1
00:00:03,960 --> 00:00:07,302
In this video, we're going to be
looking at the definitions of

2
00:00:07,302 --> 00:00:11,751
signs, cosine and tangent for
any size of angle.

3
00:00:12,710 --> 00:00:19,262
Let's first of all recall the
sine, cosine, and tangent for a

4
00:00:19,262 --> 00:00:21,125
right angle triangle.

5
00:00:21,125 --> 00:00:28,220
There's our triangle. We identify one
angle and then label the sides.

6
00:00:29,040 --> 00:00:32,366
The side that's the longest
side in the right angled triangle

7
00:00:32,366 --> 00:00:35,588
and the one that is
opposite the right angle is

8
00:00:35,588 --> 00:00:41,481
called the hypotenuse and we
write HYP hype for short.

9
00:00:41,481 --> 00:00:46,847
The side that is opposite the
angle is called the opposite side.

10
00:00:48,850 --> 00:00:54,557
OPP for short, this side the
side that is a part of the

11
00:00:54,557 --> 00:00:58,947
angle that runs alongside the
angle we call the adjacent

12
00:00:58,947 --> 00:01:01,142
side or ADJ for short.

13
00:01:03,560 --> 00:01:06,040
The sine of the angle A.

14
00:01:06,660 --> 00:01:11,156
Is defined to be the
opposite over the hypotenuse.

15
00:01:12,290 --> 00:01:19,176
The cosine of the angle A
is defined to be the adjacent

16
00:01:19,176 --> 00:01:25,436
over the hypotenuse and the
tangent of the angle A is

17
00:01:25,436 --> 00:01:30,444
defined to be the opposite
over the adjacent.

18
00:01:32,020 --> 00:01:37,636
But this is a right angle
triangle and so the angle A is

19
00:01:37,636 --> 00:01:42,388
bound to be less than 90
degrees, but more than 0.

20
00:01:43,000 --> 00:01:48,016
In other words, it's an acute
angle, so these so far are

21
00:01:48,016 --> 00:01:52,614
defined only for acute angles.
What happens if we've got an

22
00:01:52,614 --> 00:01:57,630
angle that's bigger than 90? Or
indeed if we've got an angle

23
00:01:57,630 --> 00:01:59,302
that's less than 0?

24
00:02:00,340 --> 00:02:03,838
That raises a question. To
begin with, why? How can we

25
00:02:03,838 --> 00:02:07,654
have an angle that's less than
zero? So first of all, let's

26
00:02:07,654 --> 00:02:09,562
just have a look at angles.

27
00:02:12,590 --> 00:02:14,228
Got a set of axes there.

28
00:02:14,880 --> 00:02:17,580
And based upon the origin, let

29
00:02:17,580 --> 00:02:22,473
me draw. Roughly a unit circle.

30
00:02:22,473 --> 00:02:27,856
Circle of radius
one unit so that where it

31
00:02:27,856 --> 00:02:33,424
crosses these axes, X is
one, Y is One X is minus

32
00:02:33,424 --> 00:02:36,208
one, Y is minus one there.

33
00:02:38,620 --> 00:02:46,312
Let's imagine a point P on
this circle and it moves round

34
00:02:46,312 --> 00:02:53,363
the circle in that direction. In
other words, it moves round

35
00:02:53,363 --> 00:03:00,414
anticlockwise, then this is the
angle that OP makes with OX

36
00:03:00,414 --> 00:03:02,337
the X axis.

37
00:03:03,270 --> 00:03:05,490
So there's an acute angle.

38
00:03:06,180 --> 00:03:12,459
When we get around to here we've
come right round there and that

39
00:03:12,459 --> 00:03:17,772
gives us an obtuse angle, an
angle between 90 and 180.

40
00:03:20,150 --> 00:03:24,470
Come round to here and we've
gone right the way around there.

41
00:03:25,970 --> 00:03:32,757
An angle that is greater than
180 but less than 270.

42
00:03:33,840 --> 00:03:39,208
And similarly into this
quadrant. So that's positive,

43
00:03:40,290 --> 00:03:42,930
is anticlockwise.

44
00:03:44,070 --> 00:03:49,009
So we get positive angles
if we go round in an

45
00:03:49,009 --> 00:03:49,907
anticlockwise way.

