0:00:00.960,0:00:05.616
In this video, we're going to be[br]looking at basic trig identities

0:00:05.616,0:00:09.496
and how to use them to solve[br]trigonometric equations. Trig

0:00:09.496,0:00:13.376
equation is an equation that[br]involves a trig function or

0:00:13.376,0:00:18.420
functions. When we solve it,[br]what we do is find a value for

0:00:18.420,0:00:22.688
the trig function and then find[br]the angle that corresponds to

0:00:22.688,0:00:26.956
that particular trig function.[br]But what we want to start is

0:00:26.956,0:00:31.224
with the idea of a right angle[br]triangle and go back.

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To the well known theorem of

0:00:33.854,0:00:38.953
Pythagoras. So let's begin with[br]our right angle triangle. There

0:00:38.953,0:00:43.957
the right angle in and let's[br]label the sides and the angles.

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So will have this.

0:00:46.140,0:00:50.916
As a side of length a,[br]this is a side of length

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B and the hypotenuse. The[br]side that is opposite the

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right angle will call C.

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And our label, this angle here[br]the angle a.

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Now Pythagoras theorem tells us[br]that if we take the square of

0:01:07.586,0:01:10.250
this side. And the square of

0:01:10.250,0:01:15.587
this side. Add them together.[br]We'll get the square of this

0:01:15.587,0:01:20.680
side, so Pythagoras theorem[br]tells us that A squared plus B

0:01:20.680,0:01:23.458
squared is equal to C squared.

0:01:24.250,0:01:31.180
Now. Let's divide throughout by[br]C squared, divided every term in

0:01:31.180,0:01:33.830
this equation by C squared.

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So we have a squared over C[br]squared plus B squared over C

0:01:40.694,0:01:46.430
squared is equal to and this[br]side we would have C squared

0:01:46.430,0:01:51.210
over C squared, but that of[br]course is just one.

0:01:53.120,0:02:00.439
Now we can rewrite A squared[br]over C squared as a oversee all

0:02:00.439,0:02:07.754
squared. And we can do the[br]same here with B squared over C

0:02:07.754,0:02:12.824
squared. We can rewrite that as[br]B. Oversee all squared.

0:02:12.840,0:02:14.310
And it still equal to 1.

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Let's go back to this[br]triangle again.

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What does a oversee represent?

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Well, a over C is the side[br]opposite to the angle a.

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Divided by the hypotenuse.

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And so opposite divided by[br]hypotenuses sign. So this, a

0:02:39.370,0:02:46.384
oversee is sign of a sign of[br]the angle A and we have to

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square it. Now we could write[br]it like that, sign a squared,

0:02:52.790,0:03:00.005
and for the moment I will plus[br]and let's have a look at B over

0:03:00.005,0:03:07.220
C or B is the adjacent side to[br]the angle A and C is the

0:03:07.220,0:03:11.549
hypotenuse, so B oversee is[br]adjacent over hypotenuse and

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that's cosine. So we can replace[br]B oversee by cause A and we need

0:03:18.283,0:03:20.207
to square it still.

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Equal to 1.

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Now this notation sign.

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Squared, I just said sine[br]squared, so rather than writing

0:03:31.060,0:03:36.197
it to sign a squared, which[br]might be confused with squaring

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the a, let's put the square on[br]the sign and so the notation for

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the sign of a times by the sign[br]of a is sine squared a written

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like that plus, and we use[br]exactly the same technique cost

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squared A. Equals 1.

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Now that is an identity because[br]it is true for all angles

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a like this in a right

0:04:09.756,0:04:12.330
angle triangle. However.

0:04:12.950,0:04:18.662
I could have done this for the[br]definitions of sine and cosine

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that come from a unit circle.

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In which case?

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This identity would be true for[br]all angles a no matter what

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their size, and that's the case.[br]This is a basic trig identity

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that sine squared of an angle[br]plus cost squared of an angle

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equals 1. It's true for

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all angles. What developed this

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identity now? To give[br]us two more basic

0:04:55.102,0:04:58.520
identity's. So let's

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begin. With sine[br]squared A plus cost

0:05:05.586,0:05:09.150
squared A equals 1.

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Our basic fundamental identity,[br]one that you really must learn

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and know and come to recognize[br]every time that you see it.

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What I want to do is[br]divide everything by this term

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here cause squared A.

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So sine squared a divided[br]by Cos squared A.

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Plus cost squared a divided[br]by Cos squared a is

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one over cause squared A.

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Now, because I've divided[br]everything by Cos squared, this

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is still a true equation. Still[br]in fact an identity.

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So sine squared over cost

0:06:02.370,0:06:09.533
squared. Will sign over. Cause[br]is tan and so this is in

0:06:09.533,0:06:11.745
fact TAN squared A.

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Plus cost squared divided by[br]Cos squared is just one.

