0:00:00.970,0:00:05.568
In this video, we're going to be[br]solving whole collection of
0:00:05.568,0:00:09.330
trigonometric equations now be[br]cause it's the technique of
0:00:09.330,0:00:13.510
solving the equation and in[br]ensuring that we get enough
0:00:13.510,0:00:17.272
solutions, that's important and[br]not actually looking up the
0:00:17.272,0:00:21.452
angle. All of these are designed[br]around certain special angles,
0:00:21.452,0:00:26.886
so I'm just going to list at the[br]very beginning here the special
0:00:26.886,0:00:31.484
angles and their sines, cosines,[br]and tangents that are going to
0:00:31.484,0:00:33.468
form. The basis of what[br]we're doing.
0:00:37.830,0:00:41.951
So the special angles that we're[br]going to have a look at our
0:00:41.951,0:00:49.444
zero. 30 4560[br]and 90 there in degrees.
0:00:49.444,0:00:55.236
If we're thinking about radians,[br]then there's zero.
0:00:55.940,0:00:59.348
Pie by 6.
0:01:00.130,0:01:03.718
Pie by 4.
0:01:04.670,0:01:05.738
Pie by three.
0:01:06.370,0:01:09.090
And Π by 2.
0:01:10.220,0:01:15.786
Trig ratios we're going to be[br]looking at are the sign.
0:01:15.790,0:01:21.104
The cosine. On the tangent[br]of each of these.
0:01:22.460,0:01:25.850
Sign of 0 is 0.
0:01:26.690,0:01:29.666
The sign of 30 is 1/2.
0:01:30.370,0:01:34.338
Sign of 45 is one over Route 2.
0:01:34.850,0:01:41.714
The sign of 60 is Route 3 over 2[br]and the sign of 90 is one.
0:01:43.440,0:01:45.260
Cosine of 0 is one.
0:01:45.990,0:01:52.665
Cosine of 30 is Route 3 over 2[br]cosine of 45 is one over Route
0:01:52.665,0:01:58.895
2, the cosine of 60 is 1/2, and[br]the cosine of 90 is 0.
0:01:59.570,0:02:06.096
The town of 0 is 0 the[br]town of 30 is one over
0:02:06.096,0:02:12.622
Route 3 that Anna 45 is[br]110 of 60 is Route 3 and
0:02:12.622,0:02:17.140
the town of 90 degrees is[br]infinite, it's undefined.
0:02:18.630,0:02:21.738
It's these that we're going to[br]be looking at and working with.
0:02:22.560,0:02:25.900
Let's look at our first
0:02:25.900,0:02:30.546
equation then. We're going to[br]begin with some very simple
0:02:30.546,0:02:37.844
ones. So we take sign of[br]X is equal to nought .5. Now
0:02:37.844,0:02:43.751
invariably when we get an[br]equation we get a range of
0:02:43.751,0:02:45.899
values along with it.
0:02:46.610,0:02:52.714
So in this case will take X is[br]between North and 360. So what
0:02:52.714,0:02:56.638
we're looking for is all the[br]values of X.
0:02:57.200,0:03:01.460
Husain gives us N[br].5.
0:03:03.970,0:03:10.830
Let's sketch a graph of[br]sine X over this range.
0:03:13.610,0:03:16.165
And sign looks like that[br]with 90.
0:03:17.310,0:03:18.500
180
0:03:20.120,0:03:27.152
270 and 360 and ranging between[br]one 4 sign 90 and minus
0:03:27.152,0:03:30.668
one for the sign of 270.
0:03:31.360,0:03:36.610
Sign of X is nought .5. So[br]we go there.
0:03:37.890,0:03:38.720
And there.
0:03:39.780,0:03:44.410
So there's our first angle, and[br]there's our second angle.
0:03:45.900,0:03:52.517
We know the first one is 30[br]degrees because sign of 30 is
0:03:52.517,0:03:58.625
1/2, so our first angle is 30[br]degrees. This curve is symmetric
0:03:58.625,0:04:05.242
and so because were 30 degrees[br]in from there, this one's got to
0:04:05.242,0:04:08.296
be 30 degrees back from there.
0:04:08.810,0:04:12.610
That would make it
0:04:12.610,0:04:17.080
150. There are no more answers[br]because within this range as we
0:04:17.080,0:04:18.360
go along this line.
0:04:18.890,0:04:23.453
It doesn't cross the curve at[br]any other points.
0:04:23.460,0:04:30.004
Let's have a look[br]at a cosine cause
0:04:30.004,0:04:36.548
of X is minus[br]nought .5 and the
0:04:36.548,0:04:43.092
range for this X[br]between North and 360.
0:04:43.820,0:04:49.112
So again, let's have a look at a[br]graph of the function.
0:04:50.070,0:04:53.899
Involved in the equation,[br]the cosine graph.
0:04:56.510,0:05:00.094
Looks like that. One and
0:05:00.094,0:05:04.819
minus one.[br]This is 90.
0:05:06.300,0:05:08.160
180
0:05:09.310,0:05:15.838
270 and then here[br]at the end, 360.
0:05:17.020,0:05:19.770
Minus 9.5.
0:05:20.810,0:05:26.387
Gain across there at minus[br]9.5 and down to their and
0:05:26.387,0:05:27.908
down to their.