46
00:03:53,720 --> 00:03:55,100
We draw it again.

47
00:03:57,950 --> 00:04:00,209
Same unit circle.

48
00:04:03,320 --> 00:04:06,781
And we think about our radius, OP

49
00:04:06,781 --> 00:04:10,480
What if we start to move it
around this way.

50
00:04:14,060 --> 00:04:21,480
Then this is clockwise, and
so this is a negative

51
00:04:21,480 --> 00:04:26,674
angle, so negative we're
going around clockwise.

52
00:04:28,230 --> 00:04:30,085
So of course we can go round

53
00:04:30,085 --> 00:04:35,381
from here. To the Y axis, the
negative part of the Y axis, and

54
00:04:35,381 --> 00:04:40,490
that's minus 90 degrees. If we
go right the way round to the

55
00:04:40,490 --> 00:04:44,608
negative part of the X axis,
that's minus 180 degrees.

56
00:04:44,608 --> 00:04:49,280
Effectively the same as coming
round to 180 degrees coming

57
00:04:49,280 --> 00:04:56,735
round anticlockwise. So that's
how we can have any size of

58
00:04:56,735 --> 00:05:01,506
angle. The question is can we
put these two together? Can we

59
00:05:01,506 --> 00:05:06,140
bring together these
definitions and these ideas

60
00:05:06,140 --> 00:05:11,090
about having angles which are
greater than 90 both positive

61
00:05:11,090 --> 00:05:17,030
and negative? Well, let's take
sine and have a look at that.

62
00:05:17,030 --> 00:05:20,000
So draw the same diagram again.

63
00:05:28,460 --> 00:05:32,651
Put our point P on the
circle which is going to

64
00:05:32,651 --> 00:05:36,080
move around that way in an
anti clockwise direction.

65
00:05:37,140 --> 00:05:42,279
And here mark this angle going
around that way.

66
00:05:43,630 --> 00:05:50,320
OK. Sine is opposite
over hypotenuse. Well, if

67
00:05:50,320 --> 00:05:54,520
I complete the right angle
triangle.

68
00:05:56,310 --> 00:06:02,410
This would be the side that
is opposite that angle.

69
00:06:03,190 --> 00:06:08,173
There is the right angle, so
this is the hypotenuse. And

70
00:06:08,173 --> 00:06:13,609
because this here is a unit
circle, the length of that is

71
00:06:13,609 --> 00:06:20,160
just one. So the question is,
how can I describe this line?

72
00:06:22,030 --> 00:06:27,396
If I imagine I've got my eye here
and I'm looking in that

73
00:06:27,396 --> 00:06:34,462
direction, what do I see? I see
that length as though it were

74
00:06:34,462 --> 00:06:37,810
projected onto the Y axis.

75
00:06:39,260 --> 00:06:43,889
So I'm looking that way and I
can see that length which is

76
00:06:43,889 --> 00:06:50,169
OP, as though it were
projected onto the Y axis. So

77
00:06:50,169 --> 00:06:57,296
perhaps a way of describing sine
of, let me call this theta.

78
00:06:58,460 --> 00:07:02,420
A way of describing sine theta
would be to say:

79
00:07:03,270 --> 00:07:18,791
that it is equal to the projection
of OP onto OY, the Y axis,

80
00:07:18,791 --> 00:07:25,081
divided by OP and of course
OP is then just one.

81
00:07:25,081 --> 00:07:28,819
How does it work with
any angle? Well, think what

82
00:07:28,819 --> 00:07:30,830
happens as we go round.

83
00:07:30,830 --> 00:07:33,525
As we go round as it rotates

84
00:07:33,525 --> 00:07:36,012
around. And you're still looking

85
00:07:36,012 --> 00:07:41,468
this way. Then you've still
got a projection.

86
00:07:42,210 --> 00:07:47,068
It goes down to a length of zero
and as we come back around here.

87
00:07:47,940 --> 00:07:52,059
We've still got a projection
on this axis that we can see,

88
00:07:52,059 --> 00:07:55,346
so we've still got something
that we can measure when it

89
00:07:55,346 --> 00:07:58,701
gets around to here of
course, it's on the negative

90
00:07:58,701 --> 00:08:02,430
part of the Y axis, and so
it's going to be negative.

