,Default,,0000,0000,0000,,So this is just tan a tan B.
Dialogue: 0,0:09:14.19,0:09:19.25,Default,,0000,0000,0000,,Have a look at this. Well,\Nthere's a common factor on top
Dialogue: 0,0:09:19.25,0:09:24.74,Default,,0000,0000,0000,,and bottom here of Cosby. I can\Ncancel that out. Leaves me with
Dialogue: 0,0:09:24.74,0:09:27.69,Default,,0000,0000,0000,,sign a over cause a tan a.
Dialogue: 0,0:09:28.31,0:09:32.04,Default,,0000,0000,0000,,Here the causes go out.
Dialogue: 0,0:09:32.56,0:09:39.56,Default,,0000,0000,0000,,Sign be over. Cosby\Nleaves me with Tan
Dialogue: 0,0:09:39.56,0:09:43.38,Default,,0000,0000,0000,,B so. What we end
Dialogue: 0,0:09:43.38,0:09:50.41,Default,,0000,0000,0000,,up with. Turn of\NA+B is 10A.
Dialogue: 0,0:09:51.06,0:09:54.59,Default,,0000,0000,0000,,Plus 10B.
Dialogue: 0,0:09:55.92,0:10:02.35,Default,,0000,0000,0000,,All over 1\N- 10 a
Dialogue: 0,0:10:02.35,0:10:03.42,Default,,0000,0000,0000,,10B.
Dialogue: 0,0:10:04.53,0:10:10.00,Default,,0000,0000,0000,,Now we can do the same\Nagain for tan.
Dialogue: 0,0:10:10.65,0:10:17.86,Default,,0000,0000,0000,,All A-B and we've got two ways\Nof approaching it, first of all.
Dialogue: 0,0:10:18.50,0:10:23.75,Default,,0000,0000,0000,,We could work through this again\NX set with tan of A-B is the
Dialogue: 0,0:10:23.75,0:10:26.75,Default,,0000,0000,0000,,sign of a minus B over the cause
Dialogue: 0,0:10:26.75,0:10:30.14,Default,,0000,0000,0000,,of A-B. Or we could work through
Dialogue: 0,0:10:30.14,0:10:36.03,Default,,0000,0000,0000,,it again. By making the same\Nreplacement as we did before,
Dialogue: 0,0:10:36.03,0:10:41.42,Default,,0000,0000,0000,,and rewriting this as the time\Nof A plus minus B.
Dialogue: 0,0:10:43.32,0:10:50.86,Default,,0000,0000,0000,,Let's be assured that what it is\Ngoing to give us, which ever way
Dialogue: 0,0:10:50.86,0:10:52.48,Default,,0000,0000,0000,,we do it.
Dialogue: 0,0:10:56.60,0:11:02.98,Default,,0000,0000,0000,,Is going\Nto be
Dialogue: 0,0:11:02.98,0:11:07.87,Default,,0000,0000,0000,,that. And So what we've got now\Nour our six.
Dialogue: 0,0:11:08.66,0:11:09.81,Default,,0000,0000,0000,,Addition formula.
Dialogue: 0,0:11:11.32,0:11:17.11,Default,,0000,0000,0000,,Sign of A+B sign of A-B cause of\NA+B cause of A-B. These are the
Dialogue: 0,0:11:17.11,0:11:21.74,Default,,0000,0000,0000,,ones you must know and learn\Nfrom those four you can derive
Dialogue: 0,0:11:21.74,0:11:27.14,Default,,0000,0000,0000,,these. It's a help if you know\Nthem. If you can learn them. The
Dialogue: 0,0:11:27.14,0:11:32.16,Default,,0000,0000,0000,,important thing is to be able to\Nrecognize them when you see them
Dialogue: 0,0:11:32.16,0:11:34.48,Default,,0000,0000,0000,,and recognize when you need to
Dialogue: 0,0:11:34.48,0:11:40.29,Default,,0000,0000,0000,,use them. Let's have a look at\Nthree fairly typical examples of
Dialogue: 0,0:11:40.29,0:11:45.06,Default,,0000,0000,0000,,the use of these. In many ways,\Nit's practiced that we're
Dialogue: 0,0:11:45.06,0:11:48.54,Default,,0000,0000,0000,,getting at recognizing these\Nparticular formula, 'cause we
Dialogue: 0,0:11:48.54,0:11:52.88,Default,,0000,0000,0000,,may need them more often when\Nwe're doing other, more
Dialogue: 0,0:11:52.88,0:11:57.30,Default,,0000,0000,0000,,complicated manipulations. So\Nfirst of all, let's have a look
Dialogue: 0,0:11:57.30,0:11:58.62,Default,,0000,0000,0000,,at this particular problem.