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Now, one over cause is sick, and[br]so we can rewrite this as sex

0:06:27.924,0:06:31.482
squared A. And so we have

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another identity. And normally[br]we would write this as sex

0:06:38.174,0:06:45.530
squared a is 1 + 10[br]squared a. So there's our second

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basic fundamental identity[br]that's derived directly from

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sine squared plus cost squared

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is one. Well, if we[br]can divide this equation by Cos

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squared, surely we can do the[br]same thing but with sine

0:07:04.850,0:07:10.682
squared. So we can divide the[br]whole of this equation by sine

0:07:10.682,0:07:16.270
squared. We start again by[br]writing down our basic

0:07:16.270,0:07:21.350
fundamental identity. Sine[br]squared plus cost squared is

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one. And as we said,[br]instead of dividing everything

0:07:27.074,0:07:32.504
by Cos squared, we're going to[br]divide everything by sine

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squared. So that we[br]have everything in the

0:07:39.325,0:07:46.485
identity divided by sine[br]squared. So it's still

0:07:46.485,0:07:53.645
true for all angles.[br]A sine squared divided

0:07:53.645,0:07:57.225
by sine squared. That's

0:07:57.225,0:07:59.480
just one. House.

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Now we've. 'cause squared[br]divided by sine squared. So

0:08:05.874,0:08:12.138
we've caused divided by sign all[br]squared and cons divided by sign

0:08:12.138,0:08:13.704
is just caught.

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So that is cot squared.

0:08:17.140,0:08:22.734
A. Equals and here with[br]one over sine squared.

0:08:23.240,0:08:29.910
One over sine is kosach[br]and so one over sine

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squared is cosec squared.

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And so there we have our[br]third fundamental identity, one

0:08:38.420,0:08:41.335
plus Scott squared is cosec

0:08:41.335,0:08:48.068
squared. So we've[br]now got three basic

0:08:48.068,0:08:55.425
fundamental identity's. I just[br]write them down here in

0:08:55.425,0:08:59.250
this corner sign square day.

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Cost Square day.

0:09:01.560,0:09:02.510
Is one.

0:09:03.120,0:09:10.896
1. Post[br]and squared a is sex

0:09:10.896,0:09:18.606
squared A and one plus[br]cot? Squared a? Is cosec

0:09:18.606,0:09:25.133
squared A? Now the use that[br]we're going to make of these is

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to help us solve particular[br]kinds of trigonometric

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equations. So first of all,[br]let's look at this one 2.

0:09:35.050,0:09:37.258
10 squared X.

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Is equal to sex[br]squared X.

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What we need to do is looking[br]at this equation related to

0:09:49.032,0:09:52.992
one of these three identity's[br]and it's fairly obvious that

0:09:52.992,0:09:54.576
this is the one.

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So what we need to do then is[br]get everything in terms of

0:10:01.138,0:10:03.640
either tans. Or sex.

0:10:04.690,0:10:11.008
Well, our identity says that's X[br]squared is equal to 1 + 10

0:10:11.008,0:10:16.840
squared. So let's replace the[br]sex squared here by 1 + 10

0:10:16.840,0:10:23.158
squared. So we have two 10[br]squared X is equal to 1 plus.

0:10:23.680,0:10:26.329
10 squared X.

0:10:27.290,0:10:33.036
And now we can take 10 squared[br]away from each side, which will

0:10:33.036,0:10:38.340
leave us with one 10 squared[br]this side and equals 1 there.

0:10:39.470,0:10:43.880
So now we can take the square[br]root of both sides, so Tan X is

0:10:43.880,0:10:47.702
equal to 1. And let's not forget[br]when we take a square root.

0:10:48.240,0:10:53.982
We've got 2 answers, plus or[br]minus one in this case.

0:10:54.890,0:10:58.421
Now, the one thing that we[br]didn't specify at the beginning

0:10:58.421,0:11:02.594
of this question was what was[br]the range of values that we were

0:11:02.594,0:11:04.520
going to be working with 4X.

0:11:05.240,0:11:08.683
Well, since we didn't specify at[br]the beginning, I think we're

0:11:08.683,0:11:13.065
entitled to put in any range of[br]values that we want. So for the

0:11:13.065,0:11:16.821
moment, let's say that we're[br]going to look at this for X.

0:11:17.360,0:11:23.736
Between equal to 0 but[br]less than 2π.

0:11:24.830,0:11:30.680
Sotan axes one or minus one.[br]Let's sketch the graph.

0:11:31.760,0:11:36.765
Of tanks between North and 2π,[br]so it looks like that.

0:11:37.450,0:11:43.440
Asymptotes. Like that asymptotes[br]and like that.

0:11:44.060,0:11:49.714
This is 2π this one we know[br]is π by 2.