91
00:08:03,610 --> 00:08:07,120
Well, let's have a look
what that might mean in

92
00:08:07,120 --> 00:08:08,524
terms of a graph.

93
00:08:13,770 --> 00:08:17,055
What I've got here is a
protractor and

94
00:08:17,055 --> 00:08:19,440
the middle bit of this protractor
rotates.

95
00:08:21,550 --> 00:08:26,278
Here I've got a black line which
is a fixed horizontal line.

96
00:08:27,770 --> 00:08:32,399
Along that here I've got a red
line which is going to be my OP.

97
00:08:32,399 --> 00:08:38,479
This is the point P moving
around in an anticlockwise

98
00:08:38,479 --> 00:08:40,202
direction, marking out

99
00:08:41,900 --> 00:08:45,892
positive angles, and if it went
that way around it would be

100
00:08:45,892 --> 00:08:49,241
marking out negative angles, so
they're going to start off

101
00:08:49,241 --> 00:08:55,794
together there, both pointing on
zero, the angle 0. So let's

102
00:08:55,794 --> 00:09:00,008
recall sine theta.

103
00:09:01,430 --> 00:09:08,617
The angle that OP, 
there's O, there's P,

104
00:09:08,617 --> 00:09:27,116
makes with the X axis, is defined to be
the projection of OP onto OY,

105
00:09:27,116 --> 00:09:30,173
Divided by OP.

106
00:09:30,173 --> 00:09:36,317
But if I choose to make OP
the measure, the unit then

107
00:09:36,317 --> 00:09:48,489
sine theta is just the projection
of OP onto OY.

108
00:09:49,628 --> 00:09:52,000
Now let's have a look,

109
00:09:52,310 --> 00:09:56,406
what that means in
terms of a graph?

110
00:10:00,250 --> 00:10:02,300
So this is the axis.

111
00:10:04,890 --> 00:10:09,822
Measuring marking off the
degrees, so set that to 0 so the

112
00:10:09,822 --> 00:10:14,754
first point on the graph is
there because as we look along

113
00:10:14,754 --> 00:10:19,860
there. So we look along there.
What we see is nothing. Just see

114
00:10:19,860 --> 00:10:21,435
a point. So the length.

115
00:10:22,010 --> 00:10:27,081
Of OP, the projection of OP
onto OY is 0.

116
00:10:28,210 --> 00:10:29,040
Now there.

117
00:10:30,850 --> 00:10:32,060
Halfway round.

118
00:10:33,070 --> 00:10:36,110
45 degrees it's about

119
00:10:37,100 --> 00:10:46,268
that high. So let's mark it
there. That's 45. This 90. Let's

120
00:10:46,268 --> 00:10:49,132
move that around. Do we get
to the top?

121
00:10:49,680 --> 00:10:51,279
Mark that across.

122
00:10:53,340 --> 00:10:59,698
Roughly about there. And we
start to go back down here.

123
00:11:00,640 --> 00:11:03,250
There we are 135.

124
00:11:04,830 --> 00:11:09,614
The projection is along
there and taking that

125
00:11:09,614 --> 00:11:12,006
through its to there.

126
00:11:13,470 --> 00:11:19,350
When we get round to 180 again
as we look along there, we just

127
00:11:19,350 --> 00:11:21,450
see a point, no length.

128
00:11:22,420 --> 00:11:24,310
As we come down here.

129
00:11:27,690 --> 00:11:35,530
245 there, or in fact 180 +
45, which is 225. We get a

130
00:11:35,530 --> 00:11:37,770
point which is about.

131
00:11:38,360 --> 00:11:39,210
There.

132
00:11:40,340 --> 00:11:43,884
And then as we come down to 270.

133
00:11:45,500 --> 00:11:46,540
About

134
00:11:49,640 --> 00:11:50,470
there.

135
00:11:51,810 --> 00:11:58,375
And as we come round here to
315 and through there or about

136
00:11:58,375 --> 00:12:04,086
there and then when we come
round to there we're back to 0 again

137
00:12:04,086 --> 00:12:09,485
or, having been all the
way round, 360.

138
00:12:10,500 --> 00:12:12,726
And if we join up those points.

