0:00:01.450,0:00:05.718 In this video, we're going[br]to look at a different way 0:00:05.718,0:00:06.882 of measuring angles. 0:00:08.230,0:00:12.840 Well aware.[br]In a circle. 0:00:15.300,0:00:21.500 There are 360 degrees all[br]the way around. 0:00:22.080,0:00:27.348 360 degrees in a circle, 360[br]degrees all the way around. What 0:00:27.348,0:00:32.616 we need is a new way of[br]measuring angles and this is 0:00:32.616,0:00:36.567 needed because when we look at[br]trig functions, particularly 0:00:36.567,0:00:40.518 their differentiation, we will[br]need a different measure of 0:00:40.518,0:00:45.786 angle from degrees. So let's[br]have a look at what that measure 0:00:45.786,0:00:51.802 is. So[br]we take 0:00:51.802,0:00:58.480 a circle.[br]Radius R. 0:01:00.810,0:01:07.180 And I want to mark off on the[br]circumference a length that is 0:01:07.180,0:01:11.590 actually equal tool are so that[br]distance around the 0:01:11.590,0:01:16.980 circumference is equal to R, the[br]radius. Now we join up. 0:01:17.700,0:01:24.860 There. That's our and this[br]angle here theater we describe 0:01:24.860,0:01:27.540 as being one Radian. 0:01:28.050,0:01:31.378 So Theta equals 1. 0:01:32.220,0:01:33.030 Radian 0:01:34.990,0:01:39.118 And if we want a definition of[br]that, then the definition is. 0:01:40.010,0:01:45.717 The angle of one Radian is the[br]angle subtended at the center of 0:01:45.717,0:01:51.863 the circle by an arc that is[br]equal in length to the radius of 0:01:51.863,0:01:56.692 the circle. So that gives us our[br]definition of a radium. 0:01:57.230,0:02:03.078 Let me[br]just draw. 0:02:05.020,0:02:09.790 3.[br]Circles. 0:02:11.360,0:02:18.900 Let's put that definition into[br]effect again, where we've got. 0:02:19.780,0:02:24.976 A distance equal to R, the[br]radius of the circle around the 0:02:24.976,0:02:29.866 circumference. And there's our[br]theater equals 1. 0:02:30.650,0:02:33.070 Radian now let me. 0:02:34.660,0:02:41.095 Go round the circumference a[br]distance of two are. 0:02:43.160,0:02:50.624 As the radius again, let's take[br]half way around there so that 0:02:50.624,0:02:58.088 this is Theta. In here is[br]one Radian, an theater in here 0:02:58.088,0:03:00.576 is also one Radian. 0:03:01.220,0:03:08.384 So what we can see there[br]is that this distance to our 0:03:08.384,0:03:15.548 is in fact two radians times[br]by the radius, and so that 0:03:15.548,0:03:18.533 gives us a general formula. 0:03:19.690,0:03:26.180 That if we have any angle[br]theater subtended at the center 0:03:26.180,0:03:33.260 of the circle then the arc[br]length S around there is given 0:03:33.260,0:03:36.210 by S equals R Theater. 0:03:36.930,0:03:40.442 And that's the arc 0:03:40.442,0:03:46.670 length. And it's[br]true if theater 0:03:46.670,0:03:49.850 is in radians. 0:03:50.520,0:03:55.668 So that gives us a very good way[br]of calculating the arc length 0:03:55.668,0:04:00.816 when we know the angle subtended[br]at the center, and we know its 0:04:00.816,0:04:02.004 value in radians. 0:04:02.730,0:04:10.285 So.[br]Can we get an equivalence 0:04:10.285,0:04:12.610 between our radiant measure and 0:04:12.610,0:04:16.347 our degrees? Well, let's have a[br]look at a circle again. 0:04:17.290,0:04:21.440 The circumference[br]of this circle. 0:04:22.570,0:04:29.488 We know by[br]the formula to 0:04:29.488,0:04:36.472 Π R. Now[br]let's compare that with our 0:04:36.472,0:04:38.443 formula arc length. 0:04:39.030,0:04:45.492 Our formula for arclength we[br]know, is our theater. 0:04:46.730,0:04:50.170 Well, he is, AH. 0:04:50.980,0:04:53.340 And here is our. 0:04:53.970,0:04:58.710 And what that means is that for[br]this circle, the distance all 0:04:58.710,0:05:02.660 the way round is 2π, R, and[br]therefore the angle. 