0:11:50.340,0:11:56.756
This where it crosses the X axis[br]is π, and this one here is 3 Pi

0:11:56.756,0:12:00.685
by two. 10 X equals

0:12:00.685,0:12:04.770
1. That's one of those[br]special angles.

0:12:05.850,0:12:11.660
45 degrees if we were working in[br]degrees or pie by 4 radians. If

0:12:11.660,0:12:17.470
we're working in radians. So for[br]the one bit we want Thai by 4,

0:12:17.470,0:12:23.695
but where else do we want to be?[br]Here's one, and if we go across

0:12:23.695,0:12:27.015
the tan graph we can see we meet

0:12:27.015,0:12:33.250
it here. That's the pie by 4[br]and we meet it here.

0:12:34.430,0:12:40.366
And that is going to be in there[br]halfway between pie and three Pi

0:12:40.366,0:12:47.150
by two, and so that is going to[br]be at five π by 4. So there's

0:12:47.150,0:12:48.422
our second answer.

0:12:49.030,0:12:54.503
Coming from one and then we've[br]got the minus one, so let's go

0:12:54.503,0:12:56.187
across at minus one.

0:12:57.450,0:12:58.750
Till we meet the graph.

0:13:00.040,0:13:02.038
There and there.

0:13:02.660,0:13:09.156
This is half way between pie by[br]two and Π, and so that's going

0:13:09.156,0:13:11.940
to be three π by 4.

0:13:12.720,0:13:18.670
And this one is halfway between[br]three Pi by two an 2π and so

0:13:18.670,0:13:23.770
that's going to be 7 Pi by 4,[br]and so there are.

0:13:25.550,0:13:28.270
Four solutions to this question.

0:13:29.610,0:13:35.082
Let's take another example and[br]this time I'm going to take one

0:13:35.082,0:13:40.098
that will make use of one more[br]of these particular basic

0:13:40.098,0:13:46.726
identity's. So two[br]sine squared

0:13:46.726,0:13:50.020
X. Close

0:13:50.020,0:13:55.140
call sex.[br]Equals 1.

0:13:56.230,0:13:58.600
Now this has got sine squared's

0:13:58.600,0:14:01.370
in it. And the cause?

0:14:01.930,0:14:05.230
Well, fairly obviously, I think[br]we ought to be using sine

0:14:05.230,0:14:06.730
squared plus cost. Squared is

0:14:06.730,0:14:13.278
one. But what do we replace? Do[br]we try and replace the cause or

0:14:13.278,0:14:18.946
do we try and replace the sign?[br]We've got a choice. Well, the

0:14:18.946,0:14:22.870
identity says sign squared plus[br]cost squared equals 1.

0:14:23.600,0:14:29.331
So if it's sine squared that's[br]in the identity, then perhaps

0:14:29.331,0:14:35.062
it's the sine squared that we[br]ought to replace. So let's

0:14:35.062,0:14:38.709
make that replacement instead[br]of sine squared.

0:14:39.740,0:14:46.070
Be'cause sine squared plus cost[br]squared is one sign. Squared

0:14:46.070,0:14:53.666
must be 1 minus Cos squared.[br]So in there will write 1

0:14:53.666,0:14:59.363
minus Cos squared X plus cause[br]X equals 1.

0:15:00.620,0:15:07.396
Multiply out the brackets[br]2 - 2 cost

0:15:07.396,0:15:13.325
squared X plus cause[br]X equals 1.

0:15:14.460,0:15:19.962
Well, if we simplify this, what[br]we're going to end up with is a

0:15:19.962,0:15:24.285
quadratic equation where the[br]variable is going to be cause X.

0:15:24.285,0:15:29.394
So let's move this term minus[br]two cost squared X over to this

0:15:29.394,0:15:34.503
side of the equation. By adding[br]two cost squared X to each side

0:15:34.503,0:15:40.398
so that we get it positive at[br]this side. And So what I want to

0:15:40.398,0:15:44.721
end up with is an equation[br]that's equal to 0, so.

0:15:44.780,0:15:51.812
Equals 0 at this to both[br]sides. To cost squared X, take

0:15:51.812,0:15:58.258
this away from both sides,[br]because that's plus cause X, so

0:15:58.258,0:16:00.016
minus cause X.

0:16:00.650,0:16:06.650
Take the two away from both[br]sides, so that's one. Take away

0:16:06.650,0:16:09.150
these two is minus one.

0:16:10.030,0:16:15.790
As we said before, this is[br]now just a quadratic.

0:16:16.300,0:16:20.230
So let's see if we can

0:16:20.230,0:16:26.729
factorize it. Remembering that[br]the variable is cause X, well we

0:16:26.729,0:16:32.460
want two numbers that will[br]multiply together is to give us

0:16:32.460,0:16:39.233
2 cost squared X, which suggests[br]perhaps two Cos X and cause X.

