[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.45,0:00:05.00,Default,,0000,0000,0000,,In this video, we're going to be\Nlooking at the double angle
Dialogue: 0,0:00:05.00,0:00:11.13,Default,,0000,0000,0000,,formula. But to start with,\Nwe're going to start from the
Dialogue: 0,0:00:11.13,0:00:16.70,Default,,0000,0000,0000,,addition formula. Not all of\Nthem, just the ones that deal
Dialogue: 0,0:00:16.70,0:00:22.89,Default,,0000,0000,0000,,with A+B. So let's just write\Nthose down to begin with sign of
Dialogue: 0,0:00:22.89,0:00:25.40,Default,,0000,0000,0000,,A+B, we know is sign a.
Dialogue: 0,0:00:25.95,0:00:32.84,Default,,0000,0000,0000,,Cause B. Post\Ncause a sign
Dialogue: 0,0:00:32.84,0:00:40.31,Default,,0000,0000,0000,,be. Next, we want\Nthe cause of A+B, which
Dialogue: 0,0:00:40.31,0:00:44.26,Default,,0000,0000,0000,,will be cause a calls
Dialogue: 0,0:00:44.26,0:00:51.39,Default,,0000,0000,0000,,B. Minus sign, a\Nsign be and finally
Dialogue: 0,0:00:51.39,0:00:57.80,Default,,0000,0000,0000,,the tan one tan\Nof A+B, which will
Dialogue: 0,0:00:57.80,0:01:00.100,Default,,0000,0000,0000,,be 10 A plus
Dialogue: 0,0:01:00.100,0:01:06.63,Default,,0000,0000,0000,,10B. Over\N1 -
Dialogue: 0,0:01:06.63,0:01:10.69,Default,,0000,0000,0000,,10 a\N10B.
Dialogue: 0,0:01:12.29,0:01:15.39,Default,,0000,0000,0000,,So those are three of\Nour addition formula.
Dialogue: 0,0:01:16.68,0:01:20.09,Default,,0000,0000,0000,,And each one is to do with A+B.
Dialogue: 0,0:01:20.99,0:01:27.84,Default,,0000,0000,0000,,So what happens if we let\Na be equal to be?
Dialogue: 0,0:01:28.48,0:01:34.16,Default,,0000,0000,0000,,In other words, instead of\Nhaving a plus B, we have a plus
Dialogue: 0,0:01:34.16,0:01:41.76,Default,,0000,0000,0000,,a. So that would be\Nsign of A plus a would
Dialogue: 0,0:01:41.76,0:01:44.25,Default,,0000,0000,0000,,be signed to A.
Dialogue: 0,0:01:45.72,0:01:52.97,Default,,0000,0000,0000,,What does that do to\Nthis right hand side? Well
Dialogue: 0,0:01:52.97,0:02:00.22,Default,,0000,0000,0000,,gives us sign a cause,\NA plus cause a sign
Dialogue: 0,0:02:00.22,0:02:08.16,Default,,0000,0000,0000,,a. In other words, these two\Nat the same, so we can just add
Dialogue: 0,0:02:08.16,0:02:15.64,Default,,0000,0000,0000,,them together. So sign of 2A is\N2 sign a cause A and that's our
Dialogue: 0,0:02:15.64,0:02:20.13,Default,,0000,0000,0000,,first double angle formula\Ndouble angle because it's 2A
Dialogue: 0,0:02:20.13,0:02:22.13,Default,,0000,0000,0000,,where doubling the angle.
Dialogue: 0,0:02:22.67,0:02:29.70,Default,,0000,0000,0000,,So what is it sign and so\Non. Let's do the same with
Dialogue: 0,0:02:29.70,0:02:36.74,Default,,0000,0000,0000,,cause. Let's put a equal to\Nbe. So will have cars to a
Dialogue: 0,0:02:36.74,0:02:42.69,Default,,0000,0000,0000,,is equal to what it was\Ncause a Cosby. It's now
Dialogue: 0,0:02:42.69,0:02:48.64,Default,,0000,0000,0000,,going to be cause A cause a\Nwitches caused squared A.
Dialogue: 0,0:02:49.72,0:02:56.90,Default,,0000,0000,0000,,Sign a sign be when it's now\Ngoing to be sign a sign a
Dialogue: 0,0:02:56.90,0:02:59.98,Default,,0000,0000,0000,,which is sine squared minus sign
Dialogue: 0,0:02:59.98,0:03:03.43,Default,,0000,0000,0000,,squared A. And that's how
Dialogue: 0,0:03:03.43,0:03:07.25,Default,,0000,0000,0000,,a second. Double angle formula.
Dialogue: 0,0:03:07.93,0:03:15.29,Default,,0000,0000,0000,,Doing the same with Tan\NTan 2A is equal to.
Dialogue: 0,0:03:15.94,0:03:23.74,Default,,0000,0000,0000,,10A plus 10 B this is now\Nangle a so it's 10A Plus 10A
Dialogue: 0,0:03:23.74,0:03:26.52,Default,,0000,0000,0000,,which is 2 Tab A.