139
00:12:18,100 --> 00:12:27,685
We get quite a nice smooth curve
out a bit with that one if we

140
00:12:27,685 --> 00:12:32,364
think of this going back in
this direction, what we can

141
00:12:32,364 --> 00:12:36,625
see is that we're going to get
the same ideas developing.

142
00:12:38,050 --> 00:12:42,467
Here, let's just fill in the 90.
It will be right down there, so

143
00:12:42,467 --> 00:12:47,184
it's there. And then at minus
180 right around there it's

144
00:12:47,184 --> 00:12:52,406
there and then minus 270 going
right. The way around there we

145
00:12:52,406 --> 00:12:54,734
are back up at the top again.

146
00:12:55,500 --> 00:13:00,311
And then minus 360 having
come all the way around we're

147
00:13:00,311 --> 00:13:08,150
down there. And so again we
have a nice smooth curve.

148
00:13:10,670 --> 00:13:16,676
Notice that this shape is
exactly the same as that shape

149
00:13:16,676 --> 00:13:22,980
and that we could keep on
drawing it. This block,

150
00:13:22,980 --> 00:13:28,956
This block repeats itself, it's
periodic. It keeps repeating

151
00:13:28,956 --> 00:13:35,930
itself every 360 degrees from
there to there is 360. And

152
00:13:35,930 --> 00:13:42,270
similarly from here through
here. Till here is also 360

153
00:13:42,270 --> 00:13:49,244
degrees. So now we have a
sine function if you like.

154
00:13:50,530 --> 00:13:54,545
That we can think of as being
defined by this graph.

155
00:13:55,820 --> 00:14:00,333
It covers any angle that we would
want it to cover. We can keep on

156
00:14:00,333 --> 00:14:05,636
going for 720, 1000 degrees that
way, minus 1000 that way.

157
00:14:06,340 --> 00:14:10,996
But this is always going to give
us a well defined function.

158
00:14:12,800 --> 00:14:17,946
What about something like the
cosine curve? What about that one?

159
00:14:17,946 --> 00:14:22,230
Well, let's just have a
look at that and see how we

160
00:14:22,230 --> 00:14:27,200
can make a similar graph for
our cosine function.

161
00:14:28,470 --> 00:14:30,798
I'll just stick that down again.

162
00:14:32,100 --> 00:14:35,532
And quickly draw a set of

163
00:14:35,532 --> 00:14:39,738
axes. I'm not draw this
one as accurately.

164
00:14:40,780 --> 00:14:45,996
But it should be enough for
us to be able to see the

165
00:14:45,996 --> 00:14:53,662
results that we're wanting, so
we'll mark off the divisions

166
00:14:53,662 --> 00:14:55,247
as we have before.

167
00:14:58,920 --> 00:15:07,028
0, 90, 180, 270, 360
and then -90,

168
00:15:07,028 --> 00:15:12,632
-180, -270 and -360 there.

169
00:15:13,960 --> 00:15:19,776
OK the thing that we haven't
done is made a definition.

170
00:15:20,690 --> 00:15:25,999
What do we mean
by cos theta?

171
00:15:27,590 --> 00:15:28,980
Well, let's have a look.

172
00:15:29,860 --> 00:15:33,666
We want to do the same sort of
thing as we did for sine.

173
00:15:34,030 --> 00:15:35,670
So that's the angle.

174
00:15:37,070 --> 00:15:37,870
In there.

175
00:15:39,570 --> 00:15:44,542
The adjacent side will be this
side, which is the projection.

176
00:15:45,200 --> 00:15:52,310
Of OP onto the X axis, this time,

177
00:15:52,310 --> 00:16:04,740
so O cos of the angle is the
projection of OP onto OX,

178
00:16:04,740 --> 00:16:11,926
the X axis, divided by OP.

179
00:16:12,801 --> 00:16:15,474
This is a unit circle,
so OP is one.

180
00:16:15,474 --> 00:16:26,816
So cos theta is the projection
of OP onto OX

181
00:16:26,816 --> 00:16:29,700
Now we need to
look at that and see how

182
00:16:29,700 --> 00:16:36,196
that grows, and varies as we
rotate around.

183
00:16:36,196 --> 00:16:41,280
So we start with zero. Remember
our eye is now looking down

184
00:16:41,280 --> 00:16:45,247
What we see is OP itself.
So what we see is

185
00:16:45,247 --> 00:16:48,214
a point there, one.

