[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:01.09,0:00:08.20,Default,,0000,0000,0000,,In this video, we're going to be\Nhaving a look at a particular Dialogue: 0,0:00:08.20,0:00:14.22,Default,,0000,0000,0000,,form of trigonometric function.\NSo in order to get into this Dialogue: 0,0:00:14.22,0:00:19.69,Default,,0000,0000,0000,,form, let's just consider 3\Ncause X +4 sign X. Dialogue: 0,0:00:20.67,0:00:23.13,Default,,0000,0000,0000,,Now it's two terms. Dialogue: 0,0:00:23.70,0:00:27.76,Default,,0000,0000,0000,,And ideally, if we were going to\Nsolve an equation or do Dialogue: 0,0:00:27.76,0:00:31.81,Default,,0000,0000,0000,,something useful with it, it\Nmight be better if it was one Dialogue: 0,0:00:31.81,0:00:35.53,Default,,0000,0000,0000,,term. So the question is, how\Ncan we bring these together? Dialogue: 0,0:00:36.09,0:00:39.76,Default,,0000,0000,0000,,Let's have a look at the graph\Nof this function. Dialogue: 0,0:00:40.79,0:00:47.17,Default,,0000,0000,0000,,So we'll get into the right mode\Non the Calculator and graph and Dialogue: 0,0:00:47.17,0:00:49.14,Default,,0000,0000,0000,,will graph three cars. Dialogue: 0,0:00:49.68,0:00:50.45,Default,,0000,0000,0000,,X. Dialogue: 0,0:00:51.52,0:00:55.25,Default,,0000,0000,0000,,Plus four sign X. Dialogue: 0,0:01:00.56,0:01:01.85,Default,,0000,0000,0000,,And now we've graphed it. Dialogue: 0,0:01:02.70,0:01:06.94,Default,,0000,0000,0000,,We can see that it's looking\Nlike a periodic function. Dialogue: 0,0:01:07.51,0:01:11.25,Default,,0000,0000,0000,,Looks In other words, rather\Nlike sign or cause. Dialogue: 0,0:01:11.81,0:01:16.18,Default,,0000,0000,0000,,So if we look at the little\Nticks on the Y axis, we can see Dialogue: 0,0:01:16.18,0:01:17.63,Default,,0000,0000,0000,,that it's approximately 5 high. Dialogue: 0,0:01:18.16,0:01:22.69,Default,,0000,0000,0000,,On the peaks and approximately 5\Ndeep on the troughs. Dialogue: 0,0:01:23.19,0:01:27.69,Default,,0000,0000,0000,,So if it looks like causal\Nsign, let's graph. Dialogue: 0,0:01:29.13,0:01:36.25,Default,,0000,0000,0000,,Cause. X. So\Nthere's cause X plowing along. Dialogue: 0,0:01:37.69,0:01:42.70,Default,,0000,0000,0000,,You can see they both look\Nperiodic and what we want to do Dialogue: 0,0:01:42.70,0:01:45.00,Default,,0000,0000,0000,,is to change cause X into. Dialogue: 0,0:01:45.53,0:01:47.17,Default,,0000,0000,0000,,What we had to begin with. Dialogue: 0,0:01:48.05,0:01:52.59,Default,,0000,0000,0000,,So let's remember that the\Nheight of the peaks was five and Dialogue: 0,0:01:52.59,0:01:56.74,Default,,0000,0000,0000,,the depth of the troughs was\Nfive. So let's multiply cosine Dialogue: 0,0:01:56.74,0:01:58.63,Default,,0000,0000,0000,,by 5, so we'll graph. Dialogue: 0,0:01:59.90,0:02:03.70,Default,,0000,0000,0000,,5 calls X. Dialogue: 0,0:02:06.97,0:02:11.81,Default,,0000,0000,0000,,And what we see is, we've gotten\Nalmost exact copy. It's just Dialogue: 0,0:02:11.81,0:02:13.42,Default,,0000,0000,0000,,displaced a little bit. Dialogue: 0,0:02:14.06,0:02:17.43,Default,,0000,0000,0000,,Things are happening a little\Nbit too soon. Dialogue: 0,0:02:18.76,0:02:22.84,Default,,0000,0000,0000,,OK, that being the case, how\Nmuch too soon it seems to come Dialogue: 0,0:02:22.84,0:02:27.55,Default,,0000,0000,0000,,if we look at the ticks on the X\Naxis and look at the distance Dialogue: 0,0:02:27.55,0:02:28.81,Default,,0000,0000,0000,,between the two peaks. Dialogue: 0,0:02:29.45,0:02:34.90,Default,,0000,0000,0000,,We seem to be about 1 unit on\Nthe X axis early. Dialogue: 0,0:02:35.49,0:02:39.32,Default,,0000,0000,0000,,So let's correct that and\Nlet's graph. Dialogue: 0,0:02:41.86,0:02:42.82,Default,,0000,0000,0000,,5. Dialogue: 0,0:02:43.84,0:02:44.74,Default,,0000,0000,0000,,Kohl's Dialogue: 0,0:02:45.86,0:02:49.06,Default,,0000,0000,0000,,Of X minus one. Dialogue: 0,0:02:52.37,0:02:56.94,Default,,0000,0000,0000,,And that's almost correct. It's\Njust covering over. Dialogue: 0,0:02:58.11,0:03:05.22,Default,,0000,0000,0000,,The graph and making it a little\Nbit thicker, so it seems that we Dialogue: 0,0:03:05.22,0:03:08.78,Default,,0000,0000,0000,,can write this as 5 cause X Dialogue: 0,0:03:08.78,0:03:15.98,Default,,0000,0000,0000,,minus one. Now my Calculator\Nwas working in radians, so Dialogue: 