04 - What is the Unit Circle? Angle Measure in Degrees, Reference Angles & More. - By Math and Science
Transcript
00:00 | Hello . Welcome back . The title of this lesson | |
00:02 | is called counting angles in degrees around the unit circle | |
00:07 | . Now truthfully this is the kind of lesson that | |
00:10 | I really wish I had when I first started learning | |
00:12 | this stuff because what's going to happen is right after | |
00:15 | you learn what an angle is very very quickly you're | |
00:18 | going to be learning something called the unit circle . | |
00:20 | And the unit circle is a big circle in your | |
00:22 | textbook or wherever you're learning and it will have all | |
00:25 | the angles written down and all of the lots of | |
00:28 | other information and it's kind of thrown at you and | |
00:30 | it's just very overwhelming at first . So what we | |
00:33 | need to learn how to do is learn what the | |
00:35 | unit circle really means and learn how to use it | |
00:38 | . And it is a process that takes a little | |
00:40 | while . It's not something you can look at and | |
00:42 | just oh I got it unit circle . If you | |
00:44 | think you got it right away , you probably don't | |
00:46 | . Okay . So what I want to do is | |
00:48 | teach you what the unit circle is . Kind of | |
00:50 | an introduction here And also teach you how to count | |
00:53 | , count in degrees to go around the unit circle | |
00:56 | . Because we're going to start learning how to do | |
00:58 | all of these things in degrees because we all kind | |
01:00 | of have a good idea of what degrees are 360° | |
01:02 | in a circle , right ? But very soon we're | |
01:05 | going to switch over to radiant measures . Radiant is | |
01:08 | a different way to measure angles . But in terms | |
01:11 | of pie basically and I don't want to get into | |
01:13 | the details of radiance yet . But it's much more | |
01:16 | difficult to understand because we don't have a good grasp | |
01:19 | of radiant measure in everyday life . So what we're | |
01:21 | gonna do is learn how to walk around this unit | |
01:24 | circle in degrees first and so you understand exactly what | |
01:27 | it means . And then we'll do a sine and | |
01:30 | cosine and tangent all of that . And then we'll | |
01:32 | come back to the unit circle and we'll talk about | |
01:34 | radiance . This idea of counting . What I call | |
01:37 | counting in degrees around the unit circle is not something | |
01:40 | you will see in most textbooks , but I personally | |
01:43 | find it to be one of the most important things | |
01:45 | you can learn because it will impact every single problem | |
01:48 | that we do from here on out . All right | |
01:50 | , so we're gonna be counting . We're gonna be | |
01:51 | learning how to count here . And also I would | |
01:53 | say one last thing . I want you to stick | |
01:55 | with me to the very end of this lesson because | |
01:57 | it's easy to think you got it . But I | |
01:59 | really want you to practice with me because it will | |
02:01 | become so critical that you understand what we're doing here | |
02:03 | . So , uh , in general , this is | |
02:05 | what we have , what we call a unit circle | |
02:07 | over here , Right ? It's got it's a circle | |
02:10 | obviously . And you have a criss cross X and | |
02:13 | y axis here , the black lines . And then | |
02:15 | we have all of these angled lines , Right ? | |
02:17 | So we're going to be getting very , very up | |
02:19 | close and personal with this unit circle idea . So | |
02:21 | we have to kind of start with the basics . | |
02:23 | Right ? So what we want to do is talk | |
02:26 | about , what does it mean to have a unit | |
02:27 | circle ? What is a unit circle ? Anyway , | |
02:30 | all it means it's a circle with a radius of | |
02:33 | one . That's all it means . And you might | |
02:35 | say , well , one what ? One centimeter ? | |
02:37 | one m one yard one light year . What ? | |
02:39 | Okay . Honestly , it doesn't matter what unit you're | |
02:42 | talking about . You could consider this thing to be | |
02:44 | one , you know , foot if you want to | |
02:46 | one m one centimeter , it doesn't matter for now | |
02:48 | . Just consider it to be a radius of one | |
02:50 | . So the distance from the center to the edge | |
02:53 | . All of these distances here is just a length | |
02:56 | of one . It doesn't matter what units you work | |
02:58 | in . So if it's convenient for you , you | |
03:00 | can think of it in terms of meters or whatever | |
03:02 | . Just think of it as a radius of one | |
03:05 | . All right now , this is a circle . | |
03:07 | So it starts at zero degrees over here . And | |
03:09 | in fact , that's the very first thing we're gonna | |
03:10 | right is that we have over here . This black | |
03:13 | line right here is zero degrees . This is the | |
03:16 | zero degree line right here . Now we said that | |
03:19 | positive angle measure goes around in this direction counterclockwise , | |
03:24 | right ? And we said negative angle measure starts from | |
03:26 | the X axis and it goes around like this . | |
03:29 | So we can label a couple of additional things on | |
03:31 | this unit circle . We can label this black line | |