Dialogue: 0,0:11:59.13,0:12:02.81,Default,,0000,0000,0000,,If we know that sign of
Dialogue: 0,0:12:02.81,0:12:05.73,Default,,0000,0000,0000,,A. Is 3/5.
Dialogue: 0,0:12:07.04,0:12:08.52,Default,,0000,0000,0000,,And that the cause.
Dialogue: 0,0:12:09.12,0:12:13.03,Default,,0000,0000,0000,,Of B is 5
Dialogue: 0,0:12:13.03,0:12:20.15,Default,,0000,0000,0000,,thirteenths. Then\NWhat's the sign
Dialogue: 0,0:12:20.15,0:12:22.68,Default,,0000,0000,0000,,of A+B?
Dialogue: 0,0:12:24.25,0:12:27.58,Default,,0000,0000,0000,,What's the cause
Dialogue: 0,0:12:27.58,0:12:35.40,Default,,0000,0000,0000,,of A-B?\NOK.
Dialogue: 0,0:12:36.45,0:12:40.62,Default,,0000,0000,0000,,We've got sign a but we don't\Nknow anything about cause a
Dialogue: 0,0:12:40.62,0:12:43.67,Default,,0000,0000,0000,,seemingly. We've got Cosby.
Dialogue: 0,0:12:44.58,0:12:47.35,Default,,0000,0000,0000,,And we really don't know\Nanything about sign be.
Dialogue: 0,0:12:49.52,0:12:55.23,Default,,0000,0000,0000,,We don't know much about A and\Nbe really because a could be
Dialogue: 0,0:12:55.23,0:13:00.93,Default,,0000,0000,0000,,either an obtuse angle or an\Nacute angle. So we really need a
Dialogue: 0,0:13:00.93,0:13:05.76,Default,,0000,0000,0000,,little bit more information. So\Nlet's say that A&B are acute.
Dialogue: 0,0:13:06.29,0:13:09.81,Default,,0000,0000,0000,,In alerts that both less than 90
Dialogue: 0,0:13:09.81,0:13:14.27,Default,,0000,0000,0000,,degrees. Well, if the boat less\Nthan 90 degrees, one of the
Dialogue: 0,0:13:14.27,0:13:16.06,Default,,0000,0000,0000,,things we can do is represent
Dialogue: 0,0:13:16.06,0:13:21.28,Default,,0000,0000,0000,,the angle. And the sign with a\Nright angle triangle. So I just
Dialogue: 0,0:13:21.28,0:13:23.12,Default,,0000,0000,0000,,have a look at that.
Dialogue: 0,0:13:24.48,0:13:27.51,Default,,0000,0000,0000,,Right angle triangle.
Dialogue: 0,0:13:27.51,0:13:34.46,Default,,0000,0000,0000,,Right angle there the angle a\Nsign a is 3/5 opposite over
Dialogue: 0,0:13:34.46,0:13:40.83,Default,,0000,0000,0000,,hypotenuse, so that's three. The\Nside opposite the angle A and
Dialogue: 0,0:13:40.83,0:13:46.62,Default,,0000,0000,0000,,that's five the hypotenuse which\Nis always opposite the right
Dialogue: 0,0:13:46.62,0:13:49.51,Default,,0000,0000,0000,,angle, always the longest side.
Dialogue: 0,0:13:50.26,0:13:55.51,Default,,0000,0000,0000,,Pythagoras tells us that this\Nother side has to be 4.
Dialogue: 0,0:13:58.20,0:14:01.96,Default,,0000,0000,0000,,And so now that we know the\Nadjacent side, we know
Dialogue: 0,0:14:01.96,0:14:05.72,Default,,0000,0000,0000,,everything about angle a. We\Nknow it sign, we know it's
Dialogue: 0,0:14:05.72,0:14:09.14,Default,,0000,0000,0000,,cosine and if we want we can\Nfind its tangent.
Dialogue: 0,0:14:10.68,0:14:11.96,Default,,0000,0000,0000,,Let's do the same.