0:05:03.580,0:05:10.345 In radians all the way round[br]is this number here 2π. 0:05:10.940,0:05:18.396 So what we've got[br]is that 2π radians. 0:05:19.350,0:05:25.307 Is exactly the same[br]as 360 degrees. 0:05:26.670,0:05:32.742 Now this equivalence enables us[br]to write down a whole series of 0:05:32.742,0:05:40.568 Equalities. Between angles, so[br]we've got 2π radians. 0:05:40.570,0:05:48.250 Is the same as 360[br]degrees, so Pi radians is 0:05:48.250,0:05:55.930 the same as 180 degrees.[br]Just dividing both sides by 0:05:55.930,0:06:02.840 two. If we divide by[br]two, again π by two radians. 0:06:03.400,0:06:10.438 Notes, so I said that Π by 2 not[br]π / 2, but pie by two. It's a 0:06:10.438,0:06:15.130 shorthand, so Pi by two radians[br]is 90 degrees, and that's an 0:06:15.130,0:06:17.085 important one, because that's a 0:06:17.085,0:06:24.130 right angle. Pie by 4[br]/ 2 again high by 0:06:24.130,0:06:26.830 4 radians is 45. 0:06:27.350,0:06:34.075 Degrees. So these are the[br]important angle, so to speak. 0:06:34.075,0:06:39.070 Some other quite important[br]angles in trigonometry are 60 0:06:39.070,0:06:46.285 degrees, well, 60 degrees is 1/3[br]of 180 degrees. So Pi by three 0:06:46.285,0:06:51.438 radians. Is the same as 60[br]degrees, another angle that's 0:06:51.438,0:06:53.598 important is the angle 30 0:06:53.598,0:07:00.490 degrees. And 30 degrees is[br]1/6 of 180 degrees, so π 0:07:00.490,0:07:02.242 by 6 radians. 0:07:03.320,0:07:06.530 Is the same as 30 degrees. 0:07:07.130,0:07:12.113 OK, we've got all of these[br]equivalences, but the one thing 0:07:12.113,0:07:18.455 we haven't got is how big is a[br]Radian in terms of degrees. One 0:07:18.455,0:07:20.720 Radian equals how many degrees? 0:07:21.430,0:07:27.649 Or will start with this[br]relationship here that Pi 0:07:27.649,0:07:30.413 radians is 180 degrees. 0:07:31.090,0:07:38.196 So.[br]Π radians 0:07:38.196,0:07:41.272 is 180 0:07:41.272,0:07:44.506 degrees. So 0:07:44.506,0:07:51.688 one Radian. Must[br]be 180 degrees divided by 0:07:51.688,0:07:58.768 pie and if we use[br]a Calculator to do that, 0:07:58.768,0:08:04.432 we find that the answer[br]is 57.296 degrees. 0:08:05.120,0:08:10.892 2 three decimal places now it's[br]actually rare that we need to do 0:08:10.892,0:08:16.220 this particular calculation, but[br]what it does do is it gives us 0:08:16.220,0:08:19.328 some idea of exactly how big a 0:08:19.328,0:08:22.080 radiant is. Now. 0:08:23.060,0:08:27.590 So far I've written down the[br]word Radian in fall. 0:08:28.300,0:08:33.712 But I haven't written down the[br]word degrees in full. Have used 0:08:33.712,0:08:39.124 this symbol. This little zero up[br]here to signify degrees. So what 0:08:39.124,0:08:45.438 do we do about radians? How can[br]we signify Aready? And so if we 0:08:45.438,0:08:50.850 write down 1.5 radians, how we[br]able to tell that this is 0:08:50.850,0:08:57.164 radians? One way is to put a[br]little see up there at the top, 0:08:57.164,0:08:58.517 usually in italic. 0:08:58.550,0:09:04.820 Print if it's published in a[br]book, so 1.5 radians. Another 0:09:04.820,0:09:11.660 way is to write 1.5 ramps where[br]Rams is obviously short for 0:09:11.660,0:09:18.030 radians. What we've done does[br]not affect trig ratios. So for 0:09:18.030,0:09:24.387 instance, the cosine of π by[br]three is exactly the same as the 0:09:24.387,0:09:30.255 cosine of 60 degrees because the[br]two angles are the same, the 0:09:30.255,0:09:37.101 tangent of Π by 4 is exactly the[br]same as the tangent of 45 0:09:37.101,0:09:44.798 degrees. And the sign of three[br]Pi by two is exactly the same 0:09:44.798,0:09:48.122 as the sign of 270 degrees. 