Dialogue: 0,0:03:26.54,0:03:33.36,Default,,0000,0000,0000,,All over 1 - 10, eight and be.\NBut this is now a instead of B,
Dialogue: 0,0:03:33.36,0:03:34.63,Default,,0000,0000,0000,,so it's tanae.
Dialogue: 0,0:03:35.29,0:03:41.17,Default,,0000,0000,0000,,10 eight times by\N10 A is 10 squared.
Dialogue: 0,0:03:41.17,0:03:44.43,Default,,0000,0000,0000,,1 - 10 squared A.
Dialogue: 0,0:03:46.15,0:03:51.22,Default,,0000,0000,0000,,And here are our three\Ndouble angle formula again
Dialogue: 0,0:03:51.22,0:03:56.85,Default,,0000,0000,0000,,to be learned to be\Nrecognized and to be used.
Dialogue: 0,0:03:58.68,0:04:04.48,Default,,0000,0000,0000,,Let's just have a look at this\None cause to A.
Dialogue: 0,0:04:05.75,0:04:10.31,Default,,0000,0000,0000,,White pick out this one. Well\Nthis right hand side which is
Dialogue: 0,0:04:10.31,0:04:14.49,Default,,0000,0000,0000,,the bit that interests because\Nit's got cost squared and sign
Dialogue: 0,0:04:14.49,0:04:17.15,Default,,0000,0000,0000,,squared in it and there is an
Dialogue: 0,0:04:17.15,0:04:21.13,Default,,0000,0000,0000,,identity. That's to do with cost\Nsquared. Plus sign squared
Dialogue: 0,0:04:21.13,0:04:26.80,Default,,0000,0000,0000,,equals 1. What that means is we\Ncan replace the sine squared.
Dialogue: 0,0:04:27.34,0:04:32.02,Default,,0000,0000,0000,,And get everything in terms of\NCos squared. Or we can do it the
Dialogue: 0,0:04:32.02,0:04:33.02,Default,,0000,0000,0000,,other way round.
Dialogue: 0,0:04:33.60,0:04:36.90,Default,,0000,0000,0000,,So I just have a\Nlook at that one.
Dialogue: 0,0:04:38.28,0:04:45.22,Default,,0000,0000,0000,,Cause to a\Ncost squared, A
Dialogue: 0,0:04:45.22,0:04:48.69,Default,,0000,0000,0000,,minus sign squared
Dialogue: 0,0:04:48.69,0:04:55.74,Default,,0000,0000,0000,,a butt. Cost\Nsquared A plus sign,
Dialogue: 0,0:04:55.74,0:04:59.52,Default,,0000,0000,0000,,squared A equals 1.
Dialogue: 0,0:05:00.20,0:05:07.34,Default,,0000,0000,0000,,In other words, sign squared a\Nis 1 minus Cos squared a so
Dialogue: 0,0:05:07.34,0:05:13.92,Default,,0000,0000,0000,,we can replace the sine squared\Nhere in our double angle formula
Dialogue: 0,0:05:13.92,0:05:19.96,Default,,0000,0000,0000,,by one minus Cos squared, so\Nwill have cause to A.
Dialogue: 0,0:05:20.56,0:05:28.14,Default,,0000,0000,0000,,Is cost squared A minus\None minus Cos squared A?
Dialogue: 0,0:05:29.15,0:05:34.16,Default,,0000,0000,0000,,Using the brackets, notice\Nto show I'm taking away all
Dialogue: 0,0:05:34.16,0:05:38.17,Default,,0000,0000,0000,,of it and now let's remove\Nthe brackets.
Dialogue: 0,0:05:39.37,0:05:46.63,Default,,0000,0000,0000,,Minus one. Minus\Nminus gives me a plus
Dialogue: 0,0:05:46.63,0:05:53.33,Default,,0000,0000,0000,,cause squared a, so I\Nnow have two cost squared,
Dialogue: 0,0:05:53.33,0:06:00.03,Default,,0000,0000,0000,,A minus one, so that's\Nanother double angle formula for
Dialogue: 0,0:06:00.03,0:06:02.04,Default,,0000,0000,0000,,cost to a.
Dialogue: 0,0:06:02.77,0:06:09.35,Default,,0000,0000,0000,,Now because I replaced the sine\Nsquared here by one minus Cos
Dialogue: 0,0:06:09.35,0:06:16.47,Default,,0000,0000,0000,,squared, I can do the same again\Nand replace the cost squared by
Dialogue: 0,0:06:16.47,0:06:23.59,Default,,0000,0000,0000,,one minus sign squared and what\Nthat will give main is cause 2A
Dialogue: 0,0:06:23.59,0:06:27.43,Default,,0000,0000,0000,,is 1 - 2 sine squared A.
Dialogue: 0,0:06:27.46,0:06:35.10,Default,,0000,0000,0000,,So lot of formally there.\NLet's just write them all
Dialogue: 0,0:06:35.10,0:06:42.31,Default,,0000,0000,0000,,down again. Sign\Nto a
Dialogue: 0,0:06:42.31,0:06:46.21,Default,,0000,0000,0000,,IS2. Find a
Dialogue: 0,0:06:46.21,0:06:50.16,Default,,0000,0000,0000,,Kohl's A. Calls
Dialogue: 0,0:06:50.16,0:06:56.88,Default,,0000,0000,0000,,to a. Is\Ncost squared A minus sign,
Dialogue: 0,0:06:56.88,0:07:03.67,Default,,0000,0000,0000,,squared A and we can\Nrewrite that as two cost
Dialogue: 0,0:07:03.67,0:07:10.46,Default,,0000,0000,0000,,square day minus one or\Nas 1 - 2 sine
Dialogue: 0,0:07:10.46,0:07:18.21,Default,,0000,0000,0000,,squared A. And then\Nturn to a is
Dialogue: 0,0:07:18.21,0:07:21.13,Default,,0000,0000,0000,,equal to 2.
