WEBVTT 00:00:03.960 --> 00:00:07.302 In this video, we're going to be looking at the definitions of 00:00:07.302 --> 00:00:11.751 signs, cosine and tangent for any size of angle. 00:00:12.710 --> 00:00:19.262 Let's first of all recall the sine, cosine, and tangent for a 00:00:19.262 --> 00:00:21.125 right angle triangle. 00:00:21.125 --> 00:00:28.220 There's our triangle. We identify one angle and then label the sides. 00:00:29.040 --> 00:00:32.366 The side that's the longest side in the right angled triangle 00:00:32.366 --> 00:00:35.588 and the one that is opposite the right angle is 00:00:35.588 --> 00:00:41.481 called the hypotenuse and we write HYP hype for short. 00:00:41.481 --> 00:00:46.847 The side that is opposite the angle is called the opposite side. 00:00:48.850 --> 00:00:54.557 OPP for short, this side the side that is a part of the 00:00:54.557 --> 00:00:58.947 angle that runs alongside the angle we call the adjacent 00:00:58.947 --> 00:01:01.142 side or ADJ for short. 00:01:03.560 --> 00:01:06.040 The sine of the angle A. 00:01:06.660 --> 00:01:11.156 Is defined to be the opposite over the hypotenuse. 00:01:12.290 --> 00:01:19.176 The cosine of the angle A is defined to be the adjacent 00:01:19.176 --> 00:01:25.436 over the hypotenuse and the tangent of the angle A is 00:01:25.436 --> 00:01:30.444 defined to be the opposite over the adjacent. 00:01:32.020 --> 00:01:37.636 But this is a right angle triangle and so the angle A is 00:01:37.636 --> 00:01:42.388 bound to be less than 90 degrees, but more than 0. 00:01:43.000 --> 00:01:48.016 In other words, it's an acute angle, so these so far are 00:01:48.016 --> 00:01:52.614 defined only for acute angles. What happens if we've got an 00:01:52.614 --> 00:01:57.630 angle that's bigger than 90? Or indeed if we've got an angle 00:01:57.630 --> 00:01:59.302 that's less than 0? 00:02:00.340 --> 00:02:03.838 That raises a question. To begin with, why? How can we 00:02:03.838 --> 00:02:07.654 have an angle that's less than zero? So first of all, let's 00:02:07.654 --> 00:02:09.562 just have a look at angles. 00:02:12.590 --> 00:02:14.228 Got a set of axes there. 00:02:14.880 --> 00:02:17.580 And based upon the origin, let 00:02:17.580 --> 00:02:22.473 me draw. Roughly a unit circle. 00:02:22.473 --> 00:02:27.856 Circle of radius one unit so that where it 00:02:27.856 --> 00:02:33.424 crosses these axes, X is one, Y is One X is minus 00:02:33.424 --> 00:02:36.208 one, Y is minus one there. 00:02:38.620 --> 00:02:46.312 Let's imagine a point P on this circle and it moves round 00:02:46.312 --> 00:02:53.363 the circle in that direction. In other words, it moves round 00:02:53.363 --> 00:03:00.414 anticlockwise, then this is the angle that OP makes with OX 00:03:00.414 --> 00:03:02.337 the X axis. 00:03:03.270 --> 00:03:05.490 So there's an acute angle. 00:03:06.180 --> 00:03:12.459 When we get around to here we've come right round there and that 00:03:12.459 --> 00:03:17.772 gives us an obtuse angle, an angle between 90 and 180. 00:03:20.150 --> 00:03:24.470 Come round to here and we've gone right the way around there. 00:03:25.970 --> 00:03:32.757 An angle that is greater than 180 but less than 270. 00:03:33.840 --> 00:03:39.208 And similarly into this quadrant. So that's positive, 00:03:40.290 --> 00:03:42.930 is anticlockwise. 00:03:44.070 --> 00:03:49.009 So we get positive angles if we go round in an 00:03:49.009 --> 00:03:49.907 anticlockwise way. 00:03:53.720 --> 00:03:55.100 We draw it again. 