0:00:02.350,0:00:08.974
This video is going to look at[br]three knew trig functions. Cosec
0:00:08.974,0:00:10.630
Zack and caught.
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However, they're not entirely[br]knew, because they are derived
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from the three that we know[br]about already sign calls and
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tan. So let's have a look.
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The first one that we want to[br]have a look at is cosec that is
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defined to be one over sine. So[br]one over sine Theta is equal to.
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Now to give it its full name it[br]is the cosecant.
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Home theater But we[br]shorten that till cosec theater.
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Second one. Follows the[br]same line one over 'cause
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Theater and it's full name is[br]the secant of Theta. But again,
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we shorten that to set theater[br]and the final one, one over
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10 theater. Equals and it's[br]full name is the cotangent of
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Theta, and again we shorten that[br]to caught theater.
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Now, why do we need these? Well,[br]first of all, they will help us
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to solve trig equations.[br]Secondly, there involved in
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identity's and 3rd they come up[br]when we do calculus,
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particularly when we do
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integration. Let's just have a[br]look at one example of where
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they might occur in terms of
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basic identity's. So the basic[br]trig identity that we've got,
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the sine squared Theta Plus Cost[br]Square theater, equals 1.
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And if I choose to divide[br]everything on both sides of this
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identity by Cos squared, then[br]I'll have sine squared Theta
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over cost Square theater plus[br]cost Square theater over Cos
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squared Theta equals one over[br]cost squared Theta.
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And so this one is sign over[br]cause all squared. So that gives
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us stand square theater plus[br]cost squared into Cos squared is
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one equals and then one over[br]cost squared is one over cause
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all squared. So that is set
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squared Theta. So there's one[br]of our new trick functions
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popping up in an identity this[br]time, and there is a similar
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one that we can get if we[br]divide throughout by sine
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squared, and if we do that, we[br]end up with one plus cot
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squared Theta is equal to[br]cosec squared Theta, so there
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the other two trig functions[br]that we've just introduced
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again pop up.
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Let's have a look at what might[br]happen when we reached the later
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stages of solving a trig
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equation. So let's take[br]cot squared Theta equals
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3 four theater between[br]360 and 0 degrees.
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Well, we begin to solve this by[br]taking the square root. So
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caught theater equals Route 3 or[br]minus Route 3. Remember, we take
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a square root, it has to be plus[br]or minus. Now we might think,
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well, let's just look this up in[br]some tables or let's take our
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Calculator, but do we really[br]need to? We know what caught
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theater is. It's one over Tan
0:04:05.454,0:04:10.320
Theta. And that's Route 3 or[br]minus Route 3.
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Now we can turn this one upside[br]down to give us Tan Theta equals
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and we can think of each of[br]these as being root 3 over one
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or minus Route 3 over one, and[br]so we can turn these upside down
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to get one over Route 3 or minus[br]one over Route 3.
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And now it's in terms of Tan[br]Theater and this is now one of
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those special values of our trig[br]functions. In fact, one over
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Route 3 is the tangent of 30[br]degrees, so we know that this
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has one solution that is 30[br]degrees. But what about the
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other solutions? Well, let's[br]have a look at those.
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Sketch of the graph.
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Tan Theta 0 up to[br]90 from 90 up through
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180 up towards 270.
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Stopping there at[br]360, so that's not.
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9180, two, 70
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and 360. And[br]the tangent of 30 is one over
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Route 3. So somewhere here is[br]one over Route 3 coming down to
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30. So of course the next one is[br]across there and the symmetry
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tells us if this is 30 on from[br]zero. This is 30 on from 180, so
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the next one is 210 degrees[br]minus one over Route 3. Well,
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that's going to be somewhere
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along here. And again, the[br]symmetry tells us if this is 30
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on this way, then this one is 30[br]back this way. So that gives us
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150 degrees and we've got[br]another value here which is
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going to be 30 back from there,[br]which is going to be 330
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degrees. So solving equations[br]that involve things like caught,
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encek and Cosec is no different[br]to solving equations to do with
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sign causing tan because we just[br]turn them into sign calls and
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tab to conclude this, we're just[br]going to have a look at the
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graphs of these three knew trig
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functions. And in order to do[br]that, we will begin each one by
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looking at the graph of the[br]related trig function. So to
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look at Cosec, we're going to[br]look at sign first. So what does
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the graph of sign?
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Look like. Will take one[br]complete cycle between North and
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360. So 0.
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180, three, 160 and the[br]peak and trough are in
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between 1970 and that goes[br]from one down 2 -
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1 and what we're going[br]to graph now is cosec
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theater, which of course is[br]one over sine Theta. So
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let's set up similar axes.
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So mark them off, there's 90.
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180 270
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360 Now[br]here at 90
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the value of[br]sign is warm.
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So at 90 the value of cosec must[br]also be one, so I'm going to
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market their one here at 270.[br]The value of sign is minus one.
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And so at 270, the value of[br]cosec must be one over minus
0:08:14.452,0:08:16.936
one, which again is just minus
0:08:16.936,0:08:19.800
one. So there are two points.
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What about this point?
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Here at zero the sign of[br]0 is 0.
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So the value of Cosec would be[br]one over 0.