Dialogue: 0,0:07:21.13,0:07:27.54,Default,,0000,0000,0000,,Tam a over\N1 - 10
Dialogue: 0,0:07:27.54,0:07:33.26,Default,,0000,0000,0000,,squared A. So there are\Nour double angle formula
Dialogue: 0,0:07:33.26,0:07:37.36,Default,,0000,0000,0000,,formula to be learned\Nformally to be remembered
Dialogue: 0,0:07:37.36,0:07:40.94,Default,,0000,0000,0000,,and most importantly\Nrecognized and used when
Dialogue: 0,0:07:40.94,0:07:42.48,Default,,0000,0000,0000,,we need them.
Dialogue: 0,0:07:43.84,0:07:49.57,Default,,0000,0000,0000,,So let's have a look at how we\Ncan make use of these double
Dialogue: 0,0:07:49.57,0:07:57.26,Default,,0000,0000,0000,,angle formula. So sign of three\NX. Is it possible to write
Dialogue: 0,0:07:57.26,0:08:02.96,Default,,0000,0000,0000,,sign of 3X all in terms\Nof sine X?
Dialogue: 0,0:08:04.48,0:08:07.80,Default,,0000,0000,0000,,Well. Let's try and break this
Dialogue: 0,0:08:07.80,0:08:15.32,Default,,0000,0000,0000,,3X up. 3X is 2X Plus X,\Nso we can write this a sign of
Dialogue: 0,0:08:15.32,0:08:16.74,Default,,0000,0000,0000,,2X Plus X.
Dialogue: 0,0:08:18.23,0:08:24.97,Default,,0000,0000,0000,,OK, this means we can\Nuse our addition formula sign
Dialogue: 0,0:08:24.97,0:08:28.34,Default,,0000,0000,0000,,of two X cause X.
Dialogue: 0,0:08:28.93,0:08:35.57,Default,,0000,0000,0000,,Plus cause of two\NX sign X.
Dialogue: 0,0:08:36.55,0:08:44.00,Default,,0000,0000,0000,,Now I can use my double\Nangle formula here sign of two
Dialogue: 0,0:08:44.00,0:08:51.45,Default,,0000,0000,0000,,X is 2 sign X Cos\NX still to be multiplied by
Dialogue: 0,0:08:51.45,0:08:53.32,Default,,0000,0000,0000,,Cos X Plus.
Dialogue: 0,0:08:54.04,0:08:56.73,Default,,0000,0000,0000,,Now I have a choice.
Dialogue: 0,0:08:57.38,0:09:02.42,Default,,0000,0000,0000,,There are three double angle\Nformula for cause 2X, so my
Dialogue: 0,0:09:02.42,0:09:09.29,Default,,0000,0000,0000,,choice is got to be governed by\Nwhat it is I'm trying to do and
Dialogue: 0,0:09:09.29,0:09:14.78,Default,,0000,0000,0000,,we're trying to write sign 3X\Nall in terms of sine X.
Dialogue: 0,0:09:15.40,0:09:22.22,Default,,0000,0000,0000,,That means the choice I have to\Nmake here is the one that's got
Dialogue: 0,0:09:22.22,0:09:28.06,Default,,0000,0000,0000,,signs in it, not cosines, but\Nthe one that's got signs and
Dialogue: 0,0:09:28.06,0:09:35.37,Default,,0000,0000,0000,,only signs, and the one that has\Nthat is 1 - 2 sine squared X
Dialogue: 0,0:09:35.37,0:09:38.78,Default,,0000,0000,0000,,still to be times by sign X.
Dialogue: 0,0:09:39.58,0:09:46.83,Default,,0000,0000,0000,,So this front term is\Ngoing to be 2 sign
Dialogue: 0,0:09:46.83,0:09:49.73,Default,,0000,0000,0000,,X cause squared X.
Dialogue: 0,0:09:50.30,0:09:57.83,Default,,0000,0000,0000,,One times by Cynex is\Nplus sign X minus and
Dialogue: 0,0:09:57.83,0:10:05.36,Default,,0000,0000,0000,,two sine squared X times.\NBiosynex is sine cubed X.
Dialogue: 0,0:10:06.32,0:10:11.20,Default,,0000,0000,0000,,Well, we're getting there. We've\Ngot sign here sign here. Sign
Dialogue: 0,0:10:11.20,0:10:13.42,Default,,0000,0000,0000,,cubed here. Cost squared here.