0:16:40.170,0:16:44.482
We want two numbers that will[br]multiply together to give us

0:16:44.482,0:16:49.186
minus one. Well, let's put ones[br]in for the moment and worry

0:16:49.186,0:16:50.754
about the sign now.

0:16:51.490,0:16:57.561
If I take 2 cause X times[br]by one here, I will get

0:16:57.561,0:16:59.429
just to cause X.

0:17:00.830,0:17:07.086
If I take 1 by cause X here I[br]will get just cause X and I want

0:17:07.086,0:17:11.870
to end up with minus Cos X,[br]which means I've really got to

0:17:11.870,0:17:14.078
take away the result of doing

0:17:14.078,0:17:20.326
this multiplication. So the[br]minus sign there and a plus side

0:17:20.326,0:17:28.180
there. So what does this[br]tell us? If this expression is

0:17:28.180,0:17:35.908
equal to 0, then either 2[br]calls X plus one equals 0

0:17:35.908,0:17:37.840
or cause X.

0:17:38.450,0:17:44.126
Minus one equals 0. This gives[br]us a nice little equation that

0:17:44.126,0:17:50.275
says cause X is equal to. I'll[br]take one away from both sides,

0:17:50.275,0:17:56.897
so that's minus one and divide[br]both sides by two, so cause X is

0:17:56.897,0:17:59.262
equal to minus 1/2 or.

0:17:59.890,0:18:03.646
Cause X is equal to 1.

0:18:04.350,0:18:09.797
And so in order to solve this[br]equation at the top, I've now

0:18:09.797,0:18:14.406
reduced it to solving these two[br]much simpler equations at the

0:18:14.406,0:18:19.704
bottom. So I'll turn over the[br]page now and take these two with

0:18:19.704,0:18:27.144
me. So cause X[br]is equal to minus

0:18:27.144,0:18:34.298
1/2. All.[br]Kohl's X is equal to wall.

0:18:35.270,0:18:40.574
Now again, when we start at[br]solving this, we did not have a

0:18:40.574,0:18:45.878
range of values for Cos X. So[br]let's say that again we're going

0:18:45.878,0:18:50.774
to work with this range of[br]values. X is greater than or

0:18:50.774,0:18:56.894
equal to 0, but less than two[br]Pi. What we need to do first is

0:18:56.894,0:19:02.606
sketch the graph of Cos X in[br]that range. So the graph of Cos

0:19:02.606,0:19:05.462
X in that range looks like that.

0:19:07.380,0:19:09.930
This is π by 2.

0:19:10.790,0:19:13.139
This is pie.

0:19:14.160,0:19:17.672
Three Pi by two

0:19:17.672,0:19:25.496
and 2π. This is one[br]on the Y axis and minus one on

0:19:25.496,0:19:31.328
the X axis. So what are our[br]values? Well, let's take this

0:19:31.328,0:19:36.674
equation first cause X equals[br]one. We go across at one.

0:19:37.610,0:19:39.360
We've got this value here.

0:19:40.000,0:19:42.538
X equals 0.

0:19:43.420,0:19:46.801
And this value here X equals 2π,

0:19:46.801,0:19:52.570
but. This here says X is[br]strictly less than 2π, so

0:19:52.570,0:19:54.378
if I included it.

0:19:55.500,0:20:00.360
Be right across it out because[br]it's not within the range of

0:20:00.360,0:20:05.220
values. Let's now have a look[br]at this cause X equals minus

0:20:05.220,0:20:10.080
1/2. Well minus 1/2 is there.[br]So let's go across and see

0:20:10.080,0:20:13.320
where this meets the graph[br]there and there.

0:20:14.430,0:20:20.601
Right, this again, is connected[br]with one of those very special

0:20:20.601,0:20:26.723
angles. If concept X had[br]been equal to 1/2, then

0:20:26.723,0:20:32.410
X would be equal to 60[br]degrees or pie by three.

0:20:33.940,0:20:37.030
That's about there.

0:20:37.900,0:20:40.810
These curves are symmetric, so

0:20:40.810,0:20:48.330
this one. Instead of being[br]pie by three from there is π

0:20:48.330,0:20:51.140
by three back from pie.

0:20:51.850,0:20:58.077
And so this tells us that X is[br]equal to 2π by 3.

0:20:58.680,0:21:05.344
Or it's pie by three on from[br]there, which gives us four Pi by

0:21:05.344,0:21:11.056
three. So there we've got our[br]answers. Two of them there, and

0:21:11.056,0:21:12.960
one of them there.

0:21:14.190,0:21:23.152
So.[br]Let's take another example.

0:21:23.152,0:21:29.816
Three cop squared X[br]is equal to Cosec

0:21:29.816,0:21:33.410
X. Minus one.