00:03:57.950 --> 00:04:00.209 Same unit circle. 00:04:03.320 --> 00:04:06.781 And we think about our radius, OP 00:04:06.781 --> 00:04:10.480 What if we start to move it around this way. 00:04:14.060 --> 00:04:21.480 Then this is clockwise, and so this is a negative 00:04:21.480 --> 00:04:26.674 angle, so negative we're going around clockwise. 00:04:28.230 --> 00:04:30.085 So of course we can go round 00:04:30.085 --> 00:04:35.381 from here. To the Y axis, the negative part of the Y axis, and 00:04:35.381 --> 00:04:40.490 that's minus 90 degrees. If we go right the way round to the 00:04:40.490 --> 00:04:44.608 negative part of the X axis, that's minus 180 degrees. 00:04:44.608 --> 00:04:49.280 Effectively the same as coming round to 180 degrees coming 00:04:49.280 --> 00:04:56.735 round anticlockwise. So that's how we can have any size of 00:04:56.735 --> 00:05:01.506 angle. The question is can we put these two together? Can we 00:05:01.506 --> 00:05:06.140 bring together these definitions and these ideas 00:05:06.140 --> 00:05:11.090 about having angles which are greater than 90 both positive 00:05:11.090 --> 00:05:17.030 and negative? Well, let's take sine and have a look at that. 00:05:17.030 --> 00:05:20.000 So draw the same diagram again. 00:05:28.460 --> 00:05:32.651 Put our point P on the circle which is going to 00:05:32.651 --> 00:05:36.080 move around that way in an anti clockwise direction. 00:05:37.140 --> 00:05:42.279 And here mark this angle going around that way. 00:05:43.630 --> 00:05:50.320 OK. Sine is opposite over hypotenuse. Well, if 00:05:50.320 --> 00:05:54.520 I complete the right angle triangle. 00:05:56.310 --> 00:06:02.410 This would be the side that is opposite that angle. 00:06:03.190 --> 00:06:08.173 There is the right angle, so this is the hypotenuse. And 00:06:08.173 --> 00:06:13.609 because this here is a unit circle, the length of that is 00:06:13.609 --> 00:06:20.160 just one. So the question is, how can I describe this line? 00:06:22.030 --> 00:06:27.396 If I imagine I've got my eye here and I'm looking in that 00:06:27.396 --> 00:06:34.462 direction, what do I see? I see that length as though it were 00:06:34.462 --> 00:06:37.810 projected onto the Y axis. 00:06:39.260 --> 00:06:43.889 So I'm looking that way and I can see that length which is 00:06:43.889 --> 00:06:50.169 OP, as though it were projected onto the Y axis. So 00:06:50.169 --> 00:06:57.296 perhaps a way of describing sine of, let me call this theta. 00:06:58.460 --> 00:07:02.420 A way of describing sine theta would be to say: 00:07:03.270 --> 00:07:18.791 that it is equal to the projection of OP onto OY, the Y axis, 00:07:18.791 --> 00:07:25.081 divided by OP and of course OP is then just one. 00:07:25.081 --> 00:07:28.819 How does it work with any angle? Well, think what 00:07:28.819 --> 00:07:30.830 happens as we go round. 00:07:30.830 --> 00:07:33.525 As we go round as it rotates 00:07:33.525 --> 00:07:36.012 around. And you're still looking 00:07:36.012 --> 00:07:41.468 this way. Then you've still got a projection. 00:07:42.210 --> 00:07:47.068 It goes down to a length of zero and as we come back around here. 00:07:47.940 --> 00:07:52.059 We've still got a projection on this axis that we can see, 00:07:52.059 --> 00:07:55.346 so we've still got something that we can measure when it 00:07:55.346 --> 00:07:58.701 gets around to here of course, it's on the negative 00:07:58.701 --> 00:08:02.430 part of the Y axis, and so it's going to be negative. 