Dialogue: 0,0:10:14.34,0:10:19.94,Default,,0000,0000,0000,,But Cost Square can be rewritten\Nusing one of the fundamental
Dialogue: 0,0:10:19.94,0:10:24.64,Default,,0000,0000,0000,,identity's cost square plus sign\Nsquared is one so cost square
Dialogue: 0,0:10:24.64,0:10:26.77,Default,,0000,0000,0000,,can be replaced by Wang.
Dialogue: 0,0:10:27.63,0:10:32.85,Default,,0000,0000,0000,,Minus sign\Nsquared.
Dialogue: 0,0:10:34.94,0:10:40.26,Default,,0000,0000,0000,,And so we can see here.\NEverything is now in terms of
Dialogue: 0,0:10:40.26,0:10:44.24,Default,,0000,0000,0000,,sine X and all we need to do is
Dialogue: 0,0:10:44.24,0:10:51.58,Default,,0000,0000,0000,,tidied up. So we multiply out\Nthis bracket 2 sign X for
Dialogue: 0,0:10:51.58,0:10:58.54,Default,,0000,0000,0000,,the first term, 2 sign X\Ntimes by one. Then we have
Dialogue: 0,0:10:58.54,0:11:04.92,Default,,0000,0000,0000,,two sign X times Y minus\Nsign squared minus two sine
Dialogue: 0,0:11:04.92,0:11:10.72,Default,,0000,0000,0000,,cubed X plus sign X minus\Ntwo sine cubed X.
Dialogue: 0,0:11:11.73,0:11:18.92,Default,,0000,0000,0000,,2 sign X plus sign\NX that's three sign X.
Dialogue: 0,0:11:19.77,0:11:27.37,Default,,0000,0000,0000,,Minus two sine cubed minus two\Nsine cubed is minus 4 sign
Dialogue: 0,0:11:27.37,0:11:34.33,Default,,0000,0000,0000,,cubed X and that everything is\Nin terms of sine X.
Dialogue: 0,0:11:35.34,0:11:41.60,Default,,0000,0000,0000,,You can do the same with cause\Nas well cause 3X can be turned
Dialogue: 0,0:11:41.60,0:11:46.07,Default,,0000,0000,0000,,into an expression that's\Nentirely in terms of cause X.
Dialogue: 0,0:11:46.83,0:11:52.39,Default,,0000,0000,0000,,That's an example of using our\Ndouble angle formula in order to
Dialogue: 0,0:11:52.39,0:11:57.48,Default,,0000,0000,0000,,reduce if we like to use that\Nexpression and multiple angle
Dialogue: 0,0:11:57.48,0:12:03.96,Default,,0000,0000,0000,,sign 3X is a multiple angle down\Nto a single angle in terms of
Dialogue: 0,0:12:03.96,0:12:09.98,Default,,0000,0000,0000,,the sign of that angle. Let's\Nhave a look now at solving an
Dialogue: 0,0:12:09.98,0:12:17.06,Default,,0000,0000,0000,,equation. Let's take cause\N2X is equal to
Dialogue: 0,0:12:17.06,0:12:23.54,Default,,0000,0000,0000,,sign X and let's\Ntake a range of
Dialogue: 0,0:12:23.54,0:12:25.96,Default,,0000,0000,0000,,values for X.
Dialogue: 0,0:12:26.54,0:12:31.98,Default,,0000,0000,0000,,Which puts X between plus\Nand minus pie.
Dialogue: 0,0:12:33.59,0:12:36.62,Default,,0000,0000,0000,,Again, I've deliberately chosen
Dialogue: 0,0:12:36.62,0:12:42.81,Default,,0000,0000,0000,,caused 2X. Be cause we have\Na choice, we have three
Dialogue: 0,0:12:42.81,0:12:45.08,Default,,0000,0000,0000,,possibilities. Which one do we
Dialogue: 0,0:12:45.08,0:12:50.64,Default,,0000,0000,0000,,choose? Well, if I want to solve\Nan equation like this, I really
Dialogue: 0,0:12:50.64,0:12:53.68,Default,,0000,0000,0000,,need it all in terms of one trig
Dialogue: 0,0:12:53.68,0:12:56.74,Default,,0000,0000,0000,,function. Not two, but one.
Dialogue: 0,0:12:58.05,0:13:03.65,Default,,0000,0000,0000,,And here I've got sine X.\NTherefore makes sense here.
Dialogue: 0,0:13:04.24,0:13:10.70,Default,,0000,0000,0000,,To replace this by\N1 - 2 sine
Dialogue: 0,0:13:10.70,0:13:13.92,Default,,0000,0000,0000,,squared, X equals sign
Dialogue: 0,0:13:13.92,0:13:19.87,Default,,0000,0000,0000,,X. Now we have a\Nquadratic equation where the
Dialogue: 0,0:13:19.87,0:13:22.13,Default,,0000,0000,0000,,variable is sign X.
Dialogue: 0,0:13:23.10,0:13:26.90,Default,,0000,0000,0000,,Let's rearrange that so that it
Dialogue: 0,0:13:26.90,0:13:32.26,Default,,0000,0000,0000,,equals 0. Add the two sine\Nsquared to each side.
Dialogue: 0,0:13:33.68,0:13:36.88,Default,,0000,0000,0000,,Plus the sign
Dialogue: 0,0:13:36.88,0:13:41.80,Default,,0000,0000,0000,,X. And take one away\Nfrom each side.