03:35 | here is what I'm labeling here . This is the | |
03:37 | X . Axis . And then this vertical black line | |
03:40 | right here we can call this uh I'm gonna put | |
03:43 | it way up at the top Y axis because I'll | |
03:45 | probably put some other numbers and markings around the circle | |
03:48 | as we go . So this is just an X | |
03:50 | . Y . Axis . That's it . This other | |
03:52 | stuff is stuck on top of it . So here | |
03:54 | is positive X values , negative X values . Here's | |
03:58 | positive Y values , here's negative Y values . It's | |
04:01 | just an Xy access . All we have done is | |
04:03 | put a circle that has a radius of one . | |
04:06 | That means the distance from here to here along the | |
04:10 | X axis is just 11 What ? Okay . Call | |
04:13 | it one m . If you want to call it | |
04:14 | one ft call it one centimeter . I don't care | |
04:16 | , it doesn't matter . But it's a distance of | |
04:18 | one . This is a distance of one . So | |
04:20 | in the Y direction , one unit . This is | |
04:23 | negative negative one in the X . Direction . And | |
04:26 | down here will be negative one in the Y direction | |
04:28 | . So it's just an X . Y grid . | |
04:30 | Now we obviously have a bunch of angled lines here | |
04:33 | right here and the way this is laid out is | |
04:36 | right here from here to here is 90 degree angle | |
04:40 | . You all know that because it's a right angle | |
04:42 | . So this is a 90 degree angle and then | |
04:44 | from here to here is another 90 degrees . And | |
04:46 | then from here down to here is another 90 degrees | |
04:48 | . And then from here over here's another 90 degrees | |
04:50 | . So if you know that from here to here | |
04:53 | is 90 degrees . Then this line is cutting this | |
04:56 | angle in half . So this angle is 45°. . | |
05:01 | And we're going to label all of this in a | |
05:02 | second . But these diagonal lines like this , this | |
05:04 | is 45 . And then this one is a distance | |
05:07 | here from here to here of 45 . The distance | |
05:11 | from here to here is 45 . And the distance | |
05:13 | from here here's 45 . And then so you have | |
05:16 | The entire thing broken up into segments of 45°. . | |
05:20 | And you might say , okay , well if this | |
05:21 | is 45 degrees , what is this angle one right | |
05:23 | here ? Well this is a 30 degree angle . | |
05:26 | And then that would mean this is a 30 degree | |
05:28 | angle and then this is another 30 degrees and another | |
05:31 | 30 degrees . And then the last thing is this | |
05:34 | angle here , since you know this one is 30 | |
05:36 | and this one is a 3rd and 30 more degrees | |
05:38 | . This is a 60 degree angle . And then | |
05:41 | from here to here is another 60 degree angle . | |
05:43 | So I haven't written anything down yet , but I | |
05:45 | just want to talk you through it . The unit | |
05:47 | circle in general is gonna have all these diagonal lines | |
05:49 | and it gets overwhelming . All you need to know | |
05:52 | is that these black lines are increments of 90 degrees | |
05:56 | And then you have 45° angles that are cutting those | |
06:00 | in half . And then you also have 30° angles | |
06:02 | and 60° angles which are also marked on the unit | |
06:05 | circle all the way around . Now the 30° angles | |
06:08 | and the 60 degree angles and the 45 degree angles | |
06:12 | and the 90 degree angles . Those are the very | |
06:14 | most important angles we learn in trigonometry . I mean | |
06:17 | we can put any angle into a calculator and get | |
06:20 | a number and we'll be doing that . But when | |
06:23 | we're doing a unit circle and doing things manually , | |
06:25 | it's going to be 30 degrees , 60 degrees , | |
06:27 | 45 degrees , 90° or some other multiple . That's | |
06:31 | why we go all the way around the unit circle | |
06:33 | like that . So now we need to start with | |
06:35 | the easiest thing possible . We want to count in | |
06:38 | degrees . We want to count in 90 degree increments | |
06:41 | , 90 degree increments . What do I mean by | |
06:44 | that ? I want you to ignore the gray lines | |
06:47 | , ignore these gray lines , ignore these gray lines | |
06:50 | and ignore these grain lines . Focus on the black | |
06:53 | lines , Right ? If this is 0° and if | |
06:57 | this one is a right angle to that , then | |
07:00 | what must this angle measure be from here ? Going | |
07:03 | up like this ? That means that this angle measure | |
07:06 | must be a 90 degrees . So as you walk | |
07:09 | around the unit circle , the angle measure from here | |
07:11 | to here is a 90 degree angle measure . All | |
07:14 | right now , if this is 90 degree angle then | |
07:17 | what must this angle b over here ? Because this | |
07:19 | is another 90 degrees over here . So if this | |
07:21 | is zero and this is 90 and this is What's | |
07:25 | the next thing , 90 plus 90 . This thing | |
07:26 | has to be 180 degrees . And that makes sense | |
07:30 | because if zero is over here , then all the | |
07:32 | way on the other side from geometry , you know | |
07:34 | , this has to be 180 degrees away . Now | |
07:37 | again , if we have another 90 degree angle down | |
07:39 | like this , what must this angle down at the | |