0,0:03:15.98,0:03:19.48,Default,,0000,0000,0000,,this one is one Radian. Dialogue: 0,0:03:19.48,0:03:27.14,Default,,0000,0000,0000,,3 calls X +4 sign. X\Nis 5 cause X minus one. Dialogue: 0,0:03:27.64,0:03:32.55,Default,,0000,0000,0000,,Well, 3, four and five or a\Npythagoreans triple. Dialogue: 0,0:03:33.81,0:03:40.22,Default,,0000,0000,0000,,3 squared +4 squared\Nequals 5 squared. Dialogue: 0,0:03:41.07,0:03:47.80,Default,,0000,0000,0000,,So that kind of size to us. If\Nwe had any function, A cause X Dialogue: 0,0:03:47.80,0:03:52.30,Default,,0000,0000,0000,,Plus B cynex, could we bring\Nthis function? These two Dialogue: 0,0:03:52.30,0:03:56.78,Default,,0000,0000,0000,,functions together as one\Nfunction like that, and if we Dialogue: 0,0:03:56.78,0:04:03.07,Default,,0000,0000,0000,,did it with the number that went\Nin front with the square root of Dialogue: 0,0:04:03.07,0:04:05.32,Default,,0000,0000,0000,,A squared plus B squared. Dialogue: 0,0:04:06.05,0:04:12.80,Default,,0000,0000,0000,,Cause of X minus something,\Nlet's call it Alpha. And Dialogue: 0,0:04:12.80,0:04:15.50,Default,,0000,0000,0000,,then what is Alpha? Dialogue: 0,0:04:16.94,0:04:21.63,Default,,0000,0000,0000,,OK, all we've done is some graph\Nwork, made some guesses what we Dialogue: 0,0:04:21.63,0:04:26.33,Default,,0000,0000,0000,,have to do now is work with\Nalgebra and work with our trig Dialogue: 0,0:04:26.33,0:04:29.94,Default,,0000,0000,0000,,functions and our trig\Nidentities to try and see if Dialogue: 0,0:04:29.94,0:04:32.10,Default,,0000,0000,0000,,this little bit here at the Dialogue: 0,0:04:32.10,0:04:38.93,Default,,0000,0000,0000,,bottom. Can actually be made\Nto work, so that's our next Dialogue: 0,0:04:38.93,0:04:46.66,Default,,0000,0000,0000,,step. So let's start with where\Nwe want. We want to end Dialogue: 0,0:04:46.66,0:04:53.17,Default,,0000,0000,0000,,up a number are multiplying\Ncause of X minus Alpha. Dialogue: 0,0:04:55.07,0:05:02.17,Default,,0000,0000,0000,,Equals are now cause of\NX minus Alpha is one Dialogue: 0,0:05:02.17,0:05:09.27,Default,,0000,0000,0000,,of our addition formula, so\Nwe can expand it as Dialogue: 0,0:05:09.27,0:05:12.82,Default,,0000,0000,0000,,cause X cause Alpha plus Dialogue: 0,0:05:12.82,0:05:15.76,Default,,0000,0000,0000,,sign. X Dialogue: 0,0:05:15.76,0:05:23.25,Default,,0000,0000,0000,,sign Alpha That's\Nmultiply out that bracket are Dialogue: 0,0:05:23.25,0:05:30.95,Default,,0000,0000,0000,,cause X cause Alpha plus\Nour sign X sign Alpha. Dialogue: 0,0:05:31.88,0:05:38.83,Default,,0000,0000,0000,,Now I want to rewrite this\Nso it's very clear what I Dialogue: 0,0:05:38.83,0:05:45.78,Default,,0000,0000,0000,,am multiplying cause X by. So\Nhere I am multiplying cause X Dialogue: 0,0:05:45.78,0:05:52.14,Default,,0000,0000,0000,,by our cause Alpha and I'm\Nmultiplying synex buy our sign Dialogue: 0,0:05:52.14,0:05:59.09,Default,,0000,0000,0000,,Alpha. So let's make that very\Nclear off 'cause Alpha times by Dialogue: 0,0:05:59.09,0:06:04.30,Default,,0000,0000,0000,,cause X plus our sign Alpha\Ntimes by sign. Dialogue: 0,0:06:04.33,0:06:12.32,Default,,0000,0000,0000,,Thanks and what we want is\Nthat this should be a cause Dialogue: 0,0:06:12.32,0:06:19.65,Default,,0000,0000,0000,,X Plus B sign X. In\Nother words, this our cause Dialogue: 0,0:06:19.65,0:06:27.64,Default,,0000,0000,0000,,Alpha will be A and this\Nare sign Alpha will be B Dialogue: 0,0:06:27.64,0:06:33.63,Default,,0000,0000,0000,,so we can make that\Nidentification are cause Alpha Dialogue: 0,0:06:33.63,0:06:39.61,Default,,0000,0000,0000,,is a. And B is\Nour sign Alpha. Dialogue: 0,0:06:40.23,0:06:46.36,Default,,0000,0000,0000,,Now, how can we bring these\Ntogether? Well, one way is to Dialogue: 0,0:06:46.36,0:06:51.98,Default,,0000,0000,0000,,notice that cost square plus\Nsign squared is one. So let's Dialogue: 0,0:06:51.98,0:06:58.12,Default,,0000,0000,0000,,Square this one, A squared\Nsquare. This one and add the two Dialogue: 0,0:06:58.12,0:07:02.71,Default,,0000,0000,0000,,together. A squared plus B\Nsquared is R-squared cost Dialogue: 0,0:07:02.71,0:07:06.29,Default,,0000,0000,0000,,squared Alpha plus R-squared,\Nsine squared Alpha. Dialogue: 0,0:07:06.89,0:07:12.54,Default,,0000,0000,0000,,And there's a common factor of\NR-squared that we can