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