00:08:03.610 --> 00:08:07.120 Well, let's have a look what that might mean in 00:08:07.120 --> 00:08:08.524 terms of a graph. 00:08:13.770 --> 00:08:17.055 What I've got here is a protractor and 00:08:17.055 --> 00:08:19.440 the middle bit of this protractor rotates. 00:08:21.550 --> 00:08:26.278 Here I've got a black line which is a fixed horizontal line. 00:08:27.770 --> 00:08:32.399 Along that here I've got a red line which is going to be my OP. 00:08:32.399 --> 00:08:38.479 This is the point P moving around in an anticlockwise 00:08:38.479 --> 00:08:40.202 direction, marking out 00:08:41.900 --> 00:08:45.892 positive angles, and if it went that way around it would be 00:08:45.892 --> 00:08:49.241 marking out negative angles, so they're going to start off 00:08:49.241 --> 00:08:55.794 together there, both pointing on zero, the angle 0. So let's 00:08:55.794 --> 00:09:00.008 recall sine theta. 00:09:01.430 --> 00:09:08.617 The angle that OP, there's O, there's P, 00:09:08.617 --> 00:09:27.116 makes with the X axis, is defined to be the projection of OP onto OY, 00:09:27.116 --> 00:09:30.173 Divided by OP. 00:09:30.173 --> 00:09:36.317 But if I choose to make OP the measure, the unit then 00:09:36.317 --> 00:09:48.489 sine theta is just the projection of OP onto OY. 00:09:49.628 --> 00:09:52.000 Now let's have a look, 00:09:52.310 --> 00:09:56.406 what that means in terms of a graph? 00:10:00.250 --> 00:10:02.300 So this is the axis. 00:10:04.890 --> 00:10:09.822 Measuring marking off the degrees, so set that to 0 so the 00:10:09.822 --> 00:10:14.754 first point on the graph is there because as we look along 00:10:14.754 --> 00:10:19.860 there. So we look along there. What we see is nothing. Just see 00:10:19.860 --> 00:10:21.435 a point. So the length. 00:10:22.010 --> 00:10:27.081 Of OP, the projection of OP onto OY is 0. 00:10:28.210 --> 00:10:29.040 Now there. 00:10:30.850 --> 00:10:32.060 Halfway round. 00:10:33.070 --> 00:10:36.110 45 degrees it's about 00:10:37.100 --> 00:10:46.268 that high. So let's mark it there. That's 45. This 90. Let's 00:10:46.268 --> 00:10:49.132 move that around. Do we get to the top? 00:10:49.680 --> 00:10:51.279 Mark that across. 00:10:53.340 --> 00:10:59.698 Roughly about there. And we start to go back down here. 00:11:00.640 --> 00:11:03.250 There we are 135. 00:11:04.830 --> 00:11:09.614 The projection is along there and taking that 00:11:09.614 --> 00:11:12.006 through its to there. 00:11:13.470 --> 00:11:19.350 When we get round to 180 again as we look along there, we just 00:11:19.350 --> 00:11:21.450 see a point, no length. 00:11:22.420 --> 00:11:24.310 As we come down here. 00:11:27.690 --> 00:11:35.530 245 there, or in fact 180 + 45, which is 225. We get a 00:11:35.530 --> 00:11:37.770 point which is about. 00:11:38.360 --> 00:11:39.210 There. 00:11:40.340 --> 00:11:43.884 And then as we come down to 270. 00:11:45.500 --> 00:11:46.540 About 00:11:49.640 --> 00:11:50.470 there. 00:11:51.810 --> 00:11:58.375 And as we come round here to 315 and through there or about 00:11:58.375 --> 00:12:04.086 there and then when we come round to there we're back to 0 again 00:12:04.086 --> 00:12:09.485 or, having been all the way round, 360. 00:12:10.500 --> 00:12:12.726 And if we join up those points. 00:12:18.100 --> 00:12:27.685 We get quite a nice smooth curve out a bit with that one if we 00:12:27.685 --> 00:12:32.364 think of this going back in this direction, what we can 00:12:32.364 --> 00:12:36.625 see is that we're going to get the same ideas developing. 