Dialogue: 0,0:14:12.48,0:14:13.78,Default,,0000,0000,0000,,The angle be.
Dialogue: 0,0:14:15.67,0:14:22.17,Default,,0000,0000,0000,,Here is the angle be in its\Nright angle triangle and were
Dialogue: 0,0:14:22.17,0:14:25.97,Default,,0000,0000,0000,,told that Cosby is 5 over 13.
Dialogue: 0,0:14:26.63,0:14:31.82,Default,,0000,0000,0000,,Cosine is adjacent over\Nhypotenuse. This is the side
Dialogue: 0,0:14:31.82,0:14:37.02,Default,,0000,0000,0000,,that's adjacent, so there's\Nfive. This is the hypotenuse,
Dialogue: 0,0:14:37.02,0:14:42.78,Default,,0000,0000,0000,,the longest side, the side\Nopposite the right angle, so
Dialogue: 0,0:14:42.78,0:14:49.60,Default,,0000,0000,0000,,that's 30. So using Pythagoras\Ntheorem, this side is 12.
Dialogue: 0,0:14:50.37,0:14:54.73,Default,,0000,0000,0000,,Now notice I said using\NPythagoras Theorem, but it was
Dialogue: 0,0:14:54.73,0:14:59.96,Default,,0000,0000,0000,,as though I knew these three\N455-1213. I do know them and
Dialogue: 0,0:14:59.96,0:15:04.76,Default,,0000,0000,0000,,you've got to get to know them\Nas well. Pythagorean triples,
Dialogue: 0,0:15:04.76,0:15:06.50,Default,,0000,0000,0000,,simple triples of integers,
Dialogue: 0,0:15:06.50,0:15:11.37,Default,,0000,0000,0000,,whole numbers. That a Bay,\NPythagoras's theorem if you have
Dialogue: 0,0:15:11.37,0:15:16.58,Default,,0000,0000,0000,,them at your fingertips, can we\Ncall them easy? Easily? You find
Dialogue: 0,0:15:16.58,0:15:20.92,Default,,0000,0000,0000,,this so much better in working\Nin trigonometry 'cause these
Dialogue: 0,0:15:20.92,0:15:26.56,Default,,0000,0000,0000,,numbers are used an awful lot.\NOK, then, let's have a look at
Dialogue: 0,0:15:26.56,0:15:32.20,Default,,0000,0000,0000,,what we can do. Sign of A+B? The\Naddition formula tells us they
Dialogue: 0,0:15:32.20,0:15:35.46,Default,,0000,0000,0000,,sign a. Cause B.
Dialogue: 0,0:15:36.05,0:15:39.68,Default,,0000,0000,0000,,Close calls a sign
Dialogue: 0,0:15:39.68,0:15:47.26,Default,,0000,0000,0000,,be. Equals.\NSign a we know is 3/5.
Dialogue: 0,0:15:47.96,0:15:55.12,Default,,0000,0000,0000,,Times, Cosby, and we know\Nthat one. It's five thirteenths
Dialogue: 0,0:15:55.12,0:16:02.39,Default,,0000,0000,0000,,Plus cause a we can\Nread cause a off here
Dialogue: 0,0:16:02.39,0:16:08.93,Default,,0000,0000,0000,,it's adjacent over hypotenuse so\Nit's 4 over 5.
Dialogue: 0,0:16:08.94,0:16:16.02,Default,,0000,0000,0000,,Times by sign B and we\Ncan read, sign be off this
Dialogue: 0,0:16:16.02,0:16:20.74,Default,,0000,0000,0000,,triangle it's opposite over\Nhypotenuse 12 over 13.
Dialogue: 0,0:16:21.62,0:16:29.20,Default,,0000,0000,0000,,Do the arithmetic 3 fives\Nare 15 and five 1365,
Dialogue: 0,0:16:29.20,0:16:36.02,Default,,0000,0000,0000,,so that's 15 over 65\N+ 4 twelve 48.
Dialogue: 0,0:16:36.80,0:16:44.05,Default,,0000,0000,0000,,And 5:13's again are 65 and\Nso that gives me a fraction
Dialogue: 0,0:16:44.05,0:16:45.86,Default,,0000,0000,0000,,63 over 65.
Dialogue: 0,0:16:46.86,0:16:50.10,Default,,0000,0000,0000,,We had cause of
Dialogue: 0,0:16:50.10,0:16:56.22,Default,,0000,0000,0000,,A-B. So let's approach that in\Nexactly the same way.