Dialogue: 0,0:13:43.05,0:13:47.20,Default,,0000,0000,0000,,This is now a quadratic\Nequation. Can I factorize it?
Dialogue: 0,0:13:47.20,0:13:48.86,Default,,0000,0000,0000,,Let's have a look.
Dialogue: 0,0:13:48.86,0:13:54.28,Default,,0000,0000,0000,,Two brackets, 2 sign X and sign\NX when multiplied together,
Dialogue: 0,0:13:54.28,0:14:00.69,Default,,0000,0000,0000,,these two will give me the two\Nsine squared I need minus one,
Dialogue: 0,0:14:00.69,0:14:07.10,Default,,0000,0000,0000,,so let's pop a one into each\Nbracket, and one of them's got
Dialogue: 0,0:14:07.10,0:14:10.06,Default,,0000,0000,0000,,to be plus and one minus.
Dialogue: 0,0:14:11.24,0:14:17.64,Default,,0000,0000,0000,,I need plus sign X in the middle\Ngoing to make this one plus one,
Dialogue: 0,0:14:17.64,0:14:24.48,Default,,0000,0000,0000,,so I get +2 sign X, make that\None minus so I get minus sign X
Dialogue: 0,0:14:24.48,0:14:29.17,Default,,0000,0000,0000,,and when I combine those two\Nterms plus sign X there.
Dialogue: 0,0:14:30.47,0:14:37.10,Default,,0000,0000,0000,,This says. A bracket,\Na lump of algebra times by
Dialogue: 0,0:14:37.10,0:14:41.61,Default,,0000,0000,0000,,another bracket. Another lump of\Nalgebra is equal to 0.
Dialogue: 0,0:14:42.63,0:14:44.24,Default,,0000,0000,0000,,And so one.
Dialogue: 0,0:14:44.85,0:14:51.74,Default,,0000,0000,0000,,Or both of these\Nbrackets must be equal
Dialogue: 0,0:14:51.74,0:14:58.11,Default,,0000,0000,0000,,to 0. And so we've\Nreduced this fairly complicated
Dialogue: 0,0:14:58.11,0:15:05.41,Default,,0000,0000,0000,,looking equation down to two\Nsimple ones, and this one tells
Dialogue: 0,0:15:05.41,0:15:13.38,Default,,0000,0000,0000,,us here. Sign X is equal\Nto add 1 to each side
Dialogue: 0,0:15:13.38,0:15:16.04,Default,,0000,0000,0000,,and divide by two.
Dialogue: 0,0:15:16.13,0:15:21.92,Default,,0000,0000,0000,,Sign X is 1/2 or this one\Nhere tells us that sign X
Dialogue: 0,0:15:21.92,0:15:24.14,Default,,0000,0000,0000,,is equal to minus one.
Dialogue: 0,0:15:25.70,0:15:29.94,Default,,0000,0000,0000,,We've now got to extract the\Nvalues of X from this
Dialogue: 0,0:15:29.94,0:15:34.56,Default,,0000,0000,0000,,information and those values of\NX must be between plus and minus
Dialogue: 0,0:15:34.56,0:15:41.30,Default,,0000,0000,0000,,pie. So let's sketch\Nthe graph of cynex between
Dialogue: 0,0:15:41.30,0:15:43.98,Default,,0000,0000,0000,,plus and minus pie.
Dialogue: 0,0:15:45.31,0:15:48.15,Default,,0000,0000,0000,,There's the graph, there's pie.
Dialogue: 0,0:15:48.95,0:15:55.88,Default,,0000,0000,0000,,Pie by 2 - π by two and\Nminus pie and it goes between
Dialogue: 0,0:15:55.88,0:16:02.81,Default,,0000,0000,0000,,one and minus one. So let's take\Nthis one. First sign X is minus
Dialogue: 0,0:16:02.81,0:16:09.74,Default,,0000,0000,0000,,one. Well that goes across there\Nand down to their, so X is minus
Dialogue: 0,0:16:09.74,0:16:13.70,Default,,0000,0000,0000,,π by two is one answer that we
Dialogue: 0,0:16:13.70,0:16:20.17,Default,,0000,0000,0000,,get there. Sign X is 1/2, half\Ngoes across there and we should
Dialogue: 0,0:16:20.17,0:16:25.34,Default,,0000,0000,0000,,recognize that this is one of\Nthose nice numbers. Sign X is
Dialogue: 0,0:16:25.34,0:16:31.38,Default,,0000,0000,0000,,1/2 for which we've got an exact\Nanswer, and So what we do know
Dialogue: 0,0:16:31.38,0:16:36.98,Default,,0000,0000,0000,,is that the sign of 30 degrees\Nis 1/2, but where working in
Dialogue: 0,0:16:36.98,0:16:43.01,Default,,0000,0000,0000,,radians. So in fact 30 degrees\Nis the same angle as pie by 6.