take out, Dialogue: 0,0:07:12.54,0:07:16.78,Default,,0000,0000,0000,,leaving us Cos squared Alpha\Nplus sign squared Alpha. Dialogue: 0,0:07:17.36,0:07:21.74,Default,,0000,0000,0000,,This is one of our basic\Nidentities, Cos squared plus Dialogue: 0,0:07:21.74,0:07:27.43,Default,,0000,0000,0000,,sign squared at the same angle\Nis always one and so that gives Dialogue: 0,0:07:27.43,0:07:32.69,Default,,0000,0000,0000,,us our squared. So the thing\Nthat we said was a possibility, Dialogue: 0,0:07:32.69,0:07:38.82,Default,,0000,0000,0000,,in fact is a result that if we\Ntake A squared plus B squared, Dialogue: 0,0:07:38.82,0:07:44.08,Default,,0000,0000,0000,,take the square root, we do\Nindeed end up with are the Dialogue: 0,0:07:44.08,0:07:48.02,Default,,0000,0000,0000,,number that goes in front here\Nof the cosine. Dialogue: 0,0:07:48.80,0:07:53.55,Default,,0000,0000,0000,,But what about the Alpha? Have\Nwe got enough information at Dialogue: 0,0:07:53.55,0:07:58.30,Default,,0000,0000,0000,,this point here to find Alpha?\NWell, let's bright these two Dialogue: 0,0:07:58.30,0:08:03.49,Default,,0000,0000,0000,,equations a equals our cause.\NAlpha and B equals R. Sign Alpha Dialogue: 0,0:08:03.49,0:08:08.67,Default,,0000,0000,0000,,down again, but this time let me\Nwrite them above one another. Dialogue: 0,0:08:08.71,0:08:14.88,Default,,0000,0000,0000,,So we write these down\Nagain, but this time. Dialogue: 0,0:08:14.89,0:08:17.12,Default,,0000,0000,0000,,Going to write. Dialogue: 0,0:08:17.12,0:08:20.72,Default,,0000,0000,0000,,The top and the other one Dialogue: 0,0:08:20.72,0:08:26.51,Default,,0000,0000,0000,,directly underneath. By doing it\Nlike that, it suggests the Dialogue: 0,0:08:26.51,0:08:31.01,Default,,0000,0000,0000,,possibility of dividing this\Nequation into this equation and Dialogue: 0,0:08:31.01,0:08:38.51,Default,,0000,0000,0000,,so we have B over A is equal\Nto R sign Alpha over our cause Dialogue: 0,0:08:38.51,0:08:45.01,Default,,0000,0000,0000,,Alpha. We can divide top and\Nbottom by R and we're just left Dialogue: 0,0:08:45.01,0:08:51.01,Default,,0000,0000,0000,,with sign out for over 'cause\NAlpha, which we know is Tan Dialogue: 0,0:08:51.01,0:08:57.24,Default,,0000,0000,0000,,Alpha. And so now we\Ncan workout what Alpha is, Dialogue: 0,0:08:57.24,0:09:03.00,Default,,0000,0000,0000,,because Alpha is the angle whose\Ntangent is B over A. Dialogue: 0,0:09:03.97,0:09:11.03,Default,,0000,0000,0000,,So what we've got now is\Nthat we can write any expression Dialogue: 0,0:09:11.03,0:09:16.91,Default,,0000,0000,0000,,of the form a Cos X\NPlus B Sign X. Dialogue: 0,0:09:16.91,0:09:23.56,Default,,0000,0000,0000,,As a function which is our\Ncause X minus Alpha. Dialogue: 0,0:09:24.10,0:09:31.21,Default,,0000,0000,0000,,And we know that all will be the\Nsquare root of A squared plus B Dialogue: 0,0:09:31.21,0:09:36.90,Default,,0000,0000,0000,,squared, and the Alpha will be\Nthe angle whose tangent is B Dialogue: 0,0:09:36.90,0:09:44.19,Default,,0000,0000,0000,,over a. And that is a\Nvery very useful tool to have Dialogue: 0,0:09:44.19,0:09:50.57,Default,,0000,0000,0000,,at ones disposal. It reduces two\Ntrig functions to one trick Dialogue: 0,0:09:50.57,0:09:55.79,Default,,0000,0000,0000,,function makes it so much easier\Nto deal with. Dialogue: 0,0:09:55.81,0:10:01.23,Default,,0000,0000,0000,,So let's have a look at how we\Ncan use this to solve equations Dialogue: 0,0:10:01.23,0:10:06.26,Default,,0000,0000,0000,,and also how we can use it to\Ndetermine Maxima and minima of Dialogue: 0,0:10:06.26,0:10:08.58,Default,,0000,0000,0000,,functions. So will begin with an Dialogue: 0,0:10:08.58,0:10:11.95,Default,,0000,0000,0000,,equation. Route 2. Dialogue: 0,0:10:12.85,0:10:19.65,Default,,0000,0000,0000,,Cause X. Plus\Nsign X equals Dialogue: 0,0:10:19.65,0:10:25.81,Default,,0000,0000,0000,,1. Now if we\Ncompare this with the standard Dialogue: 0,0:10:25.81,0:10:32.75,Default,,0000,0000,0000,,form we've been working with, A\Ncause X Plus B Sign X and Dialogue: 0,0:10:32.75,0:10:40.23,Default,,0000,0000,0000,,a is equal to two and B\Nis equal to 1. So we can Dialogue: 0,0:10:40.23,0:10:46.63,Default,,0000,0000,0000,,calculate are. All is the square\Nroot of A squared plus B Dialogue: 0,0:10:46.63,0:10:53.54,Default,,0000,0000,0000,,squared. So that's the square\Nroot of 2 all squared plus one Dialogue: 0,0:10:53.54,0:10:56.67,Default,,0000,0000,0000,,squared. 