Dialogue: 0,0:16:56.92,0:17:00.44,Default,,0000,0000,0000,,Will quote our\Nexpansion. Our
Dialogue: 0,0:17:00.44,0:17:04.66,Default,,0000,0000,0000,,addition formula cause\Nof a Cosby.
Dialogue: 0,0:17:05.76,0:17:09.24,Default,,0000,0000,0000,,Plus sign of a sign
Dialogue: 0,0:17:09.24,0:17:16.96,Default,,0000,0000,0000,,of B. Substitute our\Nvalues cause a That's four
Dialogue: 0,0:17:16.96,0:17:20.88,Default,,0000,0000,0000,,over 5 adjacent over hypotenuse.
Dialogue: 0,0:17:22.19,0:17:28.91,Default,,0000,0000,0000,,Times, Cosby. That's five\Nover 13, adjacent over
Dialogue: 0,0:17:28.91,0:17:29.75,Default,,0000,0000,0000,,hypotenuse.
Dialogue: 0,0:17:30.79,0:17:37.87,Default,,0000,0000,0000,,Plus sign a That's opposite\Nover hypotenuse, three over 5.
Dialogue: 0,0:17:39.01,0:17:45.46,Default,,0000,0000,0000,,Times by sign B.\NThat's 12 over 13,
Dialogue: 0,0:17:45.46,0:17:47.88,Default,,0000,0000,0000,,opposite over hypotenuse.
Dialogue: 0,0:17:49.04,0:17:53.55,Default,,0000,0000,0000,,Now here we've got some\Nfractions and it might be
Dialogue: 0,0:17:53.55,0:17:58.51,Default,,0000,0000,0000,,tempting to cancel the fives\Nhere. Five goals, five goes, but
Dialogue: 0,0:17:58.51,0:18:05.73,Default,,0000,0000,0000,,five times by 13 is the 65 and\Nfive times by 13 is the 65 to
Dialogue: 0,0:18:05.73,0:18:07.53,Default,,0000,0000,0000,,have the same denominator.
Dialogue: 0,0:18:08.32,0:18:15.95,Default,,0000,0000,0000,,So I'm not going to cancel.\N4 fives are 20 over five
Dialogue: 0,0:18:15.95,0:18:18.23,Default,,0000,0000,0000,,1365. Plus
Dialogue: 0,0:18:19.50,0:18:24.32,Default,,0000,0000,0000,,Three twelves\Nare 36 over that
Dialogue: 0,0:18:24.32,0:18:29.94,Default,,0000,0000,0000,,65 again giving\Nme 56 over 65.
Dialogue: 0,0:18:31.05,0:18:35.48,Default,,0000,0000,0000,,So that's one way in which these\Ncan be used.
Dialogue: 0,0:18:35.53,0:18:37.88,Default,,0000,0000,0000,,Let's have a look at another
Dialogue: 0,0:18:37.88,0:18:44.16,Default,,0000,0000,0000,,way. Supposing we were asked to\Nfind what's sign 75, but no
Dialogue: 0,0:18:44.16,0:18:49.76,Default,,0000,0000,0000,,don't reach for your Calculator,\Nyou've got to workout sign 75 as
Dialogue: 0,0:18:49.76,0:18:55.36,Default,,0000,0000,0000,,an expression, not as a set of\Ndecimals, but as an expression.
Dialogue: 0,0:18:55.36,0:19:01.90,Default,,0000,0000,0000,,What we've got to think about is\Nhow can we make 75 from angles
Dialogue: 0,0:19:01.90,0:19:04.70,Default,,0000,0000,0000,,who sign and cosine's? We know
Dialogue: 0,0:19:04.70,0:19:12.13,Default,,0000,0000,0000,,very well. Well, 45\N+ 30 gives us
Dialogue: 0,0:19:12.13,0:19:19.70,Default,,0000,0000,0000,,75. I'm 45 and 30\Nare two of the angles that
Dialogue: 0,0:19:19.70,0:19:26.34,Default,,0000,0000,0000,,we know sines and cosines for\Nexact sines and cosines.
Dialogue: 0,0:19:26.87,0:19:34.31,Default,,0000,0000,0000,,So this would be sign\Nof 45 cause of 30.