Dialogue: 0,0:16:43.52,0:16:47.87,Default,,0000,0000,0000,,And this is symmetrical.\NRemember the curve for sign is
Dialogue: 0,0:16:47.87,0:16:53.96,Default,,0000,0000,0000,,symmetrical, so if that's pie by\N6 in there, that's got to be pie
Dialogue: 0,0:16:53.96,0:17:01.36,Default,,0000,0000,0000,,by 6 in there. So this, In other\Nwords will be 5 pie by 6, and so
Dialogue: 0,0:17:01.36,0:17:08.32,Default,,0000,0000,0000,,we have our two answers for this\None pie by 6 and five pie by 6.
Dialogue: 0,0:17:08.35,0:17:14.45,Default,,0000,0000,0000,,So that we see that we've been\Nable to solve our equation using
Dialogue: 0,0:17:14.45,0:17:19.14,Default,,0000,0000,0000,,our double angle formula and by\Nmaking the right choice,
Dialogue: 0,0:17:19.14,0:17:24.76,Default,,0000,0000,0000,,particularly here when with cost\N2X, we know that we have three
Dialogue: 0,0:17:24.76,0:17:31.56,Default,,0000,0000,0000,,possibilities. So. That\Nis, have a look at another
Dialogue: 0,0:17:31.56,0:17:37.57,Default,,0000,0000,0000,,equation again using our double\Nangle formula. Sign 2 X equals
Dialogue: 0,0:17:37.57,0:17:44.67,Default,,0000,0000,0000,,sign X. And again, let's take\Nour value of X to lie between
Dialogue: 0,0:17:44.67,0:17:46.85,Default,,0000,0000,0000,,plus and minus pie.
Dialogue: 0,0:17:48.16,0:17:53.13,Default,,0000,0000,0000,,We've only one choice for sign\N2X, that's two.
Dialogue: 0,0:17:53.73,0:18:00.60,Default,,0000,0000,0000,,Sign X Cos X\Nequals sign X.
Dialogue: 0,0:18:01.50,0:18:07.60,Default,,0000,0000,0000,,Now. It's very, very tempting to\Nsay our common factor on each
Dialogue: 0,0:18:07.60,0:18:09.18,Default,,0000,0000,0000,,side. Cancel it out.
Dialogue: 0,0:18:10.50,0:18:11.86,Default,,0000,0000,0000,,And then we've lost it.
Dialogue: 0,0:18:12.71,0:18:18.45,Default,,0000,0000,0000,,And because we lose it, we might\Nlose solutions to the equation.
Dialogue: 0,0:18:19.12,0:18:25.51,Default,,0000,0000,0000,,So what's better than canceling\Nout is to get everything to
Dialogue: 0,0:18:25.51,0:18:32.76,Default,,0000,0000,0000,,one side. By taking cynex\Naway from each side.
Dialogue: 0,0:18:33.31,0:18:39.77,Default,,0000,0000,0000,,And then. Take out a\Ncommon factor, and here there's
Dialogue: 0,0:18:39.77,0:18:43.16,Default,,0000,0000,0000,,a common factor of sign X.
Dialogue: 0,0:18:43.73,0:18:50.53,Default,,0000,0000,0000,,Which will leave us\N2 cause AX minus
Dialogue: 0,0:18:50.53,0:18:58.04,Default,,0000,0000,0000,,one. Two expressions multiplied\Ntogether give us 0.
Dialogue: 0,0:18:58.96,0:19:06.52,Default,,0000,0000,0000,,So either or both of these\Nexpressions is equal to 0, so
Dialogue: 0,0:19:06.52,0:19:12.19,Default,,0000,0000,0000,,either sign X equals 0 or\Ntwo cause X.
Dialogue: 0,0:19:12.83,0:19:15.52,Default,,0000,0000,0000,,Minus one equals 0.
Dialogue: 0,0:19:16.33,0:19:22.36,Default,,0000,0000,0000,,Now we've managed to reduce this\Nequation to two smaller, simpler
Dialogue: 0,0:19:22.36,0:19:27.29,Default,,0000,0000,0000,,equations once sign and the\Nother ones for cause.
Dialogue: 0,0:19:27.82,0:19:33.12,Default,,0000,0000,0000,,But each is going to give us a\Nvalue of X. Let's take this one
Dialogue: 0,0:19:33.12,0:19:35.23,Default,,0000,0000,0000,,first, sign of X is 0.
Dialogue: 0,0:19:36.29,0:19:43.67,Default,,0000,0000,0000,,And let's draw a sketch up\Nhere of sign of X. There
Dialogue: 0,0:19:43.67,0:19:51.05,Default,,0000,0000,0000,,it is and it's zero here,\Nhere and here on the X
Dialogue: 0,0:19:51.05,0:19:57.82,Default,,0000,0000,0000,,axis minus Π Zero and Pi\Nstraightaway. We've got X equals
Dialogue: 0,0:19:57.82,0:20:05.20,Default,,0000,0000,0000,,minus π and 0, not pie,\Nbecause pie is excluded from the
Dialogue: 0,0:20:05.20,0:20:07.07,Default,,0000,0000,0000,,range that. We've got.
Dialogue: 0,0:20:07.58,0:20:14.26,Default,,0000,0000,0000,,But notice sign X equals 0 gave\Nus 2 answers for X if we have
Dialogue: 0,0:20:14.26,0:20:20.93,Default,,0000,0000,0000,,cancelled sign X out up here and\Njust got rid of it, we would not
Dialogue: 0,0:20:20.93,0:20:23.16,Default,,0000,0000,0000,,have got those two answers.