2 all squared is 2. Dialogue: 0,0:10:57.39,0:11:02.41,Default,,0000,0000,0000,,1 squared is just one, so this\Nis Route 3. Dialogue: 0,0:11:03.03,0:11:10.79,Default,,0000,0000,0000,,So what we've got is\Nnow our cause of X Dialogue: 0,0:11:10.79,0:11:18.55,Default,,0000,0000,0000,,minus Alpha, so we've got\NRoute 3 cause of X Dialogue: 0,0:11:18.55,0:11:21.65,Default,,0000,0000,0000,,minus Alpha equals 1. Dialogue: 0,0:11:21.66,0:11:26.95,Default,,0000,0000,0000,,What we have to do is we've got\Nto workout what Alpha is so. Dialogue: 0,0:11:27.73,0:11:34.73,Default,,0000,0000,0000,,We know that Tan Alpha\Nis equal to B over Dialogue: 0,0:11:34.73,0:11:41.73,Default,,0000,0000,0000,,A. And so that's\None over Route 2. Dialogue: 0,0:11:42.48,0:11:48.11,Default,,0000,0000,0000,,And so we need to know what does\Nthat tell us about Alpha? Dialogue: 0,0:11:48.62,0:11:52.82,Default,,0000,0000,0000,,Well, one of the things we\Nhaven't specified is what's the Dialogue: 0,0:11:52.82,0:11:57.79,Default,,0000,0000,0000,,range of values of X and what\Nunits are we working in. So Dialogue: 0,0:11:57.79,0:12:02.37,Default,,0000,0000,0000,,let's say we're going to be\Nworking in radians, and that we Dialogue: 0,0:12:02.37,0:12:05.05,Default,,0000,0000,0000,,want X to be between plus and Dialogue: 0,0:12:05.05,0:12:10.80,Default,,0000,0000,0000,,minus pie. So if X is in radians\Nthen Alpha's got to be in Dialogue: 0,0:12:10.80,0:12:15.31,Default,,0000,0000,0000,,radians. I will now bringing the\NCalculator in order to work that Dialogue: 0,0:12:15.31,0:12:18.88,Default,,0000,0000,0000,,out. So I want the angle Dialogue: 0,0:12:18.88,0:12:21.21,Default,,0000,0000,0000,,whose Tangent. Is. Dialogue: 0,0:12:22.40,0:12:25.67,Default,,0000,0000,0000,,1. Divided by. Dialogue: 0,0:12:27.32,0:12:29.08,Default,,0000,0000,0000,,Square root of. Dialogue: 0,0:12:30.69,0:12:31.19,Default,,0000,0000,0000,,2. Dialogue: 0,0:12:32.51,0:12:39.81,Default,,0000,0000,0000,,And that angle is nought.\N.615 lots of other figures, Dialogue: 0,0:12:39.81,0:12:45.65,Default,,0000,0000,0000,,but will stop there for\Nthe moment radiance. Dialogue: 0,0:12:45.66,0:12:50.16,Default,,0000,0000,0000,,So now I know what our is and I\Nknow what Alpha is. Dialogue: 0,0:12:50.68,0:12:54.02,Default,,0000,0000,0000,,I'm not going to write down\NNorth Point 615 for a little Dialogue: 0,0:12:54.02,0:12:55.96,Default,,0000,0000,0000,,while. I'm going to keep it as Dialogue: 0,0:12:55.96,0:13:01.69,Default,,0000,0000,0000,,Alpha. But the equation that\Nwe're now trying to solve is Dialogue: 0,0:13:01.69,0:13:08.98,Default,,0000,0000,0000,,Route 3. Cause of X\Nminus Alpha is equal to 1 Dialogue: 0,0:13:08.98,0:13:16.48,Default,,0000,0000,0000,,and we know that we want\NX to be between pie and. Dialogue: 0,0:13:16.99,0:13:24.16,Default,,0000,0000,0000,,Minus pie. Let's have a look\Nat What is this cosine want to Dialogue: 0,0:13:24.16,0:13:29.66,Default,,0000,0000,0000,,do that? We're going to divide\Nboth sides by Route 3. Dialogue: 0,0:13:30.22,0:13:37.29,Default,,0000,0000,0000,,And so we have cause of\NX minus Alpha is one over Dialogue: 0,0:13:37.29,0:13:42.00,Default,,0000,0000,0000,,Route 3. So what is X\Nminus Alpha? Dialogue: 0,0:13:43.59,0:13:45.97,Default,,0000,0000,0000,,Let's just draw a little graph. Dialogue: 0,0:13:47.11,0:13:48.95,Default,,0000,0000,0000,,To help us see what we're Dialogue: 0,0:13:48.95,0:13:54.48,Default,,0000,0000,0000,,looking at. Between\Nplus and minus pie. Dialogue: 0,0:13:55.84,0:13:57.37,Default,,0000,0000,0000,,The cosine graph. Dialogue: 0,0:13:59.27,0:14:00.53,Default,,0000,0000,0000,,Looks like that. Dialogue: 0,0:14:01.60,0:14:03.71,Default,,0000,0000,0000,,There's plus pie. Dialogue: 0,0:14:04.32,0:14:10.68,Default,,0000,0000,0000,,As minus pie, this is minus π by\Ntwo, and this is π by 2. Dialogue: 0,0:14:11.76,0:14:16.12,Default,,0000,0000,0000,,So we're looking for two\Nanswers across here. Dialogue: 0,0:14:17.38,0:14:19.77,Default,,0000,0000,0000,,Come down. To there. Dialogue: 0,0:14:20.78,0:14:25.20,Default,,0000,0000,0000,,So let's have a look for one of\Nthem. An hour Calculator will Dialogue: 