00:12:38.050 --> 00:12:42.467 Here, let's just fill in the 90. It will be right down there, so 00:12:42.467 --> 00:12:47.184 it's there. And then at minus 180 right around there it's 00:12:47.184 --> 00:12:52.406 there and then minus 270 going right. The way around there we 00:12:52.406 --> 00:12:54.734 are back up at the top again. 00:12:55.500 --> 00:13:00.311 And then minus 360 having come all the way around we're 00:13:00.311 --> 00:13:08.150 down there. And so again we have a nice smooth curve. 00:13:10.670 --> 00:13:16.676 Notice that this shape is exactly the same as that shape 00:13:16.676 --> 00:13:22.980 and that we could keep on drawing it. This block, 00:13:22.980 --> 00:13:28.956 This block repeats itself, it's periodic. It keeps repeating 00:13:28.956 --> 00:13:35.930 itself every 360 degrees from there to there is 360. And 00:13:35.930 --> 00:13:42.270 similarly from here through here. Till here is also 360 00:13:42.270 --> 00:13:49.244 degrees. So now we have a sine function if you like. 00:13:50.530 --> 00:13:54.545 That we can think of as being defined by this graph. 00:13:55.820 --> 00:14:00.333 It covers any angle that we would want it to cover. We can keep on 00:14:00.333 --> 00:14:05.636 going for 720, 1000 degrees that way, minus 1000 that way. 00:14:06.340 --> 00:14:10.996 But this is always going to give us a well defined function. 00:14:12.800 --> 00:14:17.946 What about something like the cosine curve? What about that one? 00:14:17.946 --> 00:14:22.230 Well, let's just have a look at that and see how we 00:14:22.230 --> 00:14:27.200 can make a similar graph for our cosine function. 00:14:28.470 --> 00:14:30.798 I'll just stick that down again. 00:14:32.100 --> 00:14:35.532 And quickly draw a set of 00:14:35.532 --> 00:14:39.738 axes. I'm not draw this one as accurately. 00:14:40.780 --> 00:14:45.996 But it should be enough for us to be able to see the 00:14:45.996 --> 00:14:53.662 results that we're wanting, so we'll mark off the divisions 00:14:53.662 --> 00:14:55.247 as we have before. 00:14:58.920 --> 00:15:07.028 0, 90, 180, 270, 360 and then -90, 00:15:07.028 --> 00:15:12.632 -180, -270 and -360 there. 00:15:13.960 --> 00:15:19.776 OK the thing that we haven't done is made a definition. 00:15:20.690 --> 00:15:25.999 What do we mean by cos theta? 00:15:27.590 --> 00:15:28.980 Well, let's have a look. 00:15:29.860 --> 00:15:33.666 We want to do the same sort of thing as we did for sine. 00:15:34.030 --> 00:15:35.670 So that's the angle. 00:15:37.070 --> 00:15:37.870 In there. 00:15:39.570 --> 00:15:44.542 The adjacent side will be this side, which is the projection. 00:15:45.200 --> 00:15:52.310 Of OP onto the X axis, this time, 00:15:52.310 --> 00:16:04.740 so O cos of the angle is the projection of OP onto OX, 00:16:04.740 --> 00:16:11.926 the X axis, divided by OP. 00:16:12.801 --> 00:16:15.474 This is a unit circle, so OP is one. 00:16:15.474 --> 00:16:26.816 So cos theta is the projection of OP onto OX 00:16:26.816 --> 00:16:29.700 Now we need to look at that and see how 00:16:29.700 --> 00:16:36.196 that grows, and varies as we rotate around. 00:16:36.196 --> 00:16:41.280 So we start with zero. Remember our eye is now looking down 00:16:41.280 --> 00:16:45.247 What we see is OP itself. So what we see is 00:16:45.247 --> 00:16:48.214 a point there, one. 00:16:48.214 --> 00:16:52.980 We start to move this around. Let's go to 45. 