186
00:16:48,214 --> 00:16:52,980
We start to move this around.
Let's go to 45.

187
00:16:53,530 --> 00:16:58,530
And. We're looking down, so we
see that bit there, which is

188
00:16:58,530 --> 00:17:01,930
less, which is smaller. So we
see something about there.

189
00:17:02,860 --> 00:17:07,480
As we go round to 90 when we
look straight down on this red

190
00:17:07,480 --> 00:17:11,770
line, all we see is a dot and so
the projection is 0.

191
00:17:12,650 --> 00:17:18,630
Let's go round and now to 145
and now the projection is down

192
00:17:18,630 --> 00:17:24,610
onto the negative part of the X
axis, so this is negative. Now

193
00:17:24,610 --> 00:17:28,060
down there. And at 180.

194
00:17:29,070 --> 00:17:35,728
Well, we now right round to
sitting on top of OP again of length -1.

195
00:17:39,240 --> 00:17:44,808
Start to come around again to
225 and again we get that

196
00:17:44,808 --> 00:17:49,912
projection back, so we're
starting to be here and then at

197
00:17:49,912 --> 00:17:55,016
270. Clearly again, we're going
to come round to there looking

198
00:17:55,016 --> 00:17:56,408
down that way.

199
00:17:57,550 --> 00:18:03,855
As we come round, the projection
is again at 315. Now a positive

200
00:18:03,855 --> 00:18:09,190
projection. So again we've gone
through there to there and then

201
00:18:09,190 --> 00:18:12,100
360. We're back to 0 again.

202
00:18:13,730 --> 00:18:16,943
And so the projection
is of length one, so

203
00:18:16,943 --> 00:18:18,371
let's fill that in.

204
00:18:19,790 --> 00:18:26,790
Join up the points and again we
get a nice smooth curve. And of

205
00:18:26,790 --> 00:18:32,290
course the same curve is going
to exist on this side.

206
00:18:34,680 --> 00:18:39,516
Let's just check we're going to
swing it around this way and we

207
00:18:39,516 --> 00:18:44,363
can see. Is that the projection
is positive but getting less

208
00:18:44,363 --> 00:18:50,174
until we get round till 90 when
the length of the projection is

209
00:18:50,174 --> 00:18:54,644
0? So we're going to see this
occurring round here.

210
00:18:55,270 --> 00:19:01,198
Nice smooth curve up through
there to there.

211
00:19:02,920 --> 00:19:08,250
Again, notice it's periodic.
This lump of curve here is

212
00:19:08,250 --> 00:19:15,033
repeated there and will be
every 360 as we march

213
00:19:15,033 --> 00:19:18,910
backwards and forwards along
this X axis. So again, we've got

214
00:19:18,910 --> 00:19:23,707
a function that's well
defined, got a nice curve,

215
00:19:23,707 --> 00:19:29,245
periodic nice, smooth curve,
so that's our definition for cosine.

216
00:19:30,190 --> 00:19:34,123
Notice it's contained between
plus one and minus one.

217
00:19:35,020 --> 00:19:38,670
That was something that we
didn't observe with sine, but

218
00:19:38,670 --> 00:19:43,415
that is also the case that it is
contained between plus one and

219
00:19:43,415 --> 00:19:47,430
minus one, because this
projection of OP onto OY can

220
00:19:47,430 --> 00:19:51,080
never be longer than OP
itself, which is just one.

221
00:19:54,670 --> 00:19:59,070
The other thing to notice
is that the two curves are

222
00:19:59,070 --> 00:19:59,870
the same.

223
00:20:01,310 --> 00:20:05,940
I just flipped back but
displaced by 90 degrees. We

224
00:20:05,940 --> 00:20:11,496
slide this sine curve back by
90 degrees. You can see it

225
00:20:11,496 --> 00:20:15,663
will be exactly the same as
the cosine curve.

226
00:20:16,720 --> 00:20:21,978
What about tangent? Well, let's
recall how we define the tangent

227
00:20:21,978 --> 00:20:26,758
to begin with, it was the
opposite over the adjacent.

228
00:20:27,800 --> 00:20:33,880
For sine we replaced the
opposite side by the projection

229
00:20:33,880 --> 00:20:36,920
of OP onto OY.