0,0:14:25.20,0:14:26.90,Default,,0000,0000,0000,,tell us this one here. Dialogue: 0,0:14:27.81,0:14:34.36,Default,,0000,0000,0000,,So the angle that\NI want is the Dialogue: 0,0:14:34.36,0:14:37.64,Default,,0000,0000,0000,,angle whose tangent is Dialogue: 0,0:14:37.64,0:14:43.07,Default,,0000,0000,0000,,one. Divided by the square\Nroot of 3. Dialogue: 0,0:14:45.35,0:14:46.59,Default,,0000,0000,0000,,And that is. Dialogue: 0,0:14:47.59,0:14:49.62,Default,,0000,0000,0000,,Nought Dialogue: 0,0:14:49.62,0:14:52.90,Default,,0000,0000,0000,,.95 5. Dialogue: 0,0:14:54.88,0:14:59.80,Default,,0000,0000,0000,,Now if that's not .955\Nthere, remember our Dialogue: 0,0:14:59.80,0:15:06.56,Default,,0000,0000,0000,,Calculator will give us this\None. Then the other one by Dialogue: 0,0:15:06.56,0:15:09.64,Default,,0000,0000,0000,,symmetry is minus N .955. Dialogue: 0,0:15:13.05,0:15:18.84,Default,,0000,0000,0000,,Well, now we've got X minus\NAlpha. What we need is X, so Dialogue: 0,0:15:18.84,0:15:21.95,Default,,0000,0000,0000,,this will be X equals nought .9. Dialogue: 0,0:15:22.47,0:15:27.81,Default,,0000,0000,0000,,55 plus\NAlpha or Dialogue: 0,0:15:27.81,0:15:33.14,Default,,0000,0000,0000,,minus nought.\N.955 plus Dialogue: 0,0:15:33.14,0:15:39.81,Default,,0000,0000,0000,,Alpha. And if\Nyou remember Alpha, we Dialogue: 0,0:15:39.81,0:15:47.79,Default,,0000,0000,0000,,calculated as nought. .615. So\Nwe can write the value Dialogue: 0,0:15:47.79,0:15:50.18,Default,,0000,0000,0000,,of Alpha rent. Dialogue: 0,0:15:50.93,0:15:53.45,Default,,0000,0000,0000,,And work out these two. Dialogue: 0,0:15:54.22,0:16:01.03,Default,,0000,0000,0000,,Values of X, so\Nwe add these two Dialogue: 0,0:16:01.03,0:16:06.13,Default,,0000,0000,0000,,together. We're going to\Nget 1.570. Dialogue: 0,0:16:07.50,0:16:09.67,Default,,0000,0000,0000,,And if we take these two. Dialogue: 0,0:16:10.53,0:16:17.59,Default,,0000,0000,0000,,We're going to get minus\Nnought .340 so if we Dialogue: 0,0:16:17.59,0:16:21.12,Default,,0000,0000,0000,,work to two decimal places, Dialogue: 0,0:16:21.12,0:16:28.41,Default,,0000,0000,0000,,that's 1.57. And minus\Nnought .3 four radians. Dialogue: 0,0:16:30.70,0:16:34.42,Default,,0000,0000,0000,,So let's take another example. Dialogue: 0,0:16:35.28,0:16:41.78,Default,,0000,0000,0000,,It's time you have cause\NX Minus Route 3. Dialogue: 0,0:16:42.50,0:16:46.16,Default,,0000,0000,0000,,Sign X is equal to 2. Dialogue: 0,0:16:46.81,0:16:52.11,Default,,0000,0000,0000,,And we'll take X to be in\Ndegrees and between North. Dialogue: 0,0:16:52.64,0:16:55.58,Default,,0000,0000,0000,,And 360. Dialogue: 0,0:16:56.09,0:17:02.36,Default,,0000,0000,0000,,OK. Well, slight\Ndifference here. We've got a Dialogue: 0,0:17:02.36,0:17:08.95,Default,,0000,0000,0000,,minus sign and not a plus sign,\Nso let's see what this means. A Dialogue: 0,0:17:08.95,0:17:15.54,Default,,0000,0000,0000,,must be equal to 1 because we've\Ngot one cause X&B must be equal Dialogue: 0,0:17:15.54,0:17:16.96,Default,,0000,0000,0000,,to minus three. Dialogue: 0,0:17:17.49,0:17:20.31,Default,,0000,0000,0000,,And this is going to make a\Ndifference to what happens. Dialogue: 0,0:17:20.96,0:17:26.76,Default,,0000,0000,0000,,Let's leave it on one side for\Nthe moment and go ahead with our Dialogue: 0,0:17:26.76,0:17:31.72,Default,,0000,0000,0000,,calculation. Are will be the\Nsquare root of A squared plus B Dialogue: 0,0:17:31.72,0:17:39.44,Default,,0000,0000,0000,,squared. Which will be the\Nsquare root of 1 squared plus Dialogue: 0,0:17:39.44,0:17:41.40,Default,,0000,0000,0000,,minus 3 squared. Dialogue: 0,0:17:41.96,0:17:44.77,Default,,0000,0000,0000,,1 squared is one. Dialogue: 0,0:17:45.45,0:17:52.06,Default,,0000,0000,0000,,Minus Route 3 squared is 3, so\Nit's a square root of 1 + 3, Dialogue: 0,0:17:52.06,0:17:55.59,Default,,0000,0000,0000,,which is 4 and so are is 2. Dialogue: 0,0:17:56.16,0:18:03.76,Default,,0000,0000,0000,,So what we've got now is\N2 cause X minus Alpha equals Dialogue: 0,0:18:03.76,0:18:10.09,Default,,0000,0000,0000,,2 and we can divide\Nthroughout by these two to Dialogue: 0,0:18:10.09,0:18:15.78,Default,,0000,0000,0000,,give us cause of X minus\NAlpha equals 1. Dialogue: 0,0:18:17.13,0:18:23.61,Default,,0000,0000,0000,,This should be relatively easy\Nto solve provided