0:21:34.110,0:21:41.130
So the identity that we want[br]is the one that talks to

0:21:41.130,0:21:47.565
us about cot squared and Cosec[br]squared. But which term should

0:21:47.565,0:21:54.000
we replace now? Let's recall the[br]identity is one plus cot

0:21:54.000,0:21:57.510
squared. X is cosec squared X.

0:21:58.130,0:22:02.610
Well, as we saw in the last[br]example, we want to arrive at a

0:22:02.610,0:22:06.130
quadratic that we can factorise[br]it therefore makes no sense to

0:22:06.130,0:22:10.610
try and substitute for the Cosec[br]'cause to do that we have to get

0:22:10.610,0:22:11.890
square roots in it.

0:22:12.660,0:22:17.925
But if we substitute for the cot[br]squared, we can do so much

0:22:17.925,0:22:23.804
better. Because we will just[br]have a direct substitution that

0:22:23.804,0:22:29.064
will involve cosec squared and[br]hopefully get a quadratic. So

0:22:29.064,0:22:34.324
instead of caught squared will[br]replace it. By changing this

0:22:34.324,0:22:40.110
around, that tells us that[br]caught squared X is equal to

0:22:40.110,0:22:45.896
cosine X squared X minus one, so[br]that will be 3.

0:22:46.670,0:22:53.606
Cosec squared X minus[br]one is equal to

0:22:53.606,0:22:57.074
cosec X minus one.

0:22:58.050,0:23:00.118
Multiply out the brackets.

0:23:00.770,0:23:07.564
3. Cosec squared[br]X minus three. Don't forget when

0:23:07.564,0:23:12.001
you multiply out brackets, you[br]must multiply everything inside

0:23:12.001,0:23:17.917
the bracket by what's outside,[br]so we've got to have the three

0:23:17.917,0:23:23.340
times by the minus one there[br]equals cosec X minus one.

0:23:24.260,0:23:28.112
Let's get everything on one side[br]of the equation so it says

0:23:28.112,0:23:33.436
equals 0. Keep the square term[br]to be the positive term.

0:23:33.970,0:23:37.970
That makes factorization[br]easier, so free KOs X

0:23:37.970,0:23:42.470
squared X takeaway. This[br]code set from each side.

0:23:43.600,0:23:49.216
And now I've minus three here,[br]and I've minus one here. I want

0:23:49.216,0:23:56.128
to take the minus one over to[br]that site, so I have to add 1 to

0:23:56.128,0:24:00.448
each side, so minus three plus[br]One is minus 2.

0:24:00.470,0:24:01.580
Equals 0.

0:24:02.770,0:24:06.143
Now. Factorise

0:24:06.143,0:24:12.450
this equation. The[br]variable is kosach, so we're

0:24:12.450,0:24:15.750
going to have three cosec in

0:24:15.750,0:24:21.525
here. And cosec X in there[br]because that kinds by that will

0:24:21.525,0:24:26.505
give us that term there three[br]cosec squared X. And now I've

0:24:26.505,0:24:29.410
got the minus two to deal with.

0:24:30.000,0:24:35.235
Well to itself is 2 times by[br]one. If I put the two in here

0:24:35.235,0:24:39.423
then I'm going to multiply the[br]two by the three, and that's

0:24:39.423,0:24:43.960
going to give me 6, which is a[br]very big number in Association

0:24:43.960,0:24:48.148
with the cosack. I only want[br]minus one cosec, so let's put

0:24:48.148,0:24:53.034
the two in there on the one in[br]there. Now I've got to balance

0:24:53.034,0:24:55.128
the signs I want, minus two

0:24:55.128,0:25:00.636
here. So one of these has got to[br]be negative and I see that

0:25:00.636,0:25:04.398
what's going to make the[br]decision for me. Is this minus

0:25:04.398,0:25:09.440
cosec X? So I need the bigger[br]bit in size to be negative.

0:25:10.030,0:25:16.165
Which seems to me that the three[br]cosec X has got to go with us

0:25:16.165,0:25:21.073
minus one, so three cosec times[br]by minus one is minus three

0:25:21.073,0:25:25.669
kosek. And then I've got +2[br]kocek there gives me the minus

0:25:25.669,0:25:27.943
the single cosec that I want.

0:25:28.530,0:25:34.344
Solving a quadratic two brackets[br]multiplied together equal 0.

0:25:34.344,0:25:40.804
That means one of these brackets[br]or the other one.

0:25:41.510,0:25:44.954
Has got to be

0:25:44.954,0:25:48.270
equal to. 0.

0:25:50.240,0:25:57.345
So. But this one I[br]can take two away from each side

0:25:57.345,0:26:04.135
and divide by three. So we have[br]cosec X is equal to minus two

0:26:04.135,0:26:06.560
over three or minus 2/3.