0:05:28.970,0:05:35.450
Now the one thing we do know is[br]that the cause of 60 is plus N
0:05:35.450,0:05:40.715
.5, and so that's there. So we[br]know there is 60. Now again,
0:05:40.715,0:05:45.980
this curve is symmetric, so if[br]that one is 30 back that way
0:05:45.980,0:05:51.650
this one must be 30 further on.[br]So I'll first angle must be 120
0:05:51.650,0:05:56.915
degrees. This one's got to be in[br]a similar position as this bit
0:05:56.915,0:05:59.345
of the curve is again symmetric.
0:05:59.380,0:06:07.216
So that's 270 and we need[br]to come back 30 degrees, so
0:06:07.216,0:06:12.512
that's 240. Now we're going to[br]have a look at an example where
0:06:12.512,0:06:16.494
we've got what we call on[br]multiple angle. So instead of
0:06:16.494,0:06:21.562
just being cause of X or sign of[br]X, it's going to be something
0:06:21.562,0:06:26.268
like sign of 2X or cause of[br]three X. So let's begin with
0:06:26.268,0:06:33.076
sign of. 2X is equal[br]to Route 3 over 2
0:06:33.076,0:06:40.046
and again will take X[br]to be between North and
0:06:40.046,0:06:40.743
360.
0:06:41.810,0:06:44.840
Now we've got 2X here.
0:06:45.520,0:06:52.501
So if we've got 2X and X[br]is between Norton 360, then the
0:06:52.501,0:06:59.482
total range that we're going to[br]be looking at is not to 722.
0:06:59.482,0:07:06.463
X is going to come between 0[br]and 720, and the sign function
0:07:06.463,0:07:12.370
is periodic. It repeats itself[br]every 360 degrees, so I'm going
0:07:12.370,0:07:16.129
to need 2 copies of the sine
0:07:16.129,0:07:22.410
curve. As the first one going up[br]to 360 and now I need a second
0:07:22.410,0:07:24.460
copy there going on till.
0:07:25.240,0:07:27.320
720
0:07:28.370,0:07:34.535
OK, so sign 2 X equals root, 3[br]over 2, but we know that the
0:07:34.535,0:07:41.933
sign of 60 is Route 3 over 2. So[br]if we put in Route 3 over 2 it's
0:07:41.933,0:07:46.865
there, then it's going to be[br]these along here as well. So
0:07:46.865,0:07:52.208
what have we got? Well, the[br]first one here we know is 60.
0:07:52.208,0:07:57.962
This point we know is 180 so[br]that one's got to be the same
0:07:57.962,0:08:04.184
distance. Back in due to the[br]symmetry 120, so we do know that
0:08:04.184,0:08:10.932
2X will be 60 or 120, but we[br]also now we've got these other
0:08:10.932,0:08:16.716
points on here, so let's just[br]count on where we are. There's
0:08:16.716,0:08:22.018
the 1st loop of the sign[br]function, the first copy, its
0:08:22.018,0:08:27.320
periodic and repeats itself[br]again. So now we need to know
0:08:27.320,0:08:29.248
where are these well.
0:08:29.270,0:08:36.302
This is an exact copy of[br]that, so this must be 60
0:08:36.302,0:08:43.334
further on. In other words, at[br]420, and this must be another
0:08:43.334,0:08:50.366
120 further on. In other words,[br]at 480. So we've got two
0:08:50.366,0:08:57.206
more answers. And it's X[br]that we actually want, not
0:08:57.206,0:09:00.641
2X. So this is 3060.
0:09:00.820,0:09:04.441
210 and finally
0:09:04.441,0:09:11.657
240. Let's have a[br]look at that with a tangent
0:09:11.657,0:09:17.587
function. This time tan or three[br]X is equal to.
0:09:18.160,0:09:24.864
Minus one and will[br]take X to be
0:09:24.864,0:09:28.216
between North and 180.
0:09:29.740,0:09:36.040
So we draw a graph[br]of the tangent function.
0:09:37.140,0:09:38.380
So we go up.
0:09:40.990,0:09:43.780
We've got that there. That's 90.
0:09:53.090,0:09:59.837
This is 180 and this is 270[br]now. It's 3X. X is between
0:09:59.837,0:10:07.103
Norton 180, so 3X can be between[br]North and 3 * 180 which is
0:10:07.103,0:10:13.850
540. So I need to get copies[br]of this using the periodicity of
0:10:13.850,0:10:20.597
the tangent function right up to[br]540. So let's put in some more.
0:10:21.810,0:10:26.610
That's 360. On[br]there.
0:10:27.920,0:10:30.900
That's 450.
0:10:34.820,0:10:42.470
This one here will be[br]540 and that's as near
0:10:42.470,0:10:50.120
or as far as we[br]need to go. Tanner 3X
0:10:50.120,0:10:53.945
is minus one, so here's
0:10:53.945,0:10:59.124
minus one. And we go across here[br]picking off all the ones that we
0:10:59.124,0:11:00.936
need. So we've got one there.
0:11:01.690,0:11:08.862
There there. These are our[br]values, so 3X is equal 12.
0:11:08.862,0:11:16.686
Now we know that the angle[br]whose tangent is one is 45,
0:11:16.686,0:11:24.510
which is there. So again this[br]and this are symmetric bits of
0:11:24.510,0:11:32.334
curve, so this must be 45[br]further on. In other words 130.