we can workout Dialogue: 0,0:18:23.61,0:18:30.09,Default,,0000,0000,0000,,what Alpha is. So what is\NAlpha now? Remember we calculate Dialogue: 0,0:18:30.09,0:18:37.16,Default,,0000,0000,0000,,Alpha's tan Alpha is B over\Na, which in this case is Dialogue: 0,0:18:37.16,0:18:43.64,Default,,0000,0000,0000,,minus Route 3 over 1, which\Nis just minus Route 3. Dialogue: 0,0:18:44.23,0:18:47.30,Default,,0000,0000,0000,,This is a problem to us. Dialogue: 0,0:18:47.85,0:18:51.34,Default,,0000,0000,0000,,It's a problem because\Npreviously all the answers have Dialogue: 0,0:18:51.34,0:18:55.100,Default,,0000,0000,0000,,been positive and we have had\Npositive A and positive be and Dialogue: 0,0:18:55.100,0:18:57.94,Default,,0000,0000,0000,,we've gone straight for the Dialogue: 0,0:18:57.94,0:19:04.25,Default,,0000,0000,0000,,first quadrant. Let's just\Nremind ourselves what we mean by Dialogue: 0,0:19:04.25,0:19:07.59,Default,,0000,0000,0000,,Quadrant. I'll write down again. Dialogue: 0,0:19:08.83,0:19:16.24,Default,,0000,0000,0000,,10A. Which\Nis calculated as B over a. Dialogue: 0,0:19:16.96,0:19:23.65,Default,,0000,0000,0000,,Let's just remind ourselves what\Nthese are that are sign Alpha Dialogue: 0,0:19:23.65,0:19:26.08,Default,,0000,0000,0000,,and our cause Alpha. Dialogue: 0,0:19:26.60,0:19:32.55,Default,,0000,0000,0000,,And in this question then minus\NRoute 3 over 1. Dialogue: 0,0:19:33.77,0:19:39.96,Default,,0000,0000,0000,,Revise what we know about\Nquadrants. This is the first Dialogue: 0,0:19:39.96,0:19:42.67,Default,,0000,0000,0000,,quadrant. The second quadrant. Dialogue: 0,0:19:43.17,0:19:48.89,Default,,0000,0000,0000,,The third quadrant, the 4th\Nquadrant going round Dialogue: 0,0:19:48.89,0:19:54.96,Default,,0000,0000,0000,,anticlockwise. Now in the first\Nquadrant, all of our trig Dialogue: 0,0:19:54.96,0:19:58.10,Default,,0000,0000,0000,,ratios, a for all that all Dialogue: 0,0:19:58.10,0:20:05.09,Default,,0000,0000,0000,,positive. In the second\Nquadrant, only the sine ratio is Dialogue: 0,0:20:05.09,0:20:10.36,Default,,0000,0000,0000,,positive. In the third\Nquadrant, only the tangent Dialogue: 0,0:20:10.36,0:20:16.14,Default,,0000,0000,0000,,ratio is positive, and in the\Nfourth quadrant only the Dialogue: 0,0:20:16.14,0:20:18.45,Default,,0000,0000,0000,,cosine ratio is positive. Dialogue: 0,0:20:19.57,0:20:25.93,Default,,0000,0000,0000,,Now we have 10 Alpha is\Nminus Route 3 over 1. Dialogue: 0,0:20:26.64,0:20:31.07,Default,,0000,0000,0000,,So it's negative, which means\Nthat we must either be in the Dialogue: 0,0:20:31.07,0:20:34.76,Default,,0000,0000,0000,,second quadrant or the 4th\Nquadrant, which is where tangent Dialogue: 0,0:20:34.76,0:20:39.92,Default,,0000,0000,0000,,is negative. So we have a choice\Nto make our angle Alpha is in Dialogue: 0,0:20:39.92,0:20:41.77,Default,,0000,0000,0000,,here or it's in here. Dialogue: 0,0:20:43.60,0:20:48.60,Default,,0000,0000,0000,,Let's have a look what we can\Nsee from here are sign Alpha is Dialogue: 0,0:20:48.60,0:20:50.74,Default,,0000,0000,0000,,minus Route 3 are cause, Alpha Dialogue: 0,0:20:50.74,0:20:57.18,Default,,0000,0000,0000,,is one. That means sign Alpha\Nmust be negative and cause Alpha Dialogue: 0,0:20:57.18,0:20:58.70,Default,,0000,0000,0000,,must be positive. Dialogue: 0,0:20:59.35,0:21:04.46,Default,,0000,0000,0000,,And it's in this 4th quadrant\Nwhere cause Alpha is positive Dialogue: 0,0:21:04.46,0:21:06.79,Default,,0000,0000,0000,,and sign Alpha is negative. Dialogue: 0,0:21:07.44,0:21:12.83,Default,,0000,0000,0000,,So our tan Alpha is minus Route\N3 is in here somewhere. Dialogue: 0,0:21:13.36,0:21:19.93,Default,,0000,0000,0000,,So let's draw a picture, this\Ntime looking the same, perhaps, Dialogue: 0,0:21:19.93,0:21:26.49,Default,,0000,0000,0000,,but where's the angle? Well,\Nwe've got to have minus Route Dialogue: 0,0:21:26.49,0:21:28.28,Default,,0000,0000,0000,,3 and one. Dialogue: 0,0:21:29.77,0:21:32.92,Default,,0000,0000,0000,,This is our angle here. Dialogue: 0,0:21:34.13,0:21:34.83,Default,,0000,0000,0000,,Alpha Dialogue: 0,0:21:35.93,0:21:43.08,Default,,0000,0000,0000,,So that what we can see is that\Nif tan of Alpha is equal to Dialogue: 0,0:21:43.08,0:21:49.29,Default,,0000,0000,0000,,minus Route 3, then Alpha must\Nbe equal to. Now the angle whose Dialogue: 0,0:21:49.29,0:21:55.49,Default,,0000,0000,0000,,tangent is plus Route 3 is 60\Ndegrees, so this one must be Dialogue: 0,0:21:55.49,0:21:56.92,Default,,0000,0000,0000,,minus 60 degrees. Dialogue: 0,0:21:58.25,0:22:03.77,Default,,0000,0000,0000,,OK, we've got Alpha sorted out.