Dialogue: 0,0:20:23.77,0:20:28.58,Default,,0000,0000,0000,,Let's go to this equation\Nnow. 2 cause X minus one is
Dialogue: 0,0:20:28.58,0:20:33.39,Default,,0000,0000,0000,,0, so that tells us that\Ncause X is equal to 1/2.
Dialogue: 0,0:20:34.54,0:20:39.45,Default,,0000,0000,0000,,And what we need to do is sketch\Nthe graph of cosine.
Dialogue: 0,0:20:40.43,0:20:42.49,Default,,0000,0000,0000,,And the graph of cosine.
Dialogue: 0,0:20:43.50,0:20:50.25,Default,,0000,0000,0000,,Looks. Like that between\Npie and minus pie?
Dialogue: 0,0:20:51.13,0:20:54.62,Default,,0000,0000,0000,,And here is the half.
Dialogue: 0,0:20:56.17,0:21:02.34,Default,,0000,0000,0000,,And cause X equals 1/2. This is\Nanother one of these nice
Dialogue: 0,0:21:02.34,0:21:07.99,Default,,0000,0000,0000,,relationships and we know that\Nthis one is 60 degrees or
Dialogue: 0,0:21:07.99,0:21:13.65,Default,,0000,0000,0000,,because we're working in radians\NPi by three and because of
Dialogue: 0,0:21:13.65,0:21:20.33,Default,,0000,0000,0000,,symmetry this one here has got\Nto be minus π by three, so
Dialogue: 0,0:21:20.33,0:21:25.47,Default,,0000,0000,0000,,X is minus π by 3 or\Nπ by 3.
Dialogue: 0,0:21:25.59,0:21:31.53,Default,,0000,0000,0000,,And so we've got our four\Nsolutions for this equation.
Dialogue: 0,0:21:32.53,0:21:37.36,Default,,0000,0000,0000,,In doing this work with double\Nangles, in effect, we've been
Dialogue: 0,0:21:37.36,0:21:41.75,Default,,0000,0000,0000,,looking at what are called\Nmultiple angles, and here I've
Dialogue: 0,0:21:41.75,0:21:46.58,Default,,0000,0000,0000,,been drawing sketches of a\Nsingle angle. If you like, just
Dialogue: 0,0:21:46.58,0:21:52.72,Default,,0000,0000,0000,,sign X Cos X. So the question\Nis, what does the graph of sine
Dialogue: 0,0:21:52.72,0:21:54.04,Default,,0000,0000,0000,,2X look like?
Dialogue: 0,0:21:54.67,0:22:00.43,Default,,0000,0000,0000,,What does the graph of cause 3X\Nlook like? Or for that matter,
Dialogue: 0,0:22:00.43,0:22:03.09,Default,,0000,0000,0000,,what about sign of 1/2 X?
Dialogue: 0,0:22:03.10,0:22:08.82,Default,,0000,0000,0000,,Well, let's just explore that\Nsign X. Let's just have a look
Dialogue: 0,0:22:08.82,0:22:10.26,Default,,0000,0000,0000,,at its graph.
Dialogue: 0,0:22:11.24,0:22:14.80,Default,,0000,0000,0000,,Between North And
Dialogue: 0,0:22:14.80,0:22:21.14,Default,,0000,0000,0000,,2π So\Nthat's sine
Dialogue: 0,0:22:21.14,0:22:26.28,Default,,0000,0000,0000,,X. What about\Nsign 2X?
Dialogue: 0,0:22:28.74,0:22:32.11,Default,,0000,0000,0000,,Over the same range.
Dialogue: 0,0:22:33.33,0:22:39.75,Default,,0000,0000,0000,,I just think about it. What are\Nwe doing when we're multiplying
Dialogue: 0,0:22:39.75,0:22:45.81,Default,,0000,0000,0000,,by two? Well, we're doubling\Nyes, but what that means is
Dialogue: 0,0:22:45.81,0:22:48.12,Default,,0000,0000,0000,,everything that happens on this
Dialogue: 0,0:22:48.12,0:22:52.57,Default,,0000,0000,0000,,graph. Happened twice\Nas fast on this graph.
Dialogue: 0,0:22:53.92,0:22:59.56,Default,,0000,0000,0000,,So in the time it takes this to\Ngo from North to 2π, it's done
Dialogue: 0,0:22:59.56,0:23:01.82,Default,,0000,0000,0000,,all of that in this space.
Dialogue: 0,0:23:02.97,0:23:10.37,Default,,0000,0000,0000,,So that bit of graph appears\Nin this space so that we
Dialogue: 0,0:23:10.37,0:23:17.16,Default,,0000,0000,0000,,have up down there. Then of\Ncourse it's periodic, so it
Dialogue: 0,0:23:17.16,0:23:19.01,Default,,0000,0000,0000,,does it again.
Dialogue: 0,0:23:20.70,0:23:24.36,Default,,0000,0000,0000,,So sign of 2 X.