0:09:48.890,0:09:53.606 One thing to notice here is that[br]because I've got pies in. 0:09:54.510,0:09:57.336 I haven't written down a symbol. 0:09:58.060,0:10:04.274 Because one, we've got a pie in[br]the angle it's taken as read 0:10:04.274,0:10:06.664 that these are measured in 0:10:06.664,0:10:13.482 radians, so. We've got our[br]formula S equals R Theater, 0:10:13.482,0:10:20.222 where S is the arc[br]length and theater is in 0:10:20.222,0:10:26.931 radians. What's the formula for[br]S if Vita is in degrees? Let's 0:10:26.931,0:10:30.508 just have a look at our circle. 0:10:31.440,0:10:32.100 Yeah. 0:10:33.910,0:10:40.329 Radius R.[br]Our arc length S is around the 0:10:40.329,0:10:41.490 circle like so. 0:10:43.460,0:10:48.790 Radius R and this angle[br]in here let's market as 0:10:48.790,0:10:50.389 being theater degrees. 0:10:51.910,0:10:58.890 So what's our formula? Well,[br]S must be a certain 0:10:58.890,0:11:01.682 proportion of the whole 0:11:01.682,0:11:09.480 circumference. S over 2π, R[br]and the angle theater will be 0:11:09.480,0:11:16.443 exactly the same proportion of[br]360, and so we can calculate 0:11:16.443,0:11:23.406 the arc length S as 2π[br]R that's multiplying both sides 0:11:23.406,0:11:27.204 by 2π. R times Theta over 0:11:27.204,0:11:34.181 360. So there we've got our[br]arc length formula for Theta in 0:11:34.181,0:11:39.291 radians. And here's our arc[br]length formula for theater in 0:11:39.291,0:11:44.401 degrees. Notice how much[br]simplier this one is? What about 0:11:44.401,0:11:49.000 another formula? What about the[br]area of this sector? 0:11:49.010,0:11:52.736 Well, let's have a look at 0:11:52.736,0:11:57.030 that. Is our circle[br]again? 0:11:58.140,0:12:05.420 Radius R.[br]Angle 0:12:05.420,0:12:11.805 theater[br]It's called the center of 0:12:11.805,0:12:17.035 the circle. Oh, and let's[br]label the two points A&B. 0:12:17.035,0:12:21.742 What's the area of this[br]sector of the circle? 0:12:21.742,0:12:26.972 This slice of the pie, so[br]to speak. What's its 0:12:26.972,0:12:29.064 area? Well, the area? 0:12:31.560,0:12:33.660 Of the sector. 0:12:35.370,0:12:40.302 As a proportion, or a fraction[br]of the area. 0:12:40.830,0:12:42.558 Of the circle. 0:12:43.280,0:12:50.550 Must be the same fraction,[br]the same proportion as the 0:12:50.550,0:12:53.458 angle is of 2π. 0:12:54.050,0:12:59.748 All the way around. So the[br]area of the sector considered 0:12:59.748,0:13:05.964 as a fraction of the whole[br]circle must be in the same 0:13:05.964,0:13:08.036 ratio as the angle. 0:13:09.830,0:13:15.225 Considered as a fraction of the[br]whole angle all the way round in 0:13:15.225,0:13:20.404 radians. Well, we can work[br]this out area. 0:13:21.030,0:13:23.310 Of the sector. 0:13:23.960,0:13:28.744 And we have a formula for the[br]area of the circle. We know 0:13:28.744,0:13:30.216 that's π R squared. 0:13:30.770,0:13:36.830 Equals feta over 2π, And if we[br]multiply it by this expression 0:13:36.830,0:13:43.395 here, the Pi R-squared the area[br]of the circle. Then we have the 0:13:43.395,0:13:45.415 area of the sector. 0:13:46.500,0:13:53.100 Is equal to π[br]R squared times theater 0:13:53.100,0:13:56.400 over 2π and the 0:13:56.400,0:14:00.140 pi's cancel. That leaves me with 0:14:00.140,0:14:02.913 a half. All 0:14:02.913,0:14:09.208 squared Theta. So for[br]theater in radians this is our 0:14:09.208,0:14:13.668 formula. The area of a sector is[br]1/2 R-squared Theta. 0:14:14.360,0:14:18.270 OK, we've got some formula here.[br]We've got some relationships. 0:14:18.270,0:14:23.353 Let's have a look at how we can[br]use them to do some 0:14:23.353,0:14:26.520 calculations. So in our first 0:14:26.520,0:14:29.550 example. Let's take a circle. 0:14:31.840,0:14:36.600 Of radius. 10[br]units 0:14:37.730,0:14:40.148 And let's take an arc length. 