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But we're not allowed to divide[br]by zero, but we can divide by
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something a little bit away.[br]What we can see is that would be
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a very very tiny positive number[br]that we were dividing by. So if
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we divide 1 by a very tiny[br]positive number, the answer has
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to be very big, but still[br]positive. So with a bit of curve
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there, let's have a look at 180.
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Well, at 180 sign of Theta is[br]again 0 so cosec is one over
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0 at this 180 degrees. Let's go[br]a little bit this side here of
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180 and the value of sign is[br]really very small. It's very
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close to 0.
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So again, 1 divided by something[br]very small and positive.
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Is again something very large[br]and positive, so let me put in
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an asymptotes. And we've got a[br]piece of curve there.
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Now this curve goes like that.[br]What we're seeing is that this
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curve is going to come down and[br]up like that, and it's going to
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do the same here, except because[br]what we're dividing by are
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negative numbers, it's going to
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be like. That
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So there's our graph of cosec[br]derived from the graph of sign.
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Let's take now calls feta.
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Do the same.
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Will take the graph of[br]costita between North and 360.
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At the extreme, values will[br]be minus one plus one
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9180 two 7360. Just make
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that clearer. And so let's have[br]a look here.
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Mark off the same
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points. And we're[br]graphing SEK this
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time sex theater,[br]which is one
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over 'cause theater.
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So again, let's Mark some
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points. Here when theater is 0.
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Costita is one Soucek[br]Theater is one over one
0:11:16.420,0:11:22.780
which is one. So will mark[br]the one there here at 180.
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Cost theater is minus one.[br]Soucek Theater is 1 divided
0:11:29.080,0:11:34.150
by minus one and so will[br]mark minus one here.
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Here at 90 we got exactly the[br]same problems we have before the
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value of Cos theater at 90 zero.[br]So 1 / 0 is a very big number.
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Well, in fact we're not allowed[br]to do it, so we have to go a
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little bit away from 90 to get a[br]value of Cos Theta which is very
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small, close to 0 but positive.[br]And if we divide 1 by that small
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positive number, the answer that[br]we get is very big and.
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Positive so we have a bit of[br]curve going up like that. What
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about this side of 90? Well this[br]side of 90 where dividing by
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something which the value of Cos[br]Theta is very small but
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definitely negative. So the[br]answer is going to be very big
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in size when we divide it into[br]one but negative. So a bit of
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the curve here coming down to[br]their same problem again at 270
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so we can see the curve is going[br]to go round.
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And back like that. And then[br]here again at 360, we're going
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to be able to mark that point.[br]We're going to have that one
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coming down at that.
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So there we've managed to get[br]the graph of SEK.
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Out of the graph, of course.
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Let's now have a look at the[br]graph of Tan Theater.
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These[br]off
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9180,[br]two,
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70[br]and
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360.[br]And now we'll have a look at
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caught theater, which is one[br]over Tan Theater.
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So we'll take the same graph and[br]I'll do the same as I've done
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before. Mark these off.
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So we're using the same
0:13:49.700,0:13:55.610
scale. OK, let's have a look[br]what's happening here. This bit
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of curve between North and 90.[br]We begin with something for tan
0:14:00.710,0:14:03.260
that is very small but positive.
0:14:03.960,0:14:09.056
Just above 0 and then it gets[br]bigger and bigger and bigger as
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it rises. The value of Tan Theta[br]rises towards Infinity.
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Well down here divide the value[br]of theater is very near to zero
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and so tan Theta is very small[br]but positive. So when we divide
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into one we're going to get[br]something very big and positive
0:14:28.247,0:14:30.152
self. But if curve there.
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Up here, the value of Tan[br]Theater is enormous. It's huge.
0:14:34.927,0:14:39.828
So if we divide something huge[br]into one, the answer is going to
0:14:39.828,0:14:44.729
be very nearly zero. And the[br]closer we get to 90, the closer
0:14:44.729,0:14:46.614
it would be to 0.
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So now if we look here, we can[br]see we've got something very,
0:14:51.747,0:14:56.024
very big, but negative. So the[br]answer is going to be very, very
0:14:56.024,0:14:59.972
small, but also negative. This[br]is going to be coming out of
0:14:59.972,0:15:02.960
that point there. Here 180.
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Got a problem at 180. Tan[br]Theater is 0 one over 10 theater
0:15:08.268,0:15:13.392
is there for something very very[br]big so we can put in an acid
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tote and we can see we've got[br]exactly the same problem here at
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360. So if I join up what I've[br]got in the direction of what's
0:15:23.861,0:15:27.678
happening, we're getting a very[br]similar curve and repeat it over
0:15:27.678,0:15:31.148
here, 'cause the curves are[br]repeated. We're getting a very
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similar curve, except the other
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way around. So we've seen again[br]how we can derive the graph of
0:15:38.753,0:15:42.137
coffee to directly from the[br]graph of Tan.
0:15:42.740,0:15:45.998
So remember these three[br]new functions.
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Co sack sack and caught.
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Respectively, they are one[br]over sign, one over cosine
0:15:56.020,0:15:57.940
and one over Tangent.
0:15:59.040,0:16:00.960
We can use them to solve
0:16:00.960,0:16:05.520
equations. But each time we[br]can get back to using sign
0:16:05.520,0:16:08.733
cause and tab to help us[br]workout the angles.