0:11:32.340,0:11:32.980
5.
0:11:34.170,0:11:41.954
This one here has got to be[br]45 further on, so that will be
0:11:41.954,0:11:49.670
315. This one here has got[br]to be 45 further on, so that
0:11:49.670,0:11:57.440
will be 495, but it's X that[br]we want not 3X, so let's divide
0:11:57.440,0:12:04.100
throughout by three, so freezing[br]to that is 45 threes into that
0:12:04.100,0:12:11.315
is 105 and threes into that is[br]165. Those are our three answers
0:12:11.315,0:12:14.090
for that one 45 degrees.
0:12:14.100,0:12:17.106
105 degrees under
0:12:17.106,0:12:24.680
165. Let's take cause of[br]X over 2 this time. So
0:12:24.680,0:12:31.964
instead of multiplying by two or[br]by three, were now dividing by
0:12:31.964,0:12:38.641
two. Let's see what difference[br]this might make equals minus 1/2
0:12:38.641,0:12:45.925
and will take X to be[br]between North and 360. So let's
0:12:45.925,0:12:47.746
draw the graph.
0:12:48.830,0:12:55.970
All calls X between North[br]and 360, so there we've
0:12:55.970,0:12:58.826
got it 360 there.
0:12:59.560,0:13:06.346
180 there, we've got 90 and 270[br]there in their minus. 1/2 now
0:13:06.346,0:13:08.434
that's going to be.
0:13:09.990,0:13:17.466
Their cross and then these are[br]the ones that we are after.
0:13:19.060,0:13:25.682
So let's work with that. X over[br]2 is equal tool. Now where are
0:13:25.682,0:13:32.304
we? Well, we know that the angle[br]whose cosine is 1/2 is in fact
0:13:32.304,0:13:38.926
60 degrees, which is here 30 in[br]from there. So that must be 30
0:13:38.926,0:13:45.548
further on. In other words, 120[br]and this one must be 30 back. In
0:13:45.548,0:13:49.805
other words, 240. So now we[br]multiply it by.
0:13:49.850,0:13:57.078
Two, we get 240 and 480, but[br]of course this one is outside
0:13:57.078,0:14:04.306
the given range. The range is[br]not to 360, so we do not
0:14:04.306,0:14:07.642
need that answer, just want the
0:14:07.642,0:14:14.695
240. Now we've been working with[br]a range of North 360, or in one
0:14:14.695,0:14:20.480
case not to 180, so let's change[br]the range now so it's a
0:14:20.480,0:14:26.710
symmetric range in the Y axis,[br]so the range is now going to run
0:14:26.710,0:14:29.380
from minus 180 to plus 180
0:14:29.380,0:14:36.472
degrees. So we'll begin with[br]sign of X equals 1X is to
0:14:36.472,0:14:41.892
be between 180 degrees but[br]greater than minus 180 degrees.
0:14:41.892,0:14:48.938
Let's sketch the graph of sign[br]in that range. So we want to
0:14:48.938,0:14:54.358
complete copy of it. It's going[br]to look like that.
0:14:55.200,0:15:00.192
Now we know that the angle[br]who sign is one is 90
0:15:00.192,0:15:04.352
degrees and so we know[br]that's one there and that's
0:15:04.352,0:15:09.344
90 there and we can see that[br]there is only the one
0:15:09.344,0:15:13.504
solution it meets the curve[br]once and once only, so
0:15:13.504,0:15:14.752
that's 90 degrees.
0:15:15.920,0:15:22.333
Once and once only, that is[br]within the defined range. Let's
0:15:22.333,0:15:24.082
take another one.
0:15:24.230,0:15:31.634
So now we use a multiple[br]angle cause 2 X equals 1/2
0:15:31.634,0:15:38.421
and will take X to be[br]between minus 180 degrees and
0:15:38.421,0:15:44.591
plus 180 degrees. So let's[br]sketch the graph. Let's remember
0:15:44.591,0:15:51.995
that if X is between minus[br]180 and plus one 80, then
0:15:51.995,0:15:55.697
2X will be between minus 360.
0:15:55.740,0:15:57.930
And plus 360.
0:16:02.830,0:16:07.835
So what we need to do is use the[br]periodicity of the cosine
0:16:07.835,0:16:09.375
function to sketch it.
0:16:09.980,0:16:14.630
In the range. So there's[br]the knocked 360 bit and
0:16:14.630,0:16:16.025
then we want.
0:16:19.550,0:16:25.972
To minus 360. So I just label up[br]the points. Here is 90.
0:16:26.940,0:16:28.090
180
0:16:29.240,0:16:36.540
Two 7360 and then back[br]this way minus 90 -
0:16:36.540,0:16:40.344
180. Minus 270 and
0:16:40.344,0:16:47.634
minus 360. Now cause[br]2X is 1/2, so here's a half.
0:16:48.260,0:16:52.745
Membrane that this goes between[br]plus one and minus one and if we
0:16:52.745,0:16:56.540
draw a line across to see where[br]it meets the curve.