Dialogue: 0,0:19:35.26,0:19:42.82,Default,,0000,0000,0000,,Close calls of\N45. Sign of
Dialogue: 0,0:19:42.82,0:19:50.54,Default,,0000,0000,0000,,30. Sign 45\Nthat's one over Route
Dialogue: 0,0:19:50.54,0:19:57.62,Default,,0000,0000,0000,,2. Times by cost 30\Nwill cost 30 is Route 3
Dialogue: 0,0:19:57.62,0:19:58.71,Default,,0000,0000,0000,,over 2.
Dialogue: 0,0:19:59.86,0:20:05.24,Default,,0000,0000,0000,,Plus cause of 45 is one\Nover Route 2.
Dialogue: 0,0:20:05.82,0:20:11.78,Default,,0000,0000,0000,,Times by sign of 30, which\Nis just a half.
Dialogue: 0,0:20:12.52,0:20:19.24,Default,,0000,0000,0000,,So we have Route 3. That's one\Ntimes by Route 3 over 2 Route 2.
Dialogue: 0,0:20:19.86,0:20:26.83,Default,,0000,0000,0000,,Plus one over and again two\Ntimes route 2 is 2 Route 2.
Dialogue: 0,0:20:26.83,0:20:31.65,Default,,0000,0000,0000,,We've got the same denominator,\Nso 2 Route 2.
Dialogue: 0,0:20:32.23,0:20:39.34,Default,,0000,0000,0000,,To Route 2 and on the\Ntop Route 3 + 1.
Dialogue: 0,0:20:39.92,0:20:44.32,Default,,0000,0000,0000,,Now leave not won't like that we\Nmight try and get rid of this
Dialogue: 0,0:20:44.32,0:20:48.08,Default,,0000,0000,0000,,Route 2 in the denominator, but\Njust for the moment, let's leave
Dialogue: 0,0:20:48.08,0:20:52.16,Default,,0000,0000,0000,,it like that. There's nothing to\Nbe gained by doing it at this
Dialogue: 0,0:20:52.16,0:20:53.74,Default,,0000,0000,0000,,stage, and let's take another
Dialogue: 0,0:20:53.74,0:20:56.88,Default,,0000,0000,0000,,example. Let's have a look.
Dialogue: 0,0:20:56.88,0:21:03.54,Default,,0000,0000,0000,,An 50 What we\Nneed to be able to do is to find
Dialogue: 0,0:21:03.54,0:21:07.38,Default,,0000,0000,0000,,15 in terms of angles that we\Nknow lots of things about.
Dialogue: 0,0:21:09.13,0:21:11.16,Default,,0000,0000,0000,,And of course.
Dialogue: 0,0:21:11.68,0:21:18.27,Default,,0000,0000,0000,,One of the ways of doing this,\Nbut only one is to do 60 - 45.
Dialogue: 0,0:21:19.61,0:21:26.09,Default,,0000,0000,0000,,So that means that we need the\Naddition formula that's to do
Dialogue: 0,0:21:26.09,0:21:33.11,Default,,0000,0000,0000,,with tangent and the one that's\Nto do with the tan of A-B.
Dialogue: 0,0:21:33.11,0:21:36.35,Default,,0000,0000,0000,,So this is 1060 - 1045.
Dialogue: 0,0:21:37.10,0:21:43.82,Default,,0000,0000,0000,,Over 1\N+ 10
Dialogue: 0,0:21:43.82,0:21:51.04,Default,,0000,0000,0000,,sixty 10:45.\NSo now let's put the numerical
Dialogue: 0,0:21:51.04,0:21:58.10,Default,,0000,0000,0000,,values in. Now the tangent of 60\Nis Route 3 and the tangent of
Dialogue: 0,0:21:58.10,0:21:59.61,Default,,0000,0000,0000,,45 is one.
Dialogue: 0,0:22:00.40,0:22:06.19,Default,,0000,0000,0000,,Over. One plus\NRoute 3.
Dialogue: 0,0:22:07.35,0:22:09.57,Default,,0000,0000,0000,,Times by one.
Dialogue: 0,0:22:10.12,0:22:17.71,Default,,0000,0000,0000,,So we'll have root 3 - 1\Nover Route 3 times by one is
Dialogue: 0,0:22:17.71,0:22:19.33,Default,,0000,0000,0000,,just Route 3.