0:14:40.830,0:14:44.118 Of 25 units. 0:14:45.390,0:14:51.162 Now if I make this a little more[br]representative, 25 more likely 0:14:51.162,0:14:57.415 to be around there 10 and 10,[br]and the question is what's this 0:14:57.415,0:15:03.187 angle here in terms of radians[br]and also in terms of degrees? 0:15:03.790,0:15:10.554 So. What's our formula[br]for Arclength? That's arc length 0:15:10.554,0:15:17.034 equals are times by theater. If[br]theater is in radians. 0:15:17.670,0:15:24.670 So as we know[br]to be 25. 0:15:24.950,0:15:32.138 We know to be 10 times[br]by theater, so theater is 25 0:15:32.138,0:15:39.326 over 10 which is 2.5 and[br]I can put a little see 0:15:39.326,0:15:45.915 up there to stand for[br]radians. Or I could write it 0:15:45.915,0:15:47.712 as 2.5 rats. 0:15:48.960,0:15:55.860 So that's our angle in terms[br]of radians. What will our angle 0:15:55.860,0:16:02.760 be in terms of degrees? Well,[br]one way might be to convert 0:16:02.760,0:16:08.510 this answer here. These two[br]point 5 radians into degrees. 0:16:08.510,0:16:11.960 Let's remember that we had Pi 0:16:11.960,0:16:14.830 radians. Was 0:16:14.830,0:16:21.598 180 degrees.[br]So one Radian 0:16:21.598,0:16:28.414 was 180 /[br]π degrees, so 0:16:28.414,0:16:30.686 2.5 radians. 0:16:31.740,0:16:39.468 Must be 180[br]over π times 0:16:39.468,0:16:46.958 by 2.5. And[br]we can use a Calculator to work 0:16:46.958,0:16:50.054 this out. And the answer is 0:16:50.054,0:16:55.876 143.2 degrees. Now let's have a[br]look at another example. 0:16:56.950,0:17:00.987 Start from the same point. That[br]is, we've got a circle. 0:17:02.940,0:17:08.297 We've got circle of radius 10[br]centimeters, and this time let's 0:17:08.297,0:17:13.654 say we're going to take an arc[br]length of 15 centimeters. 0:17:14.290,0:17:21.610 The question is, what is the[br]angle in radians? Here at the 0:17:21.610,0:17:28.320 center, but also, what's the[br]area of this sector oab? So 0:17:28.320,0:17:35.640 let's begin with our formula for[br]arc length S equals R Theater. 0:17:36.350,0:17:43.550 Our arc length is 15 hour,[br]radius is 10 and the angle 0:17:43.550,0:17:50.750 is theater. So we divide both[br]sides by 10 to give us 0:17:50.750,0:17:56.150 theater is 15 over 10, which[br]is 1.5 radians. 0:17:56.840,0:18:01.097 Now area of the sector. We have[br]a formula for that. 0:18:02.250,0:18:08.874 Area of sector. We[br]know that that is 0:18:08.874,0:18:11.358 1/2 R-squared Theta. 0:18:12.230,0:18:19.646 So we can take our values.[br]We know that R is 10, 0:18:19.646,0:18:26.444 so that's 10 squared for the[br]R-squared, times by angle we've 0:18:26.444,0:18:28.298 just calculated 1.5. 0:18:28.900,0:18:36.052 So that's 1/2 times by 100[br]times by 1.5 now 100 times 0:18:36.052,0:18:43.204 by 1.5 is 150 and we[br]want half of it, so that 0:18:43.204,0:18:49.760 is 75 centimeters squared. So[br]that's the area of this sector. 0:18:51.210,0:18:56.875 Further questions we could ask[br]is what's the area of the 0:18:56.875,0:19:03.055 segment the minor segment? If[br]you take a line across a circle, 0:19:03.055,0:19:09.235 it divides a circle into two[br]parts, a big bit, the major 0:19:09.235,0:19:14.900 segment and a smaller piece.[br]They minus segment, and we can 0:19:14.900,0:19:17.990 ask what's the area of that 0:19:17.990,0:19:23.233 minor segment? Let's just draw[br]another diagram and look at that 0:19:23.233,0:19:28.394 again, 'cause it was quite a lot[br]of vocabulary in that quite a 0:19:28.394,0:19:33.555 lot of words. So we're going to[br]take a line across the circle. 0:19:33.555,0:19:38.716 We're keeping this as 50. We've[br]got the center of the circle and 0:19:38.716,0:19:44.274 we know what the angle is, so[br]there's our as are we know that 0:19:44.274,0:19:45.465 to be 10. 