0:16:58.530,0:17:04.900
Then we can see it meets it in[br]four places. There, there there
0:17:04.900,0:17:11.270
and there we know that the angle[br]where it meets here is 60
0:17:11.270,0:17:16.660
degrees. So our first value is 2[br]X equals 60 degrees.
0:17:17.600,0:17:23.768
By symmetry, this one back here[br]has got to be minus 60.
0:17:24.350,0:17:30.494
What about this one here? Well,[br]again, symmetry says that we are
0:17:30.494,0:17:37.662
60 from here, so we've got to[br]be 60 back from there, so this
0:17:37.662,0:17:44.318
must be 300 and our symmetry of[br]the curve says that this one
0:17:44.318,0:17:51.486
must be minus 300, and so we[br]have X is 30 degrees minus 30
0:17:51.486,0:17:54.046
degrees, 150 degrees and minus
0:17:54.046,0:18:00.824
150 degrees. Working with[br]the tangent function tan, two
0:18:00.824,0:18:07.744
X equals Route 3 and[br]again will place X between
0:18:07.744,0:18:14.664
180 degrees and minus 180[br]degrees. We want to sketch
0:18:14.664,0:18:21.584
the function for tangent and[br]we want to be aware
0:18:21.584,0:18:24.352
that we've got 2X.
0:18:24.970,0:18:32.245
So since X is between minus 118[br]+ 182, X is got to be between
0:18:32.245,0:18:34.670
minus 360 and plus 360.
0:18:38.670,0:18:40.756
So if we take the bit between.
0:18:45.860,0:18:48.630
North And 360.
0:18:49.530,0:18:55.130
Which is that bit of the curve[br]we need a copy of that between
0:18:55.130,0:18:59.130
minus 360 and 0 because again[br]the tangent function is
0:18:59.130,0:19:01.530
periodic, so we need this bit.
0:19:07.580,0:19:08.510
That
0:19:14.620,0:19:18.680
And we need that and it's Mark[br]off this axis so we know where
0:19:18.680,0:19:20.130
we are. This is 90.
0:19:21.610,0:19:23.050
180
0:19:24.280,0:19:31.790
270 and 360. So this[br]must be minus 90 -
0:19:31.790,0:19:36.296
180 - 270 and minus[br]360.
0:19:37.420,0:19:42.672
Now 2X is Route 3, the angle[br]whose tangent is Route 3. We
0:19:42.672,0:19:49.136
know is 60, so we go across here[br]at Route 3 and we meet the curve
0:19:49.136,0:19:50.348
there and there.
0:19:51.440,0:19:57.656
And we come back this way. We[br]meet it there and we meet there.
0:19:57.656,0:20:00.320
So our answers are down here.
0:20:01.090,0:20:07.586
Working with this one, first we[br]know that that is 60, so 2X is
0:20:07.586,0:20:14.082
equal to 60 and so that that one[br]is 60 degrees on from that
0:20:14.082,0:20:19.650
point. Symmetry says there for[br]this one is also 60 degrees on
0:20:19.650,0:20:22.434
from there. In other words, it's
0:20:22.434,0:20:28.690
240. Let's work our way[br]backwards. This one must be 60
0:20:28.690,0:20:36.054
degrees on from minus 180, so it[br]must be at minus 120. This one
0:20:36.054,0:20:38.158
is 60 degrees on.
0:20:39.220,0:20:46.425
From minus 360 and so therefore[br]it must be minus 300.
0:20:47.120,0:20:54.188
And so if we divide throughout[br]by two, we have 31120 -
0:20:54.188,0:21:01.256
60 and minus 150 degrees. We[br]want to put degree signs on
0:21:01.256,0:21:06.557
all of these, so there are[br]four solutions there.
0:21:07.470,0:21:12.860
Trick equations often come up as[br]a result of having expressions
0:21:12.860,0:21:17.270
or other equations which are[br]rather more complicated than
0:21:17.270,0:21:19.230
that and depends upon
0:21:19.230,0:21:26.360
identity's. So I'm going to[br]have a look at a couple
0:21:26.360,0:21:31.805
of equations. These equations[br]both dependa pawn two identity's
0:21:31.805,0:21:37.855
that is expressions involving[br]trig functions that are true for
0:21:37.855,0:21:40.275
all values of X.
0:21:40.840,0:21:45.812
So the first one is sine[br]squared of X plus cost
0:21:45.812,0:21:51.688
squared of X is one. This is[br]true for all values of X.
0:21:52.890,0:21:56.130
The second one we derive from
0:21:56.130,0:22:02.119
this one. How we derive it[br]doesn't matter at the moment,
0:22:02.119,0:22:09.175
but what it tells us is that sex[br]squared X is equal to 1 + 10
0:22:09.175,0:22:15.343
squared X. So these are the two[br]identity's that I'm going to be
0:22:15.343,0:22:21.966
using. Sine squared X plus cost[br]squared X is one and sex squared
0:22:21.966,0:22:25.550
of X is 1 + 10 squared of
0:22:25.550,0:22:33.070
X OK. So how do we go[br]about using one of those to do
0:22:33.070,0:22:35.770
an equation like this? Cos
0:22:35.770,0:22:42.462
squared X? Plus cause[br]of X is equal
0:22:42.462,0:22:49.078
to sine squared of[br]X&X is between 180
0:22:49.078,0:22:51.559
and 0 degrees.