Dialogue: 0,0:22:20.47,0:22:26.92,Default,,0000,0000,0000,,Plus one. Now that's\Nall right, and it's correct.
Dialogue: 0,0:22:27.77,0:22:32.24,Default,,0000,0000,0000,,Doesn't look very nice and we\Ntend to have a tradition of not
Dialogue: 0,0:22:32.24,0:22:35.68,Default,,0000,0000,0000,,leaving things like root 3 plus\None in the denominator.
Dialogue: 0,0:22:36.24,0:22:39.75,Default,,0000,0000,0000,,So one of the ways of tidying
Dialogue: 0,0:22:39.75,0:22:46.86,Default,,0000,0000,0000,,that up. Is to multiply top\Nand bottom of this fraction by
Dialogue: 0,0:22:46.86,0:22:48.41,Default,,0000,0000,0000,,the same thing?
Dialogue: 0,0:22:49.25,0:22:54.83,Default,,0000,0000,0000,,Multiplying top and bottom by\Nthe same thing keeps it the same
Dialogue: 0,0:22:54.83,0:22:59.94,Default,,0000,0000,0000,,value, but what should we\Nmultiply it by? Well, I'm going
Dialogue: 0,0:22:59.94,0:23:03.20,Default,,0000,0000,0000,,to choose to multiply by Route 3
Dialogue: 0,0:23:03.20,0:23:10.03,Default,,0000,0000,0000,,- 1. Not be cause that's what's\Nin the numerator there, but
Dialogue: 0,0:23:10.03,0:23:16.52,Default,,0000,0000,0000,,because this is the difference\Nof two squares A+B, A-B and
Dialogue: 0,0:23:16.52,0:23:23.01,Default,,0000,0000,0000,,A+B times by A-B, let's just\Nwrite that down up here.
Dialogue: 0,0:23:23.04,0:23:26.78,Default,,0000,0000,0000,,Is A squared minus B
Dialogue: 0,0:23:26.78,0:23:33.99,Default,,0000,0000,0000,,squared? So that means that\Non the bottom I have Route 3
Dialogue: 0,0:23:33.99,0:23:36.54,Default,,0000,0000,0000,,times by Route 3A squared.
Dialogue: 0,0:23:37.61,0:23:43.06,Default,,0000,0000,0000,,Which is just three minus B\Nsquared. B was just one, so
Dialogue: 0,0:23:43.06,0:23:49.87,Default,,0000,0000,0000,,that's just minus one and 3 - 1\Nis 2 and integer not absurd. Not
Dialogue: 0,0:23:49.87,0:23:55.32,Default,,0000,0000,0000,,one of these things with a\Nsquare root sign attached to it.
Dialogue: 0,0:23:55.32,0:24:00.76,Default,,0000,0000,0000,,What have we got on top? We two\Nbrackets that we're multiplying
Dialogue: 0,0:24:00.76,0:24:07.57,Default,,0000,0000,0000,,together route 3 by Route 3 is\N3, route 3 by minus one is minus
Dialogue: 0,0:24:07.57,0:24:14.79,Default,,0000,0000,0000,,Route 3. 1 by minus one\Nby Route 3 is minus Route 3
Dialogue: 0,0:24:14.79,0:24:18.47,Default,,0000,0000,0000,,and minus one with minus one is
Dialogue: 0,0:24:18.47,0:24:24.81,Default,,0000,0000,0000,,plus one. 3 plus One is 4 -\N2 RT. Threes were taking away
Dialogue: 0,0:24:24.81,0:24:30.97,Default,,0000,0000,0000,,all over 2 and now there's a\Ncommon factor of 2 on the top
Dialogue: 0,0:24:30.97,0:24:38.01,Default,,0000,0000,0000,,and on the bottom 2 into 4 goes\Nto an two into minus 2. Route 3
Dialogue: 0,0:24:38.01,0:24:39.77,Default,,0000,0000,0000,,goes minus Route 3.
Dialogue: 0,0:24:40.34,0:24:44.78,Default,,0000,0000,0000,,And two into two goes wall, so\Nwe needn't worry about that one.