Dialogue: 0,0:23:26.22,0:23:31.16,Default,,0000,0000,0000,,Gets through everything twice as\Nquickly as sign X and sign of
Dialogue: 0,0:23:31.16,0:23:36.93,Default,,0000,0000,0000,,three. X will get through it 3\Ntimes as quickly, so I'll have 3
Dialogue: 0,0:23:36.93,0:23:41.46,Default,,0000,0000,0000,,copies of this graph in the same\Nspace, not to pie.
Dialogue: 0,0:23:42.06,0:23:48.72,Default,,0000,0000,0000,,And the same with cause 2X and\Ncause 3X and cost 4X.
Dialogue: 0,0:23:49.29,0:23:54.52,Default,,0000,0000,0000,,What if I take sign of\N1/2 of X?
Dialogue: 0,0:23:54.52,0:24:01.43,Default,,0000,0000,0000,,Do that. Sign of\N1/2 of X.
Dialogue: 0,0:24:02.21,0:24:09.42,Default,,0000,0000,0000,,I'll do sign of X first just\Nto give us the picture again.
Dialogue: 0,0:24:09.94,0:24:13.76,Default,,0000,0000,0000,,That's our graph between North
Dialogue: 0,0:24:13.76,0:24:17.27,Default,,0000,0000,0000,,and 2π. So what about?
Dialogue: 0,0:24:17.84,0:24:23.20,Default,,0000,0000,0000,,Sign. Of half of X\Non the same scale.
Dialogue: 0,0:24:24.16,0:24:27.38,Default,,0000,0000,0000,,Well, things are happening half
Dialogue: 0,0:24:27.38,0:24:34.31,Default,,0000,0000,0000,,as quickly. 1/2 of two\Npies just pie, so we will only
Dialogue: 0,0:24:34.31,0:24:40.83,Default,,0000,0000,0000,,have got through that bit of the\Ncurve by the time we've got to
Dialogue: 0,0:24:40.83,0:24:47.36,Default,,0000,0000,0000,,2π, so that in effect the graph\Nof sign a half pie looks like
Dialogue: 0,0:24:47.36,0:24:49.22,Default,,0000,0000,0000,,that and continues on.
Dialogue: 0,0:24:49.89,0:24:52.45,Default,,0000,0000,0000,,In double the space.
Dialogue: 0,0:24:53.90,0:24:58.51,Default,,0000,0000,0000,,So graphs of multiple angles\Nlook a little bit different to
Dialogue: 0,0:24:58.51,0:24:59.77,Default,,0000,0000,0000,,the ordinary angle.
Dialogue: 0,0:25:00.48,0:25:05.49,Default,,0000,0000,0000,,But the thing that we have to\Nremember is that if we have a
Dialogue: 0,0:25:05.49,0:25:06.92,Default,,0000,0000,0000,,multiple here, that's greater
Dialogue: 0,0:25:06.92,0:25:11.78,Default,,0000,0000,0000,,than one. Then it's going to get\Nthrough it much quicker and
Dialogue: 0,0:25:11.78,0:25:13.73,Default,,0000,0000,0000,,we're going to see the graph
Dialogue: 0,0:25:13.73,0:25:18.26,Default,,0000,0000,0000,,repeated. If we've got a\Nmultiple here, that's less than
Dialogue: 0,0:25:18.26,0:25:21.93,Default,,0000,0000,0000,,one, it's going to take longer\Nto get through.
Dialogue: 0,0:25:22.81,0:25:28.28,Default,,0000,0000,0000,,The graph and we're going to see\Nthe graph extended and drawn
Dialogue: 0,0:25:28.28,0:25:34.45,Default,,0000,0000,0000,,out. Let's have\Na look at the
Dialogue: 0,0:25:34.45,0:25:40.07,Default,,0000,0000,0000,,graph of cause\NX and cause 2X.
Dialogue: 0,0:25:42.76,0:25:45.45,Default,,0000,0000,0000,,Again between North.
Dialogue: 0,0:25:45.95,0:25:48.15,Default,,0000,0000,0000,,And. 2π
Dialogue: 0,0:25:49.43,0:25:51.43,Default,,0000,0000,0000,,So there's our graph.
Dialogue: 0,0:25:53.98,0:26:00.22,Default,,0000,0000,0000,,What about? Mark the points in\Nthe same positions. This is
Dialogue: 0,0:26:00.22,0:26:06.01,Default,,0000,0000,0000,,going to go through twice as\Nquickly, so we're going to see
Dialogue: 0,0:26:06.01,0:26:11.33,Default,,0000,0000,0000,,that shape repeated in this\Nspace here, so we're going to
Dialogue: 0,0:26:11.33,0:26:18.09,Default,,0000,0000,0000,,see that come down and go up,\Nand then we're going to see it
Dialogue: 0,0:26:18.09,0:26:21.90,Default,,0000,0000,0000,,repeated again. I thought so.\NSketching these graphs of
Dialogue: 0,0:26:21.90,0:26:25.99,Default,,0000,0000,0000,,multiple angles is quite easy.\NAll you need to do is look at
Dialogue: 0,0:26:25.99,0:26:29.46,Default,,0000,0000,0000,,the original graph and judge\Nthe number of times that it's
Dialogue: 0,0:26:29.46,0:26:31.98,Default,,0000,0000,0000,,going to be repeated over the\Ngiven range.