00:16:53.530 --> 00:16:58.530 And. We're looking down, so we see that bit there, which is 00:16:58.530 --> 00:17:01.930 less, which is smaller. So we see something about there. 00:17:02.860 --> 00:17:07.480 As we go round to 90 when we look straight down on this red 00:17:07.480 --> 00:17:11.770 line, all we see is a dot and so the projection is 0. 00:17:12.650 --> 00:17:18.630 Let's go round and now to 145 and now the projection is down 00:17:18.630 --> 00:17:24.610 onto the negative part of the X axis, so this is negative. Now 00:17:24.610 --> 00:17:28.060 down there. And at 180. 00:17:29.070 --> 00:17:35.728 Well, we now right round to sitting on top of OP again of length -1. 00:17:39.240 --> 00:17:44.808 Start to come around again to 225 and again we get that 00:17:44.808 --> 00:17:49.912 projection back, so we're starting to be here and then at 00:17:49.912 --> 00:17:55.016 270. Clearly again, we're going to come round to there looking 00:17:55.016 --> 00:17:56.408 down that way. 00:17:57.550 --> 00:18:03.855 As we come round, the projection is again at 315. Now a positive 00:18:03.855 --> 00:18:09.190 projection. So again we've gone through there to there and then 00:18:09.190 --> 00:18:12.100 360. We're back to 0 again. 00:18:13.730 --> 00:18:16.943 And so the projection is of length one, so 00:18:16.943 --> 00:18:18.371 let's fill that in. 00:18:19.790 --> 00:18:26.790 Join up the points and again we get a nice smooth curve. And of 00:18:26.790 --> 00:18:32.290 course the same curve is going to exist on this side. 00:18:34.680 --> 00:18:39.516 Let's just check we're going to swing it around this way and we 00:18:39.516 --> 00:18:44.363 can see. Is that the projection is positive but getting less 00:18:44.363 --> 00:18:50.174 until we get round till 90 when the length of the projection is 00:18:50.174 --> 00:18:54.644 0? So we're going to see this occurring round here. 00:18:55.270 --> 00:19:01.198 Nice smooth curve up through there to there. 00:19:02.920 --> 00:19:08.250 Again, notice it's periodic. This lump of curve here is 00:19:08.250 --> 00:19:15.033 repeated there and will be every 360 as we march 00:19:15.033 --> 00:19:18.910 backwards and forwards along this X axis. So again, we've got 00:19:18.910 --> 00:19:23.707 a function that's well defined, got a nice curve, 00:19:23.707 --> 00:19:29.245 periodic nice, smooth curve, so that's our definition for cosine. 00:19:30.190 --> 00:19:34.123 Notice it's contained between plus one and minus one. 00:19:35.020 --> 00:19:38.670 That was something that we didn't observe with sine, but 00:19:38.670 --> 00:19:43.415 that is also the case that it is contained between plus one and 00:19:43.415 --> 00:19:47.430 minus one, because this projection of OP onto OY can 00:19:47.430 --> 00:19:51.080 never be longer than OP itself, which is just one. 00:19:54.670 --> 00:19:59.070 The other thing to notice is that the two curves are 00:19:59.070 --> 00:19:59.870 the same. 00:20:01.310 --> 00:20:05.940 I just flipped back but displaced by 90 degrees. We 00:20:05.940 --> 00:20:11.496 slide this sine curve back by 90 degrees. You can see it 00:20:11.496 --> 00:20:15.663 will be exactly the same as the cosine curve. 00:20:16.720 --> 00:20:21.978 What about tangent? Well, let's recall how we define the tangent 00:20:21.978 --> 00:20:26.758 to begin with, it was the opposite over the adjacent. 