Dialogue: 0,0:24:44.78,0:24:49.23,Default,,0000,0000,0000,,So there we've got a second\Nexample to add to the first one
Dialogue: 0,0:24:49.23,0:24:51.62,Default,,0000,0000,0000,,that we did. Let's take one more
Dialogue: 0,0:24:51.62,0:24:57.87,Default,,0000,0000,0000,,type. Of example, where we can\Nmake you solve the addition
Dialogue: 0,0:24:57.87,0:25:04.31,Default,,0000,0000,0000,,formula and this is where we get\Nto simplify an expression. So
Dialogue: 0,0:25:04.31,0:25:09.15,Default,,0000,0000,0000,,supposing we've got the sign of\N90 plus A.
Dialogue: 0,0:25:10.07,0:25:15.50,Default,,0000,0000,0000,,Could we have this simpler?\NCould we just have, say, sign a
Dialogue: 0,0:25:15.50,0:25:21.39,Default,,0000,0000,0000,,or minus sign a or cause a or\Nsomething like that? Does it
Dialogue: 0,0:25:21.39,0:25:27.74,Default,,0000,0000,0000,,have to be 90 plus a? Well,\Nlet's have a look. It's sign of
Dialogue: 0,0:25:27.74,0:25:31.07,Default,,0000,0000,0000,,90. Calls of
Dialogue: 0,0:25:31.07,0:25:37.82,Default,,0000,0000,0000,,a. Plus the\Ncause of 90
Dialogue: 0,0:25:37.82,0:25:40.73,Default,,0000,0000,0000,,sign of A.
Dialogue: 0,0:25:42.01,0:25:44.36,Default,,0000,0000,0000,,Sign 90 is one.
Dialogue: 0,0:25:44.98,0:25:50.72,Default,,0000,0000,0000,,Cause I just cause I so first of\Nall we have one times cause a
Dialogue: 0,0:25:50.72,0:25:52.64,Default,,0000,0000,0000,,which is just cause a.
Dialogue: 0,0:25:53.76,0:25:55.97,Default,,0000,0000,0000,,But cause of 90.
Dialogue: 0,0:25:56.50,0:26:02.97,Default,,0000,0000,0000,,Is 0. Times by sign a\Nwell, anything times by zero is
Dialogue: 0,0:26:02.97,0:26:06.64,Default,,0000,0000,0000,,0 so we just left with cause a.
Dialogue: 0,0:26:07.60,0:26:15.03,Default,,0000,0000,0000,,So an expression such a sign\Nof 90 plus a reducers tool
Dialogue: 0,0:26:15.03,0:26:21.84,Default,,0000,0000,0000,,cause a what about something\Nlike the cosine of 180 minus
Dialogue: 0,0:26:21.84,0:26:28.73,Default,,0000,0000,0000,,a? Well, this\Nis cause 180
Dialogue: 0,0:26:28.73,0:26:34.38,Default,,0000,0000,0000,,cause a.\NPlus sign
Dialogue: 0,0:26:34.38,0:26:37.31,Default,,0000,0000,0000,,180 sign
Dialogue: 0,0:26:37.31,0:26:44.42,Default,,0000,0000,0000,,a. The cosine of\N180. That's minus one, so minus
Dialogue: 0,0:26:44.42,0:26:52.09,Default,,0000,0000,0000,,one times by cause a is minus\Ncause a sign of 180 zero and
Dialogue: 0,0:26:52.09,0:26:58.67,Default,,0000,0000,0000,,sign is just sign a but zero\Ntimes by anything is 0.
Dialogue: 0,0:26:59.28,0:27:05.82,Default,,0000,0000,0000,,So we're just left with minus\Ncalls a so we can see that these
Dialogue: 0,0:27:05.82,0:27:10.95,Default,,0000,0000,0000,,addition formula help us to very\Nquickly simplify and get in
Dialogue: 0,0:27:10.95,0:27:13.29,Default,,0000,0000,0000,,terms of this angle a.
Dialogue: 0,0:27:14.11,0:27:16.48,Default,,0000,0000,0000,,Possibly quite complicated
Dialogue: 0,0:27:16.48,0:27:23.36,Default,,0000,0000,0000,,expressions. But the basic thing\Nis that these 6 addition
Dialogue: 0,0:27:23.36,0:27:29.43,Default,,0000,0000,0000,,formula. Again, I stress it. You\Nhave to learn you have to know.
Dialogue: 0,0:27:29.43,0:27:33.24,Default,,0000,0000,0000,,But most importantly, you've got\Nto recognize them and recognize
Dialogue: 0,0:27:33.24,0:27:37.81,Default,,0000,0000,0000,,their use when you see them and\Nwhen you see the situation.