0:19:46.170,0:19:50.427 And we've calculated what this[br]angle is. So here's the line 0:19:50.427,0:19:54.684 that we drew across the circle,[br]and it divides circling two 0:19:54.684,0:19:59.328 parts of big part like that[br]which we call the major segment 0:19:59.328,0:20:03.972 and a small part like that which[br]we call the minor segment. 0:20:04.510,0:20:11.734 The question is, what's the area[br]of this minor segment? How can 0:20:11.734,0:20:18.958 we work it out? Well, we[br]know the area of the sector 0:20:18.958,0:20:21.366 of the circle area. 0:20:21.530,0:20:28.530 All sector 'cause we calculated[br]that and that was 75 0:20:28.530,0:20:35.911 centimeters squared. What we[br]need to be able to do is 0:20:35.911,0:20:38.706 workout the area of this 0:20:38.706,0:20:46.227 triangle. Area of the triangle[br]AOB. The area of the 0:20:46.227,0:20:49.319 triangle will be 1/2. 0:20:49.860,0:20:53.848 All squared sine theater. 0:20:55.420,0:21:02.492 So that's 1/2 times by 10[br]squared times by the sign of 1.5 0:21:02.492,0:21:09.020 radians. And again we can work[br]this out on a Calculator. We 0:21:09.020,0:21:16.636 have to be very careful that are[br]Calculator a set up to work in 0:21:16.636,0:21:23.708 radians, not in degrees, to work[br]in radians. And so here we have 0:21:23.708,0:21:25.884 1/2 times by 100. 0:21:25.930,0:21:33.210 Times by and the sign of this[br]angle is nought. .997 and then 0:21:33.210,0:21:40.490 lots of decimal places. And if[br]we work that out on a Calculator 0:21:40.490,0:21:47.770 you get 49.8747 and so our final[br]answer, which we want is the 0:21:47.770,0:21:50.010 shaded area here area. 0:21:50.020,0:21:53.290 All the minor. 0:21:54.080,0:21:59.868 Segment. And that's[br]got to be the difference between 0:21:59.868,0:22:01.648 the area of the sector. 0:22:02.570,0:22:06.140 And the area of the triangle so 0:22:06.140,0:22:12.944 that 75. Take away[br]49.8747 and to a 0:22:12.944,0:22:19.496 suitable degree of accuracy[br]2 decimal places. That's 0:22:19.496,0:22:21.953 25.13 centimeters squared. 0:22:23.540,0:22:29.018 Let's have a look at one final[br]example. We have converted 0:22:29.018,0:22:34.994 radians into degrees, but one of[br]the things that we haven't done 0:22:34.994,0:22:39.974 is to convert an angle that's in[br]degrees into radians. 0:22:40.110,0:22:47.630 So let's take an angle[br]120 degrees. What is this 0:22:47.630,0:22:55.150 angle in radians? Let's start[br]off with our relationship that 0:22:55.150,0:22:58.910 Pi radians is 180 degrees. 0:22:59.430,0:23:06.714 Now we were able to say[br]that one Radian was 180 over 0:23:06.714,0:23:13.998 π, dividing both sides by pie.[br]So if we want 1 degree, 0:23:13.998,0:23:21.282 then we can divide both sides[br]by 180. So π over 180 0:23:21.282,0:23:28.566 radians is equal to 1 degree.[br]We want 120 degrees, so π 0:23:28.566,0:23:35.860 over 180. Times 120 is equal[br]to 120 degrees and again we 0:23:35.860,0:23:43.108 need a Calculator to work this[br]out. And when we work it 0:23:43.108,0:23:50.356 out we get 2.09 radians and[br]that's to two decimal places. Is 0:23:50.356,0:23:53.376 the same as 120 degrees. 0:23:54.860,0:24:00.150 Important things to take away[br]from this video are this 0:24:00.150,0:24:06.253 relationship. Π radians is 180[br]degrees and the two formally 0:24:06.253,0:24:07.834 that we had. 0:24:08.360,0:24:15.030 One for the arc length,[br]which was S equals R 0:24:15.030,0:24:21.700 theater and one for the[br]area of the sector which 0:24:21.700,0:24:28.370 was area of the sector[br]is equal to 1/2 R-squared 0:24:28.370,0:24:35.040 Theater and both of these[br]to apply when theater is 0:24:35.040,0:24:36.374 in radiance.