230
00:20:38,130 --> 00:20:44,741
And for cosine, we replaced the
adjacent side as the projection

231
00:20:44,741 --> 00:20:47,746
of OP onto OX.

232
00:20:49,150 --> 00:20:52,614
What does that mean then
for our definition?

233
00:21:02,720 --> 00:21:06,210
We draw our unit circle.

234
00:21:08,110 --> 00:21:08,830
Take.

235
00:21:12,590 --> 00:21:17,666
OP moving around in that
direction through an angle

236
00:21:17,666 --> 00:21:24,184
theta and complete right
angle triangle. This here

237
00:21:24,184 --> 00:21:27,254
is the opposite side. It's
opposite, the angle that

238
00:21:27,254 --> 00:21:28,946
we're talking about.

239
00:21:30,900 --> 00:21:35,553
tan theta equals, so the
opposite side, we have

240
00:21:35,553 --> 00:21:47,662
replaced by the projection
of OP onto OY

241
00:21:49,060 --> 00:21:52,757
divided by, now for a right
angle triangle it would be

242
00:21:52,757 --> 00:21:58,097
the adjacent side and that
adjacent side has been

243
00:21:58,097 --> 00:22:07,288
replaced by the projection of
OP on till OX. Well let's

244
00:22:07,288 --> 00:22:14,344
remember OY is the Y
axis OX is the X axis.

245
00:22:16,350 --> 00:22:20,326
This again gives us a definition
that's going to work as that

246
00:22:20,326 --> 00:22:23,710
radius vector runs around the
circle like that,

247
00:22:24,340 --> 00:22:27,937
in either direction. So again,
it's going to give us a

248
00:22:27,937 --> 00:22:32,896
definition that will work
for any size of angle.

249
00:22:33,350 --> 00:22:37,404
One of the things we can notice
straight away about this is that

250
00:22:37,404 --> 00:22:45,024
it means that tan theta is of
course sine theta divided by cos theta

251
00:22:45,024 --> 00:22:49,052
and that gives us an
identity which we need to learn

252
00:22:49,052 --> 00:22:54,902
and remember. tan theta is sine
theta divided by cos theta.

253
00:22:56,690 --> 00:23:00,470
But what does the graph of
tangent look like?

254
00:23:04,190 --> 00:23:06,290
It's a little bit
trickier to draw.

255
00:23:11,320 --> 00:23:15,841
Let's see if we can justify what
we're going to get.

256
00:23:28,710 --> 00:23:31,908
Now we're looking
at tan theta.

257
00:23:33,510 --> 00:23:39,414
where theta is the angle between
the X axis and the Y Axis.

258
00:23:40,320 --> 00:23:45,598
And we know that the tangent is
defined to be the projection of

259
00:23:45,598 --> 00:23:47,628
OP on the Y axis,

260
00:23:48,210 --> 00:23:52,110
divided by the projection of
OP on the X axis.

261
00:23:53,030 --> 00:23:57,494
So when we begin
here at theta is 0.

262
00:23:59,180 --> 00:24:05,631
Then we know that the projection
onto the Y axis looking this way

263
00:24:05,631 --> 00:24:11,660
is zero and onto the X axis is
one, so we've got 0 / 1 which is

264
00:24:11,660 --> 00:24:17,484
0. So we start there, for the
angle zero, we start there. As we

265
00:24:17,484 --> 00:24:21,682
move around. When we come to 45
degrees then the projections are

266
00:24:21,682 --> 00:24:27,228
equal. So let's just mark 45 and
if the two projections are equal

267
00:24:27,228 --> 00:24:29,964
then that must be 1.

268
00:24:30,840 --> 00:24:33,366
What happens is we come up

269
00:24:33,366 --> 00:24:40,230
towards 90. Well, the projection
on to the Y axis is getting

270
00:24:40,230 --> 00:24:42,278
bigger and bigger approaching

271
00:24:42,278 --> 00:24:48,578
one. But the projection onto the
X axis is getting smaller and

272
00:24:48,578 --> 00:24:54,074
smaller and smaller, and it's
that that we are dividing by, so

273
00:24:54,074 --> 00:24:57,280
we're dividing something
approaching one by something

274
00:24:57,280 --> 00:25:01,860
that is getting smaller and
smaller and smaller. So our

275
00:25:01,860 --> 00:25:06,440
answer is getting bigger and
bigger and bigger. It's becoming

276
00:25:06,440 --> 00:25:11,936
infinite and we have a way of
showing that on a graph.