00:20:27.800 --> 00:20:33.880 For sine we replaced the opposite side by the projection 00:20:33.880 --> 00:20:36.920 of OP onto OY. 00:20:38.130 --> 00:20:44.741 And for cosine, we replaced the adjacent side as the projection 00:20:44.741 --> 00:20:47.746 of OP onto OX. 00:20:49.150 --> 00:20:52.614 What does that mean then for our definition? 00:21:02.720 --> 00:21:06.210 We draw our unit circle. 00:21:08.110 --> 00:21:08.830 Take. 00:21:12.590 --> 00:21:17.666 OP moving around in that direction through an angle 00:21:17.666 --> 00:21:24.184 theta and complete right angle triangle. This here 00:21:24.184 --> 00:21:27.254 is the opposite side. It's opposite, the angle that 00:21:27.254 --> 00:21:28.946 we're talking about. 00:21:30.900 --> 00:21:35.553 tan theta equals, so the opposite side, we have 00:21:35.553 --> 00:21:47.662 replaced by the projection of OP onto OY 00:21:49.060 --> 00:21:52.757 divided by, now for a right angle triangle it would be 00:21:52.757 --> 00:21:58.097 the adjacent side and that adjacent side has been 00:21:58.097 --> 00:22:07.288 replaced by the projection of OP on till OX. Well let's 00:22:07.288 --> 00:22:14.344 remember OY is the Y axis OX is the X axis. 00:22:16.350 --> 00:22:20.326 This again gives us a definition that's going to work as that 00:22:20.326 --> 00:22:23.710 radius vector runs around the circle like that, 00:22:24.340 --> 00:22:27.937 in either direction. So again, it's going to give us a 00:22:27.937 --> 00:22:32.896 definition that will work for any size of angle. 00:22:33.350 --> 00:22:37.404 One of the things we can notice straight away about this is that 00:22:37.404 --> 00:22:45.024 it means that tan theta is of course sine theta divided by cos theta 00:22:45.024 --> 00:22:49.052 and that gives us an identity which we need to learn 00:22:49.052 --> 00:22:54.902 and remember. tan theta is sine theta divided by cos theta. 00:22:56.690 --> 00:23:00.470 But what does the graph of tangent look like? 00:23:04.190 --> 00:23:06.290 It's a little bit trickier to draw. 00:23:11.320 --> 00:23:15.841 Let's see if we can justify what we're going to get. 00:23:28.710 --> 00:23:31.908 Now we're looking at tan theta. 00:23:33.510 --> 00:23:39.414 where theta is the angle between the X axis and the Y Axis. 00:23:40.320 --> 00:23:45.598 And we know that the tangent is defined to be the projection of 00:23:45.598 --> 00:23:47.628 OP on the Y axis, 00:23:48.210 --> 00:23:52.110 divided by the projection of OP on the X axis. 00:23:53.030 --> 00:23:57.494 So when we begin here at theta is 0. 00:23:59.180 --> 00:24:05.631 Then we know that the projection onto the Y axis looking this way 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 00:24:11.660 --> 00:24:17.484 0. So we start there, for the angle zero, we start there. As we 00:24:17.484 --> 00:24:21.682 move around. When we come to 45 degrees then the projections are 00:24:21.682 --> 00:24:27.228 equal. So let's just mark 45 and if the two projections are equal 00:24:27.228 --> 00:24:29.964 then that must be 1. 00:24:30.840 --> 00:24:33.366 What happens is we come up 00:24:33.366 --> 00:24:40.230 towards 90. Well, the projection on to the Y axis is getting 00:24:40.230 --> 00:24:42.278 bigger and bigger approaching 00:24:42.278 --> 00:24:48.578 one. But the projection onto the X axis is getting smaller and 00:24:48.578 --> 00:24:54.074 smaller and smaller, and it's that that we are dividing by, so 00:24:54.074 --> 00:24:57.280 we're dividing something approaching one by something 00:24:57.280 --> 00:25:01.860 that is getting smaller and smaller and smaller. So our 00:25:01.860 --> 00:25:06.440 answer is getting bigger and bigger and bigger. It's becoming 00:25:06.440 --> 00:25:11.936 infinite and we have a way of showing that on a graph. 