\NLet's just remember what our Dialogue: 0,0:22:03.77,0:22:09.80,Default,,0000,0000,0000,,equation was. Our equation. We\Nhad got down to cause of X Dialogue: 0,0:22:09.80,0:22:12.81,Default,,0000,0000,0000,,minus Alpha was equal to 1. Dialogue: 0,0:22:13.45,0:22:18.44,Default,,0000,0000,0000,,So now we need to put these two\Npieces of information together Dialogue: 0,0:22:18.44,0:22:23.43,Default,,0000,0000,0000,,to help us solve the full\Nequation. Help us find the value Dialogue: 0,0:22:23.43,0:22:25.51,Default,,0000,0000,0000,,of X so it's turnover. Dialogue: 0,0:22:26.59,0:22:33.15,Default,,0000,0000,0000,,And write these down. Alpha we\Nknow is minus 60 degrees and Dialogue: 0,0:22:33.15,0:22:39.17,Default,,0000,0000,0000,,we're looking to solve the\Nequation cause of X minus Alpha Dialogue: 0,0:22:39.17,0:22:41.36,Default,,0000,0000,0000,,is equal to 1. Dialogue: 0,0:22:42.17,0:22:48.65,Default,,0000,0000,0000,,And we know that X is to\Nbe between North and 360 Dialogue: 0,0:22:48.65,0:22:54.05,Default,,0000,0000,0000,,degrees, so let's sketch our\Ncosine curve between North and Dialogue: 0,0:22:54.05,0:22:59.99,Default,,0000,0000,0000,,360 degrees. We know that it\Nlooks like that there's 360. Dialogue: 0,0:22:59.99,0:23:06.47,Default,,0000,0000,0000,,There's 180, and here we have\N90, and here 270 to squeezing Dialogue: 0,0:23:06.47,0:23:14.03,Default,,0000,0000,0000,,that in there. We know that this\Ngoes down as far as minus one. Dialogue: 0,0:23:14.04,0:23:21.14,Default,,0000,0000,0000,,And these maximum points are up\Nat one. So what we can see Dialogue: 0,0:23:21.14,0:23:27.69,Default,,0000,0000,0000,,there is that the angle whose\Ncosine is one is either zero Dialogue: 0,0:23:27.69,0:23:34.24,Default,,0000,0000,0000,,degrees or 360 degrees, so X\Nminus Alpha is equal to 0 Dialogue: 0,0:23:34.24,0:23:36.43,Default,,0000,0000,0000,,degrees or 360 degrees. Dialogue: 0,0:23:36.93,0:23:44.32,Default,,0000,0000,0000,,What is Alpha? Well, it's minus\N60, so X minus minus 60 Dialogue: 0,0:23:44.32,0:23:46.79,Default,,0000,0000,0000,,is equal to 0. Dialogue: 0,0:23:47.60,0:23:51.53,Default,,0000,0000,0000,,All 360 degrees. Dialogue: 0,0:23:52.31,0:23:59.25,Default,,0000,0000,0000,,That's X plus 60 is equal\Nto 0 or 360 degrees and Dialogue: 0,0:23:59.25,0:24:06.18,Default,,0000,0000,0000,,now we can take 60 away\Nfrom both of these. So X Dialogue: 0,0:24:06.18,0:24:09.65,Default,,0000,0000,0000,,is minus 60 degrees or 300 Dialogue: 0,0:24:09.65,0:24:16.28,Default,,0000,0000,0000,,degrees. Remember the original\Nrange of values of X was between Dialogue: 0,0:24:16.28,0:24:22.90,Default,,0000,0000,0000,,Norton 360, so this is not an\Nanswer minus 60 degrees. This Dialogue: 0,0:24:22.90,0:24:30.08,Default,,0000,0000,0000,,one is the only answer within\Nour range of values of XX is Dialogue: 0,0:24:30.08,0:24:32.29,Default,,0000,0000,0000,,equal to 300 degrees. Dialogue: 0,0:24:34.00,0:24:36.27,Default,,0000,0000,0000,,Let's turn over. Dialogue: 0,0:24:37.06,0:24:41.84,Default,,0000,0000,0000,,Now let's have a look at what\Nmight be termed a calculus type Dialogue: 0,0:24:41.84,0:24:45.52,Default,,0000,0000,0000,,example. In other words, a\Nquestion that's going to involve Dialogue: 0,0:24:45.52,0:24:47.36,Default,,0000,0000,0000,,us with Maxima and minima. Dialogue: 0,0:24:47.95,0:24:55.70,Default,,0000,0000,0000,,So what about this function\NF of X is 4? Dialogue: 0,0:24:56.22,0:24:59.51,Default,,0000,0000,0000,,Cause X +3. Dialogue: 0,0:25:01.27,0:25:05.01,Default,,0000,0000,0000,,Sign X. Minus 3. Dialogue: 0,0:25:06.74,0:25:11.17,Default,,0000,0000,0000,,Water, it's turning points. What\Nare its maximum and minimum Dialogue: 0,0:25:11.17,0:25:17.99,Default,,0000,0000,0000,,values? Well. Knowing what\Nwe do know from the previous Dialogue: 0,0:25:17.99,0:25:24.90,Default,,0000,0000,0000,,work that we've done, we know\Nthat we can express this bit for Dialogue: 0,0:25:24.90,0:25:31.82,Default,,0000,0000,0000,,cause X plus three