0:26:07.070,0:26:09.749
And here kosek.

0:26:10.300,0:26:16.069
X is equal to 1, so[br]again we've reduced.

0:26:16.800,0:26:23.550
An equation like this to[br]solving two much smaller, much

0:26:23.550,0:26:29.938
simpler equations. So taking[br]these two over the

0:26:29.938,0:26:35.804
page cosec X is[br]minus 2/3 or?

0:26:36.500,0:26:40.220
Cosec X is one.

0:26:41.320,0:26:48.420
Now what is cosec? Cosack[br]is one over sine X.

0:26:48.470,0:26:52.256
So let's write

0:26:52.256,0:26:58.760
that down. If you just[br]look at this one over sine X

0:26:58.760,0:27:03.817
equals 1. While that can only[br]mean cynex itself is equal to 1,

0:27:03.817,0:27:09.263
and if we can turn this upside[br]down, we can do the same this

0:27:09.263,0:27:13.931
side. So turning that back[br]upside down, sign X is equal to

0:27:13.931,0:27:15.487
minus three over 2.

0:27:16.650,0:27:21.785
Now when we began this equation[br]and we began to solve it, we

0:27:21.785,0:27:26.920
didn't state a range of values[br]of X, so let's use the range

0:27:26.920,0:27:32.055
again that we've been using and[br]that is X greater than or equal

0:27:32.055,0:27:34.425
to 0, but less than 2π.

0:27:35.520,0:27:41.556
Let's just sketch the graph of[br]cynex over that range of values.

0:27:42.670,0:27:46.910
Graph of Cynex will look like[br]that going from North.

0:27:47.800,0:27:49.788
Through pie by two.

0:27:50.960,0:27:52.130
Pie.

0:27:53.570,0:27:55.338
Three π by 2.

0:27:55.940,0:28:02.141
And 2π and it will range between[br]plus one and minus one. So

0:28:02.141,0:28:07.388
again, if we look at this[br]solution here, we can see

0:28:07.388,0:28:13.112
straightaway. We've got the one[br]solution here at X equals π by

0:28:13.112,0:28:19.570
2. Let's have a look at this[br]one. Sign X equals minus three

0:28:19.570,0:28:21.670
over two. Well, that's here.

0:28:22.380,0:28:24.756
And of course there's a problem.

0:28:26.180,0:28:30.723
This doesn't mean the graph[br]anywhere. There are no values of

0:28:30.723,0:28:33.201
X that will produce minus three

0:28:33.201,0:28:38.080
over 2. Be'cause sign is[br]contained between plus one

0:28:38.080,0:28:43.080
and minus one. That doesn't[br]mean that we've done it

0:28:43.080,0:28:49.080
wrong. All it means is that[br]there are no values of X

0:28:49.080,0:28:53.580
that come from this[br]equation, and so the only

0:28:53.580,0:28:57.580
solutions are those that[br]come from this equation

0:28:57.580,0:28:59.080
here, so this.

0:29:00.210,0:29:03.549
Is our answer and[br]our only answer.

0:29:04.620,0:29:06.660
Will take one more.

0:29:07.410,0:29:14.626
Example, this one is cost[br]squared X minus sign squared X

0:29:14.626,0:29:17.250
is equal to 0.

0:29:18.970,0:29:22.460
Now we've got an identity[br]that says cost squared plus

0:29:22.460,0:29:25.950
sign squared is one, so I[br]could choose to replace

0:29:25.950,0:29:28.742
either the cost squared all[br]the sine squared.

0:29:30.510,0:29:35.190
But The reason I've chosen[br]this example is that you can

0:29:35.190,0:29:36.558
do it another way.

0:29:37.820,0:29:40.634
So let's have a look at the

0:29:40.634,0:29:47.310
other way. This is cost[br]squared minus sign squared. So

0:29:47.310,0:29:53.860
in algebra terms it's the[br]difference of two squares. It's

0:29:53.860,0:30:00.410
A squared minus B squared and[br]that has a standard

0:30:00.410,0:30:03.030
factorization of A-B A+B.

0:30:03.770,0:30:09.836
So this factorizes[br]as cause X

0:30:09.836,0:30:12.869
minus sign X.

0:30:12.870,0:30:19.334
And cause X plus[br]sign X equals 0.

0:30:20.950,0:30:28.009
So one of these two brackets,[br]one or the other, is equal to

0:30:28.009,0:30:33.439
0, so cause X minus sign X[br]equals 0 or.

0:30:34.440,0:30:40.677
Cause X plus sign[br]X equals 0.

0:30:41.520,0:30:46.788
Let's develop this one first,[br]cause X minus sign X equals 0

0:30:46.788,0:30:53.373
means that they must be the same[br]cause X and sign X are the same.