00:25:14.490 --> 00:25:19.190 I've deliberately put in a dotted line. Now a graph 00:25:19.190 --> 00:25:23.310 approaches that dotted line, but does not cross it, and that's an 00:25:23.310 --> 00:25:28.337 asymptote. That's what we call an asymptote. What about the 00:25:28.337 --> 00:25:34.679 next bit of the graph up to 180 degrees? Well, as we tip over 00:25:34.679 --> 00:25:41.280 into this quadrant, then the projection on to the X axis 00:25:41.280 --> 00:25:45.040 becomes negative, but he's still very, very small. 00:25:46.320 --> 00:25:51.060 The projection onto the Y axis is still positive, still near 1, 00:25:51.060 --> 00:25:55.010 so we're dividing something that's positive and near 1 by 00:25:55.010 --> 00:25:58.960 something that's negative but very very small. So the answer 00:25:58.960 --> 00:26:03.700 must be very, very big, but negative, and so there's a bit 00:26:03.700 --> 00:26:09.920 of graph. Down there on the other side of the asymptote, and 00:26:09.920 --> 00:26:15.644 now we run this round to 180 degrees and what happens then? 00:26:15.644 --> 00:26:21.368 Well, when we've got round to 180, the projection onto the X 00:26:21.368 --> 00:26:27.175 axis is then of length one, the same as OP, but the projection 00:26:27.175 --> 00:26:35.201 onto the Y axis is 0, so we've got 0 / 1 again, which gives us 00:26:35.201 --> 00:26:41.003 0 at 180 degrees. So this comes up to there like that. 00:26:42.030 --> 00:26:46.551 Think about what's going to happen now as it comes around 00:26:46.551 --> 00:26:48.606 here. Let's draw one in. 00:26:49.840 --> 00:26:54.868 And we can see that the projections on to both axes are 00:26:54.868 --> 00:26:58.120 both negative. And a negative divided by a 00:26:58.120 --> 00:26:59.360 negative gives a positive. 00:27:02.460 --> 00:27:06.223 You can also see that when we get here again, we've got 00:27:06.223 --> 00:27:09.953 exactly the same problems as we had when we got here. We're 00:27:09.953 --> 00:27:13.812 dividing by something very, very small into something that's 00:27:13.812 --> 00:27:18.284 around about one, so again I'll answer is going to be very, very 00:27:18.284 --> 00:27:22.412 big, and so again, we're going to get a climb like that. 00:27:23.620 --> 00:27:27.022 As we move through this, it's the same as if we move 00:27:27.022 --> 00:27:31.038 through there, so again we will start back down here in 00:27:31.038 --> 00:27:35.005 the graph, will climb and then off up again there. 00:27:36.570 --> 00:27:38.470 What about back this way? 00:27:39.020 --> 00:27:41.168 When we've got negative 00:27:44.420 --> 00:27:49.249 angles what's going to happen then we must have the same 00:27:49.249 --> 00:27:54.078 things occuring in terms of our asymptotes. And so we're 00:27:54.078 --> 00:27:59.346 going to have the same things occur in with the graph there. 00:28:00.930 --> 00:28:05.390 There and so on. So again, notice we get a periodic 00:28:05.390 --> 00:28:11.036 function. That bit of graph is repeated again there. 00:28:12.070 --> 00:28:17.108 Every 360 degrees we get a periodic function. We get a 00:28:17.108 --> 00:28:19.856 repeat of this section of the 00:28:19.856 --> 00:28:23.860 graph. So that's our function tangent. 00:28:25.840 --> 00:28:28.327 And we can think of the function as being defined if 00:28:28.327 --> 00:28:30.583 you like by that particular graph.