0:22:52.690,0:22:53.310
Well.
0:22:54.750,0:22:57.863
We've got a cost squared, A[br]cause and a sine squared.
0:22:58.680,0:23:03.852
If we were to use our identity[br]sine squared plus cost squared
0:23:03.852,0:23:08.593
is one to replace the sine[br]squared. Here I'd have a
0:23:08.593,0:23:15.058
quadratic in terms of Cos X, and[br]if I got a quadratic then I know
0:23:15.058,0:23:19.799
I can solve it either by[br]Factorizing or by using the
0:23:19.799,0:23:24.540
formula. So let me write down[br]sign squared X plus cost
0:23:24.540,0:23:29.281
squared. X is equal to 1, from[br]which we can see.
0:23:29.310,0:23:36.522
Sine squared X is equal to[br]1 minus Cos squared of X,
0:23:36.522,0:23:43.734
so I can take this and[br]plug it into their. So my
0:23:43.734,0:23:49.744
equation now becomes cost[br]squared X Plus X is equal
0:23:49.744,0:23:53.350
to 1 minus Cos squared X.
0:23:54.090,0:24:00.723
I want to get this as[br]a quadratic square term linear
0:24:00.723,0:24:07.959
term. Constant term equals 0, so[br]I begin by adding cost squared
0:24:07.959,0:24:09.768
to both sides.
0:24:09.870,0:24:16.331
So adding on a cost squared[br]there makes 2 Cos squared X plus
0:24:16.331,0:24:23.289
cause X equals 1. 'cause I added[br]cost square to get rid of that
0:24:23.289,0:24:30.247
one. Now I need to take one away[br]from both sides to cost squared
0:24:30.247,0:24:33.726
X Plus X minus one equals 0.
0:24:34.850,0:24:38.964
Now this is just a quadratic[br]equation, so the first question
0:24:38.964,0:24:43.826
I've got to ask is does it[br]factorize? So let's see if we
0:24:43.826,0:24:45.696
can get it to factorize.
0:24:46.550,0:24:51.295
I'll put two calls X in there[br]and cause X in there because
0:24:51.295,0:24:56.770
that 2 cause X times that cause[br]X gives Me 2 cost squared and I
0:24:56.770,0:25:01.880
put a one under one there 'cause[br]one times by one gives me one
0:25:01.880,0:25:07.355
and now I know to get a minus[br]sign. One's got to be minus and
0:25:07.355,0:25:13.195
one's got to be plus now I want[br]plus cause X so if I make this
0:25:13.195,0:25:16.845
one plus I'll have two cause X[br]times by one.
0:25:16.880,0:25:21.664
Is to cause X if I make this one[br]minus I'll have minus Cos X from
0:25:21.664,0:25:26.330
there. Taking those two[br]together, +2 cause X minus Cos X
0:25:26.330,0:25:31.296
is going to give me the plus[br]Kozaks in there, so that equals
0:25:31.296,0:25:36.290
0. Now, if not equal 0, I'm[br]multiplying 2 numbers together.
0:25:36.290,0:25:42.290
This one 2 cause X minus one and[br]this one cause X plus one, so
0:25:42.290,0:25:47.890
one of them or both of them have[br]got to be equal to 0.
0:25:48.770,0:25:54.714
So 2 calls X minus[br]one is 0.
0:25:55.560,0:26:02.952
All cause of X Plus One is[br]0, so this one tells me that
0:26:02.952,0:26:06.648
cause of X is equal to 1/2.
0:26:07.660,0:26:13.224
And this one tells Maine that[br]cause of X is equal to minus
0:26:13.224,0:26:17.932
one, and both of these are[br]possibilities. So I've got to
0:26:17.932,0:26:22.640
solve both equations to get the[br]total solution to the original
0:26:22.640,0:26:28.204
equation. So let's begin with[br]this cause of X is equal to 1/2.
0:26:28.830,0:26:35.526
And if you remember the range[br]of values was nought to 180
0:26:35.526,0:26:41.664
degrees, so let me sketch[br]cause of X between North and
0:26:41.664,0:26:46.686
180 degrees, and it looks[br]like that zero 9180.
0:26:47.740,0:26:53.096
We go across there at half and[br]come down there and there is
0:26:53.096,0:26:58.864
only one answer in the range, so[br]that's X is equal to 60 degrees.
0:27:00.220,0:27:06.892
But this one again let's sketch[br]cause of X between North and
0:27:06.892,0:27:13.560
180. There and there between[br]minus one and plus one and we
0:27:13.560,0:27:20.140
want cause of X equal to minus[br]one just at one point there and
0:27:20.140,0:27:26.250
so therefore X is equal to 180[br]degrees. So those are our two
0:27:26.250,0:27:30.010
answers to the full equation[br]that we had.