0:30:54.750,0:31:01.386
Divide both sides by Cos X and[br]we get sign over calls which is

0:31:01.386,0:31:07.224
tan. And so Tan X is equal to[br]dividing both sides by Cos X Cos

0:31:07.224,0:31:09.480
divided by cause is just one.

0:31:10.410,0:31:17.121
Or Look at this[br]one. Take kozaks away from each

0:31:17.121,0:31:23.967
side and we have sine X is equal[br]to minus Cos X. Divide both

0:31:23.967,0:31:29.835
sides by cause X sign X over[br]cause exusia gain tanks and

0:31:29.835,0:31:34.725
minus cause X divided by Cos X[br]is minus one.

0:31:35.770,0:31:41.378
We've broken that down into[br]two separate equations.

0:31:42.210,0:31:46.566
Let's have a look at how we[br]solve them. Again, let's assume

0:31:46.566,0:31:50.559
that the range of values of X is[br]not to pie.

0:31:52.160,0:31:56.380
And let's sketch the graph of[br]tan in that range.

0:31:57.050,0:32:02.011
Sketches don't have to be[br]accurate, just enough to give us

0:32:02.011,0:32:07.874
a picture of the symmetry of the[br]curve to help us solve the

0:32:07.874,0:32:14.570
equation. Tan X is one we know[br]that this is one of those

0:32:14.570,0:32:19.270
special angles that its 45[br]degrees in degrees. But since

0:32:19.270,0:32:25.380
we're working in radians, it's X[br]equals π by 4. In other words,

0:32:25.380,0:32:26.790
were across here.

0:32:27.690,0:32:34.746
One there is π by 4,[br]halfway between North and pie by

0:32:34.746,0:32:37.686
two. So again, this is.

0:32:38.210,0:32:45.042
Pie and the one we want is there[br]that will be halfway between pie

0:32:45.042,0:32:52.850
and three Pi by two. So this is[br]going to give us the one that is

0:32:52.850,0:32:58.706
halfway between pie and three Pi[br]by two, five π by 4.

0:32:59.480,0:33:02.378
With this one, where minus one.

0:33:03.250,0:33:08.290
So we're down here, meet, sit[br]there half way between pie by

0:33:08.290,0:33:14.590
two and Π, and so X there is[br]going to be three π by 4.

0:33:15.160,0:33:20.740
And meets the curve again here[br]halfway between three Pi by two

0:33:20.740,0:33:25.855
an 2π. So that's going to be 7[br]Pi by 4.

0:33:26.670,0:33:31.400
So by spotting that we could[br]factorise this equation, we

0:33:31.400,0:33:36.603
didn't need to use the[br]identity and we came up with

0:33:36.603,0:33:37.549
these solutions.

0:33:39.150,0:33:42.825
If you want, you can use the

0:33:42.825,0:33:48.172
identity. Notice what we get[br]here are the all the pie by

0:33:48.172,0:33:49.740
force if you like.

0:33:50.770,0:33:58.178
Only old pie by falls pie by 4[br]five. PI43 Pi 4 Seven π by 4.

0:34:00.020,0:34:04.244
So let's have a look at this[br]equation again, Cos squared X.

0:34:04.850,0:34:08.306
Minus sign squared X equals 0.

0:34:09.070,0:34:15.398
And we're going to use our basic[br]trig identity to solve it, so we

0:34:15.398,0:34:21.274
know that sine squared X plus[br]cost squared X is equal to 1.

0:34:22.290,0:34:27.450
I'm going to replace the sine[br]squared here, so let's have a

0:34:27.450,0:34:31.750
look what is sine squared[br]according to our identity sign

0:34:31.750,0:34:37.340
squared X? If we take away Cos[br]squared from each side is 1

0:34:37.340,0:34:42.930
minus Cos squared X. So I'm[br]going to take that, put it in

0:34:42.930,0:34:50.363
there. Cos squared X minus,[br]then a bracket 1 minus

0:34:50.363,0:34:52.694
Cos squared X.

0:34:52.700,0:34:57.032
And I use the bracket because[br]I'm taking away all of this

0:34:57.032,0:35:02.086
expression, not just a little[br]bit of it, but all of it. So the

0:35:02.086,0:35:05.696
bracket show that now I need to[br]remove the brackets.

0:35:05.800,0:35:13.312
Cos squared X minus one and[br]minus minus gives me a plus.

0:35:14.020,0:35:17.160
Cos squared X equals 0.

0:35:17.690,0:35:22.915
So now I've cost squared plus[br]cost squared. That's two of

0:35:22.915,0:35:25.765
them. Two cost squared X equals.

0:35:26.280,0:35:33.210
Wall by adding this one to both[br]sides. Now let me divide by two.

0:35:33.810,0:35:37.268
Cost squared X is one over 2.