sign X as\N5 cause of X minus Alpha. Dialogue: 0,0:25:32.44,0:25:35.69,Default,,0000,0000,0000,,And then the minus three where. Dialogue: 0,0:25:36.66,0:25:44.44,Default,,0000,0000,0000,,Tan Alpha is 3\Nover 4B over A. Dialogue: 0,0:25:45.03,0:25:50.29,Default,,0000,0000,0000,,Now this function is now very\Neasy to deal with because we Dialogue: 0,0:25:50.29,0:25:52.48,Default,,0000,0000,0000,,know all about a cosine Dialogue: 0,0:25:52.48,0:25:56.92,Default,,0000,0000,0000,,function. The maximum\Nvalue of cause. Dialogue: 0,0:25:58.61,0:26:01.00,Default,,0000,0000,0000,,Is Dialogue: 0,0:26:01.00,0:26:08.01,Default,,0000,0000,0000,,warm. Never\Ngets any bigger than one, and so Dialogue: 0,0:26:08.01,0:26:09.46,Default,,0000,0000,0000,,the maximum value. Dialogue: 0,0:26:11.13,0:26:13.82,Default,,0000,0000,0000,,All F of X. Dialogue: 0,0:26:14.96,0:26:21.26,Default,,0000,0000,0000,,Is. Well, it must\Nbe 5 times by one. Dialogue: 0,0:26:21.26,0:26:25.26,Default,,0000,0000,0000,,Minus three, which is just two. Dialogue: 0,0:26:25.88,0:26:31.11,Default,,0000,0000,0000,,And when does that occur? Well,\Nlet's think the maximum value of Dialogue: 0,0:26:31.11,0:26:32.86,Default,,0000,0000,0000,,cosine is one when. Dialogue: 0,0:26:35.10,0:26:38.72,Default,,0000,0000,0000,,The angle. Equals Dialogue: 0,0:26:38.72,0:26:44.79,Default,,0000,0000,0000,,0. Well, in this case\Nthe angle will equal 0. Dialogue: 0,0:26:45.62,0:26:51.31,Default,,0000,0000,0000,,When X equals Alpha, so very\Nquickly we found out exactly Dialogue: 0,0:26:51.31,0:26:55.96,Default,,0000,0000,0000,,where this maximum value is\Nexactly at what point? Dialogue: 0,0:26:56.62,0:26:59.60,Default,,0000,0000,0000,,This function F of X has its Dialogue: 0,0:26:59.60,0:27:02.83,Default,,0000,0000,0000,,maximum value. What about the Dialogue: 0,0:27:02.83,0:27:06.78,Default,,0000,0000,0000,,minimum value? The least Dialogue: 0,0:27:06.78,0:27:12.81,Default,,0000,0000,0000,,value. Well, let's think about\NCo sign the least value of Dialogue: 0,0:27:12.81,0:27:18.63,Default,,0000,0000,0000,,cosine is minus one never goes\Nany lower than minus one. When Dialogue: 0,0:27:18.63,0:27:22.100,Default,,0000,0000,0000,,does that happen? That happens\Nwhen the angle is. Dialogue: 0,0:27:23.53,0:27:27.28,Default,,0000,0000,0000,,Pie. Or 180. Dialogue: 0,0:27:27.87,0:27:32.19,Default,,0000,0000,0000,,So what we know there is that\Nthe minimum value. Dialogue: 0,0:27:32.96,0:27:40.06,Default,,0000,0000,0000,,Of F of X most occur when\Nthis is at its minimum. In other Dialogue: 0,0:27:40.06,0:27:47.16,Default,,0000,0000,0000,,words, minus one, so it be 5\Ntimes by minus 1 - 3 which Dialogue: 0,0:27:47.16,0:27:49.18,Default,,0000,0000,0000,,is just minus 8. Dialogue: 0,0:27:50.00,0:27:56.79,Default,,0000,0000,0000,,And when will that happen?\NWill happen when the angle Dialogue: 0,0:27:56.79,0:28:03.58,Default,,0000,0000,0000,,X minus Alpha equals π.\NIn other words, when X Dialogue: 0,0:28:03.58,0:28:06.30,Default,,0000,0000,0000,,equals π plus Alpha. Dialogue: 0,0:28:07.11,0:28:11.70,Default,,0000,0000,0000,,So very quickly and with a\Nminimum amount of work, we've Dialogue: 0,0:28:11.70,0:28:15.45,Default,,0000,0000,0000,,established a maximum value\Nunder minimum value and based Dialogue: 0,0:28:15.45,0:28:20.04,Default,,0000,0000,0000,,upon this information we could\Ngo ahead and rapidly sketch the Dialogue: 0,0:28:20.04,0:28:25.04,Default,,0000,0000,0000,,curve. So again, we see that\Nthis form our cause X minus Dialogue: 0,0:28:25.04,0:28:30.88,Default,,0000,0000,0000,,Alpha is a very powerful form\Nfor us to know how to use and Dialogue: 0,0:28:30.88,0:28:35.88,Default,,0000,0000,0000,,for us to be able to formulate\Nfrom expressions such As for Dialogue: 0,0:28:35.88,0:28:38.38,Default,,0000,0000,0000,,cause X plus three sign X. Dialogue: 0,0:28:39.35,0:28:42.99,Default,,0000,0000,0000,,It's a form that you really\Ndo need to work with, and a Dialogue: 0,0:28:42.99,0:28:45.79,Default,,0000,0000,0000,,form that you really do need\Nto practice. Initially. It's Dialogue: 0,0:28:45.79,0:28:49.15,Default,,0000,0000,0000,,not so easy to get your mind\Naround, but it is possible Dialogue: 0,0:28:49.15,0:28:52.51,Default,,0000,0000,0000,,to do so and it will pay\Ngreat dividends if you can.