0:27:30.060,0:27:33.819
So it's now have a look at
0:27:33.819,0:27:41.220
three. 10 squared X is[br]equal to two sex squared X
0:27:41.220,0:27:44.904
Plus One and this time will
0:27:44.904,0:27:50.790
take X. To be between North and[br]180 degrees. Now, the identity
0:27:50.790,0:27:56.146
that we want is obviously the[br]one, the second one of the two
0:27:56.146,0:28:01.090
that we had before. In other[br]words, the one that tells us
0:28:01.090,0:28:08.094
that sex squared X is equal to 1[br]+ 10 squared X and we want to be
0:28:08.094,0:28:14.274
able to take this 1 + 10 squared[br]and put it into their. So we've
0:28:14.274,0:28:21.392
got 3. 10 squared[br]X is equal to 2
0:28:21.392,0:28:29.052
* 1 + 10 squared[br]X Plus one. Multiply out
0:28:29.052,0:28:36.712
this bracket. 310 squared X[br]is 2 + 210 squared
0:28:36.712,0:28:39.010
X plus one.
0:28:39.570,0:28:44.549
We can combine the two and the[br]one that will give us 3.
0:28:45.070,0:28:51.024
And we can take the 210 squared[br]X away from the three times
0:28:51.024,0:28:57.436
squared X there. That will give[br]us 10 squared X. Now we take the
0:28:57.436,0:29:03.848
square root of both sides so we[br]have 10X is equal to plus Route
0:29:03.848,0:29:06.138
3 or minus Route 3.
0:29:07.890,0:29:11.474
And we need to look at each of
0:29:11.474,0:29:15.180
these separately. So.
0:29:15.770,0:29:19.042
Time X equals Route
0:29:19.042,0:29:26.067
3. And Tan X[br]equals minus Route 3.
0:29:26.690,0:29:34.074
Access to be between North and[br]180, so let's have a sketch of
0:29:34.074,0:29:40.322
the graph of tan between those[br]values, so there is 90.
0:29:41.930,0:29:48.573
And there is 180 the angle whose[br]tangent is Route 3, we know.
0:29:49.820,0:29:56.780
Is there at 60 so we[br]know that X is equal to
0:29:56.780,0:30:02.167
60 degrees? Here we've[br]got minus Route 3, so
0:30:02.167,0:30:03.730
again, little sketch.
0:30:04.970,0:30:10.129
Between North and 180 range over[br]which were working here, we've
0:30:10.129,0:30:15.757
got minus Route 3 go across[br]there and down to their and
0:30:15.757,0:30:22.323
symmetry says it's got to be the[br]same as this one. Over here it's
0:30:22.323,0:30:29.358
got to be the same either side.[br]So in fact if that was 60 there
0:30:29.358,0:30:34.986
this must be 120 here, so X is[br]equal to 120 degrees.
0:30:35.160,0:30:39.670
So far we've been working in[br]degrees, but it makes little
0:30:39.670,0:30:43.360
difference if we're actually[br]working in radians and let's
0:30:43.360,0:30:49.100
just have a look at one or two[br]examples where in fact the range
0:30:49.100,0:30:55.250
of values that we've got is in[br]radians. So if we take Tan, X is
0:30:55.250,0:31:00.580
minus one and we take X to be[br]between plus or minus pie.
0:31:00.870,0:31:06.162
Another way of looking at[br]that would be if we were in
0:31:06.162,0:31:10.572
degrees. It will be between[br]plus and minus 180. Let's
0:31:10.572,0:31:14.100
sketch the graph of tangent[br]within that range.
0:31:15.780,0:31:17.178
Up to there.
0:31:18.060,0:31:20.728
That's π by 2.
0:31:22.950,0:31:25.668
Up to their which is π.
0:31:26.280,0:31:33.208
Minus Π[br]by 2.
0:31:36.370,0:31:39.870
Their minus
0:31:39.870,0:31:45.266
pie. Ton of X is[br]minus one, so somewhere
0:31:45.266,0:31:49.193
across here it's going to[br]meet the curve and we can
0:31:49.193,0:31:50.978
see that means it here.
0:31:52.120,0:31:56.060
And here giving us these[br]solutions at these points. Well,
0:31:56.060,0:32:01.576
we know that the angle whose[br]tangent is plus one is π by 4.
0:32:02.330,0:32:08.960
So this must be pie by 4 further[br]on, and so we have X is equal to
0:32:08.960,0:32:16.370
pie by 2 + π by 4. That will be[br]3/4 of Π or three π by 4, and
0:32:16.370,0:32:18.320
this one here must be.
0:32:19.080,0:32:25.295
Minus Π by 4 back there, so[br]minus π by 4.
0:32:26.010,0:32:30.385
Let's take one with[br]a multiple angle.
0:32:32.240,0:32:39.104
So we'll have a look cause[br]of two X is equal to Route
0:32:39.104,0:32:40.688
3 over 2.
0:32:41.730,0:32:47.826
I will take[br]X between North
0:32:47.826,0:32:54.560
and 2π. Now if[br]X is between North and 2π, and
0:32:54.560,0:32:55.850
we've got 2X.
0:32:56.680,0:33:01.372
And that means that 2X can be[br]between North and four π.
0:33:02.270,0:33:07.270
So again, we've got to make use[br]of the periodicity.
0:33:08.080,0:33:12.484
Of the graph of cosine to get a[br]second copy of it.
0:33:14.380,0:33:20.704
So there's the first copy[br]between North and 2π, and now we
0:33:20.704,0:33:27.028
want a second copy that goes[br]from 2π up till four π.
0:33:28.380,0:33:32.880
We can mark these off that one[br]will be pie by two.
0:33:33.450,0:33:34.220
Pie.
0:33:35.290,0:33:37.310
Three π by 2.