0:35:38.140,0:35:42.732
Now at this point I could say[br]one over 2 and half. That's not

0:35:42.732,0:35:46.668
.5 and get my Calculator out[br]because I'm going to have to

0:35:46.668,0:35:47.980
take a square root.

0:35:48.590,0:35:55.814
But I don't want to do that,[br]why not? Well, half is a nice

0:35:55.814,0:36:01.898
number and I happen to know, for[br]instance, that sign 30 is 1/2

0:36:01.898,0:36:08.918
cost, 60 is 1/2. I also know[br]that sign of 45 and cause of 45

0:36:08.918,0:36:15.002
are both one over the square[br]root of 2, so there are enough

0:36:15.002,0:36:20.150
indications here to suggest to[br]me that there is a nice

0:36:20.150,0:36:24.280
relationship. Between the angle[br]and the cosine that I'm going to

0:36:24.280,0:36:29.404
get when I take the square root,[br]so I don't want to spoil that

0:36:29.404,0:36:33.796
relationship by messing it up[br]with a lot of decimals through a

0:36:33.796,0:36:38.188
Calculator. So let's take that[br]square root cause of X is one

0:36:38.188,0:36:40.384
over the square root of 2.

0:36:41.330,0:36:47.434
But I've taken a square root so[br]that means not only must I have

0:36:47.434,0:36:50.050
plus, but I must have minus.

0:36:50.840,0:36:55.988
Now we didn't say at the[br]beginning what was the range

0:36:55.988,0:37:01.604
of values of X, so let's take[br]the range that we've been

0:37:01.604,0:37:04.880
working with, namely between[br]North and 2π.

0:37:06.080,0:37:10.616
So a sketch of the graph just to[br]help us see where we are.

0:37:11.700,0:37:14.828
Here. Pie by two.

0:37:15.400,0:37:17.440
Here pie.

0:37:18.640,0:37:25.084
Three Pi by two there and 2π[br]there and our cosine function

0:37:25.084,0:37:27.769
ranges between one and minus

0:37:27.769,0:37:34.432
one. We know that the cosine of[br]X is one over Route 2. We know

0:37:34.432,0:37:39.606
that's 45 degrees or radians π[br]by 4, so we know that we're

0:37:39.606,0:37:46.320
here. Arc PY by 4[br]and of course right across there

0:37:46.320,0:37:52.776
and again halfway between these[br]two, so this bit is telling us

0:37:52.776,0:37:59.232
X is π by 4 or its[br]partner is here halfway between

0:37:59.232,0:38:04.612
three. Pi by two and 2π Seven[br]Π by 4.

0:38:05.350,0:38:10.600
And then 4 - 1 over Route 2,[br]we're going to be about there.

0:38:11.660,0:38:13.300
Go across to the graph.

0:38:14.970,0:38:20.528
Up to the X axis and again this[br]by the symmetry of the curve

0:38:20.528,0:38:26.086
must be half way between pivi 2[br]and Π, and so that gives us

0:38:26.086,0:38:27.674
three π by 4.

0:38:28.560,0:38:33.504
And here again, halfway between[br]pie and three Pi by two. So

0:38:33.504,0:38:39.272
again, that gives us five π by[br]4. So in solving this equation a

0:38:39.272,0:38:44.628
different way we've got the same[br]set of answers. And again we can

0:38:44.628,0:38:49.572
recognize them, because these[br]are the odd pie by force pie by

0:38:49.572,0:38:55.340
4, three π by 4, five π by 4,[br]and Seven π by 4.

0:38:56.200,0:39:00.400
So whether we do this solution[br]of the equation by.

0:39:01.020,0:39:02.409
Using this method.

0:39:03.170,0:39:08.198
Using the identity or the method[br]that we had before where we

0:39:08.198,0:39:12.388
factorized it doesn't matter.[br]And that's true in solving any

0:39:12.388,0:39:16.578
of these trig equations. The[br]method that you use shouldn't

0:39:16.578,0:39:20.768
matter. It should always give[br]the same set of answers.

0:39:21.680,0:39:28.445
But let's just recap where we[br]started from these three basic

0:39:28.445,0:39:34.595
and fundamental identity's sign[br]squared X plus cost squared X

0:39:34.595,0:39:37.055
is equal to 1.

0:39:38.000,0:39:40.365
1

0:39:40.365,0:39:47.320
plus. Hot[br]squared X is equal

0:39:47.320,0:39:50.992
tool cosec squared X.

0:39:52.130,0:39:59.642
And 1 + 10 squared X[br]is equal to sex squared X.

0:40:00.360,0:40:03.348
Those are our three[br]fundamental trigonometric

0:40:03.348,0:40:08.328
identities, and they must be[br]learned. They must be known,

0:40:08.328,0:40:13.308
and you must be able to[br]recognize them whenever you

0:40:13.308,0:40:14.304
see them.