277
00:25:14,490 --> 00:25:19,190
I've deliberately put in a
dotted line. Now a graph

278
00:25:19,190 --> 00:25:23,310
approaches that dotted line, but
does not cross it, and that's an

279
00:25:23,310 --> 00:25:28,337
asymptote. That's what we call
an asymptote. What about the

280
00:25:28,337 --> 00:25:34,679
next bit of the graph up to 180
degrees? Well, as we tip over

281
00:25:34,679 --> 00:25:41,280
into this quadrant, then the
projection on to the X axis

282
00:25:41,280 --> 00:25:45,040
becomes negative, but he's still
very, very small.

283
00:25:46,320 --> 00:25:51,060
The projection onto the Y axis
is still positive, still near 1,

284
00:25:51,060 --> 00:25:55,010
so we're dividing something
that's positive and near 1 by

285
00:25:55,010 --> 00:25:58,960
something that's negative but
very very small. So the answer

286
00:25:58,960 --> 00:26:03,700
must be very, very big, but
negative, and so there's a bit

287
00:26:03,700 --> 00:26:09,920
of graph. Down there on the
other side of the asymptote, and

288
00:26:09,920 --> 00:26:15,644
now we run this round to 180
degrees and what happens then?

289
00:26:15,644 --> 00:26:21,368
Well, when we've got round to
180, the projection onto the X

290
00:26:21,368 --> 00:26:27,175
axis is then of length one, the
same as OP, but the projection

291
00:26:27,175 --> 00:26:35,201
onto the Y axis is 0, so we've
got 0 / 1 again, which gives us

292
00:26:35,201 --> 00:26:41,003
0 at 180 degrees. So this comes
up to there like that.

293
00:26:42,030 --> 00:26:46,551
Think about what's going to
happen now as it comes around

294
00:26:46,551 --> 00:26:48,606
here. Let's draw one in.

295
00:26:49,840 --> 00:26:54,868
And we can see that the
projections on to both axes are

296
00:26:54,868 --> 00:26:58,120
both negative. And a
negative divided by a

297
00:26:58,120 --> 00:26:59,360
negative gives a positive.

298
00:27:02,460 --> 00:27:06,223
You can also see that when we
get here again, we've got

299
00:27:06,223 --> 00:27:09,953
exactly the same problems as we
had when we got here. We're

300
00:27:09,953 --> 00:27:13,812
dividing by something very, very
small into something that's

301
00:27:13,812 --> 00:27:18,284
around about one, so again I'll
answer is going to be very, very

302
00:27:18,284 --> 00:27:22,412
big, and so again, we're going
to get a climb like that.

303
00:27:23,620 --> 00:27:27,022
As we move through this, it's
the same as if we move

304
00:27:27,022 --> 00:27:31,038
through there, so again we
will start back down here in

305
00:27:31,038 --> 00:27:35,005
the graph, will climb and
then off up again there.

306
00:27:36,570 --> 00:27:38,470
What about back this way?

307
00:27:39,020 --> 00:27:41,168
When we've got negative

308
00:27:44,420 --> 00:27:49,249
angles what's going to happen
then we must have the same

309
00:27:49,249 --> 00:27:54,078
things occuring in terms of
our asymptotes. And so we're

310
00:27:54,078 --> 00:27:59,346
going to have the same things
occur in with the graph there.

311
00:28:00,930 --> 00:28:05,390
There and so on. So again,
notice we get a periodic

312
00:28:05,390 --> 00:28:11,036
function. That bit of graph is
repeated again there.

313
00:28:12,070 --> 00:28:17,108
Every 360 degrees we get a
periodic function. We get a

314
00:28:17,108 --> 00:28:19,856
repeat of this section of the

315
00:28:19,856 --> 00:28:23,860
graph. So that's our
function tangent.

316
00:28:25,840 --> 00:28:28,327
And we can think of the
function as being defined if

317
00:28:28,327 --> 00:28:30,583
you like by that particular
graph.