0:33:38.150,0:33:46.018
This one will be 5 Pi by[br]two. This one three Pi and this
0:33:46.018,0:33:48.828
one Seven π by 2.
0:33:49.790,0:33:55.362
So where are we with this cost?[br]2 X equals. Well, in fact we
0:33:55.362,0:34:00.934
know cost to access Route 3 over[br]2. We know that the angle that
0:34:00.934,0:34:06.904
gives us the cosine that is[br]Route 3 over 2 is π by 6. So
0:34:06.904,0:34:12.874
I'll first one is π Phi six,[br]root 3 over 2. Up here we go
0:34:12.874,0:34:18.446
across we meet the curve we come[br]down. We know that this one here
0:34:18.446,0:34:20.038
is π by 6.
0:34:20.080,0:34:25.540
Let's keep going across the[br]curves and see where we come to,
0:34:25.540,0:34:32.365
what we come to one here which[br]is π by 6 short of 2π. So
0:34:32.365,0:34:39.645
let me write it down as 2π -[br]Π by 6, and then again we come
0:34:39.645,0:34:45.105
to one here. Symmetry suggests[br]it should be pie by 6 further
0:34:45.105,0:34:51.020
on, so that's 2π + π by 6,[br]and then this one here.
0:34:51.030,0:34:57.606
Is symmetry would suggest his[br]pie by 6 short of four Pi,
0:34:57.606,0:35:04.730
so four π - π by 6.[br]So let's do that arithmetic 2X
0:35:04.730,0:35:06.922
is π by 6.
0:35:07.530,0:35:12.954
Now, how many sixths are there[br]in two? Well, the answer. Is
0:35:12.954,0:35:19.282
there a 12 of them and we're[br]going to take one of them away,
0:35:19.282,0:35:26.062
so that's eleven π by 6. We're[br]going to now add a 6th on, so
0:35:26.062,0:35:28.322
that's 13 Pi by 6.
0:35:29.630,0:35:37.022
How many 6th are there in four[br]or there are 24 of them? We're
0:35:37.022,0:35:43.886
going to take one away, so[br]that's 23. Pi over 6. Now we
0:35:43.886,0:35:51.278
want X, so we divide each of[br]these by 2π by 1211 Pi by
0:35:51.278,0:35:58.670
12:13, pie by 12, and 20, three[br]π by 12, and there are our
0:35:58.670,0:36:05.900
four solutions. Let's have a[br]look at one where we've got the
0:36:05.900,0:36:12.572
X divided by two rather than[br]multiplied by two. So the sign
0:36:12.572,0:36:18.132
of X over 2 is minus Route[br]3 over 2.
0:36:18.640,0:36:26.272
And let's take X to be[br]between pie and minus π. So
0:36:26.272,0:36:33.268
will sketch the graph of sign[br]between those limited, so it's
0:36:33.268,0:36:40.149
there. And their π[br]zero and minus pie.
0:36:40.750,0:36:46.535
Where looking for minus three[br]over 2. Now the one thing we do
0:36:46.535,0:36:53.210
know is that the angle who sign[br]is 3 over 2 is π by 3.
0:36:53.750,0:36:59.150
But we want minus Route 3 over[br]2, so that's down there.
0:36:59.740,0:37:01.510
We go across.
0:37:02.160,0:37:04.512
And we meet the curve these two
0:37:04.512,0:37:08.724
points. Now this curve is[br]symmetric with this one.
0:37:09.230,0:37:12.070
So if we know that.
0:37:12.710,0:37:14.900
Plus Route 3 over 2.
0:37:15.450,0:37:21.190
This one was Pi by three. Then[br]we know that this one must be
0:37:21.190,0:37:22.830
minus π by 3.
0:37:23.350,0:37:30.998
This one is π by three back, so[br]it's at 2π by three, so this one
0:37:30.998,0:37:38.168
must be minus 2π by three, and[br]so we have X over 2 is equal
0:37:38.168,0:37:45.338
to minus 2π by three and minus,[br]π by three, but it's X that we
0:37:45.338,0:37:52.030
want, so we multiply up X equals[br]minus four Pi by three and minus
0:37:52.030,0:37:53.464
2π by 3.
0:37:54.210,0:37:59.622
Let's just check on these[br]values. How do they fit with the
0:37:59.622,0:38:05.936
given range? Well, this 1 - 2π[br]by three is in that given range.
0:38:06.540,0:38:11.060
This one is outside, so we don't[br]want that one.
0:38:12.010,0:38:18.918
A final example here, working[br]with the idea again of using
0:38:18.918,0:38:24.570
those identities and will take 2[br]cost squared X.
0:38:25.490,0:38:31.167
Plus sign X is[br]equal to 1.
0:38:31.970,0:38:37.874
And we'll take X between[br]North and 2π.
0:38:38.780,0:38:43.060
We've got causes and signs,[br]so the identity that we're
0:38:43.060,0:38:47.768
going to want to help us[br]will be sine squared plus
0:38:47.768,0:38:49.908
cost. Squared X equals 1.
0:38:51.000,0:38:52.560
Cost squared here.
0:38:54.500,0:38:59.725
Cost squared here. Let's use[br]this identity to tell us that
0:38:59.725,0:39:05.900
cost squared X is equal to 1[br]minus sign squared X and make
0:39:05.900,0:39:08.750
the replacement up here for cost
0:39:08.750,0:39:14.460
squared. Because that as we will[br]see when we do it.
0:39:14.630,0:39:22.624
Leads to a quadratic in sign X,[br]so it's multiply this out 2 -
0:39:22.624,0:39:30.618
2 sine squared X plus sign X[br]is equal to 1 and I want
0:39:30.618,0:39:37.470
it as a quadratic, so I want[br]positive square term and then
0:39:37.470,0:39:44.893
the linear term and then the[br]constant term. So I need to add.
0:39:44.920,0:39:51.262
This to both sides of 0 equals 2[br]sine squared X. Adding it to
0:39:51.262,0:39:57.604
both sides. Now I need to take[br]this away minus sign X from both
0:39:57.604,0:40:03.946
sides and I need to take the two[br]away from both sides. So one
0:40:03.946,0:40:06.211
takeaway two is minus one.
0:40:07.040,0:40:10.930
And now does this factorize?[br]It's clearly a quadratic. Let's
0:40:10.930,0:40:16.765
look to see if we can make it[br]factorize 2 sign X and sign X.
0:40:16.765,0:40:20.655
Because multiplied together,[br]these two will give Me 2 sine
0:40:20.655,0:40:24.156
squared one and one because[br]multiplied together, these two
0:40:24.156,0:40:29.602
will give me one, but one of[br]them needs to be minus. To make
0:40:29.602,0:40:34.659
this a minus sign here. So I[br]think I'll have minus there and
0:40:34.659,0:40:39.327
plus there because two sign X[br]times by minus one gives me.
0:40:39.390,0:40:45.598
Minus 2 sign X one times by sign[br]X gives me sign X and if I
0:40:45.598,0:40:50.254
combine sign X with minus two[br]sign XI get minus sign X.
0:40:50.770,0:40:55.291
I have two numbers multiplied[br]together. This number 2 sign X
0:40:55.291,0:40:59.812
Plus One and this number sign X[br]minus one. They multiply
0:40:59.812,0:41:05.977
together to give me 0, so one or[br]both of them must be 0. Let's
0:41:05.977,0:41:07.210
write that down.
0:41:07.940,0:41:15.604
2 sign X Plus One is equal to[br]0 and sign X minus one is equal
0:41:15.604,0:41:23.268
to 0, so this tells me that sign[br]of X is equal. To take one away
0:41:23.268,0:41:29.974
from both sides and divide by[br]two. So sign X is minus 1/2 and
0:41:29.974,0:41:35.243
this one tells me that sign X is[br]equal to 1.
0:41:35.810,0:41:40.386
I'm now in a position to solve[br]these two separate equations.
0:41:40.910,0:41:43.360
So let me take this one first.
0:41:43.980,0:41:51.123
Now. We were working between[br]North and 2π, so we'll have a
0:41:51.123,0:41:53.488
sketch between North and 2π.
0:41:53.990,0:41:59.528
Of the sine curve and we want[br]sign X equals one. Well, there's
0:41:59.528,0:42:05.492
one and there's where it meets,[br]and that's pie by two, so we can
0:42:05.492,0:42:08.900
see that X is equal to pie by
0:42:08.900,0:42:15.744
two. Sign X equals minus 1/2.[br]Again, the range that we've been
0:42:15.744,0:42:21.618
given is between North and 2π.[br]So let's sketch between Norton
0:42:21.618,0:42:23.220
2π There's 2π.
0:42:25.450,0:42:27.090
Three π by 2.
0:42:27.810,0:42:33.966
Pie pie by two 0 - 1/2,[br]so that's coming along between
0:42:33.966,0:42:39.609
minus one and plus one that's[br]going to come along there.
0:42:40.890,0:42:45.869
And meet the curve there and[br]there. Now the one thing that we
0:42:45.869,0:42:48.933
do know is the angle who sign is
0:42:48.933,0:42:55.576
plus 1/2. Is π by 6, so we're[br]looking at plus 1/2. It will be
0:42:55.576,0:42:58.792
there and it would be pie by 6.
0:42:59.870,0:43:06.520
So it's π by 6 in from there,[br]so symmetry tells us that this
0:43:06.520,0:43:14.120
must be pie by 6 in from there,[br]so we've got X is equal to π
0:43:14.120,0:43:21.720
+ π by 6, and symmetry tells us[br]it's pie by 6 in. From there, 2π
0:43:21.720,0:43:23.620
- Π by 6.
0:43:25.340,0:43:32.634
There are six sixths in pie, so[br]that's Seven π by 6. There is
0:43:32.634,0:43:39.407
1216, two Pi. We're taking one[br]of them away, so it will be
0:43:39.407,0:43:41.491
11 Pi over 6.
0:43:41.840,0:43:46.910
So we've shown there how to[br]solve some trig equations.
0:43:46.910,0:43:51.980
The important thing is the[br]sketch the graph. Find the
0:43:51.980,0:43:56.543
initial value and then[br]workout where the others are
0:43:56.543,0:44:01.106
from the graphs. Remember,[br]the graphs are all symmetric
0:44:01.106,0:44:05.669
and they're all periodic, so[br]they repeat themselves every
0:44:05.669,0:44:08.204
2π or every 360 degrees.