07:41 | bottom as measured from the X axis ? All the | |
07:44 | way around . What must this angle be ? Well | |
07:46 | , it would be this plus another 90 degrees . | |
07:48 | So what this angle measure is is 270 degrees . | |
07:53 | These are numbers that you're going to have to remember | |
07:56 | . Okay . And if you take from 270 degrees | |
07:59 | and you walk another 90 degrees because this is another | |
08:01 | 90 degree angle 270 plus 90 . What do you | |
08:04 | get ? You get to 360 degrees . So notice | |
08:09 | the way I'm writing this , the zero degree angle | |
08:11 | is written right above . That's telling you that if | |
08:13 | I start at zero then that's where I start . | |
08:16 | And as I walk around I do one entire circle | |
08:18 | of 360°. . I get back to where I start | |
08:21 | from . And the angle that I and end on | |
08:23 | . Is it positive 360 degrees . That's exactly what | |
08:26 | you would expect when you go one circumference , one | |
08:29 | circle around you always go 3 60 you get back | |
08:32 | to where you started from . All right . So | |
08:34 | so far we just kind of wrote the angle measures | |
08:37 | down here on the on the unit circle . But | |
08:39 | we didn't do any counting yet . And that's really | |
08:41 | what I want to talk to you about . I | |
08:43 | want to talk and you might say this is trivial | |
08:46 | . Okay , that's cool . But when we get | |
08:47 | to more complicated angles and radiance , this is going | |
08:50 | to be very , very helpful . So what we | |
08:52 | want to do is count bye 90°. . And what | |
08:57 | I mean by that is I want to count in | |
08:59 | increments of 90°. . Count in increments of 90°. . | |
09:04 | All right . And what I mean by that is | |
09:05 | this angle measure from here to here . This is | |
09:08 | a 90° angle . So think about the 90° angle | |
09:11 | is being a quantity of something . It's a slice | |
09:14 | of the circle . I want to count around that | |
09:17 | circle . In chunks of 90°, , in increments of | |
09:20 | 90° , In units of 90°. . I want you | |
09:23 | to think of this 90° wedge of a circle as | |
09:25 | a thing . It's an object , it's a solid | |
09:28 | pie shaped wedge . And we're gonna now count by | |
09:31 | increments of 90 degrees . So this is a 90 | |
09:34 | degree increments and we're gonna count around the unit circle | |
09:36 | . So let's see what happens if we start at | |
09:41 | zero , then the first angle we have a zero | |
09:43 | but then we go and increment by one times 90 | |
09:46 | degrees . Okay , one times 90 degrees . And | |
09:50 | we're counting by 90 degrees increments . So if we | |
09:52 | increment again then we'll increment another time . So it'll | |
09:56 | be two times 90 degrees to get to the next | |
09:58 | location . So again , this is zero , this | |
10:01 | is one chunk of 90 degrees , This is two | |
10:03 | chunks of 90 degrees , This is three chunks of | |
10:05 | 90 degrees and this is four chunks of 90 degrees | |
10:08 | . The numbers are all here . But I don't | |
10:11 | want you to think of those numbers now , I | |
10:12 | want you to think about this being an increment of | |
10:14 | 90 , another increment of 90 , another increment of | |
10:16 | 90 another increments 90 . I can keep going and | |
10:19 | saying , here's one increments to increments , three increments | |
10:23 | four increments of 90 five increments of 96 increments of | |
10:26 | 97 and eight increments of 90 and I can keep | |
10:29 | going and going 10 , 11 , 12 and 13 | |
10:32 | increments of 90 degrees . So what will I get | |
10:34 | ? One increment of 92 increments of 93 increments of | |
10:38 | 90 . I'm gonna drop the degree symbol , four | |
10:41 | increments of 90 and so on . This is four | |
10:45 | increments of 90 degrees . And what would come next | |
10:48 | ? I mean obviously we can continue this game . | |
10:50 | Let me go down to the next line , we | |
10:51 | would have five increments of 90 and then six increments | |
10:55 | of 90 . And then I'm gonna go all the | |
10:57 | way around seven increments of 90 and then eight increments | |
11:01 | of 90 . I'm counting by chunks of only 90 | |
11:04 | degrees , you might say , why is he doing | |
11:06 | this ? Okay . Of course we know this . | |
11:07 | Yes , it's because when we get to radiance it's | |
11:09 | gonna be so helpful to count in turn chunks of | |
11:12 | radiant measure . Right ? So then what happens if | |
11:14 | we if we count by nineties , what do we | |
11:16 | get then ? We know that this corresponds to zero | |
11:19 | . This corresponds to 90 . This corresponds to 180 | |
11:23 | because two times 90 is 1 80 this corresponds to | |
11:26 | 270 . This corresponds four times nine is 36 So | |
11:29 | 360 I'm gonna keep on going five times nine . | |
11:33 | This is going to correspond to 450 . This six | |
11:37 | times 90 is going to correspond to 540° and then | |
11:42 | this is going to correspond to 630° And then this | |
11:47 | is going to correspond to 720°. . All right . | |
11:50 | So I put my little degree symbols here to try | |
11:53 | to keep it organized . Okay , so why am | |
11:55 | I doing all this stuff ? It's because when you | |
11:57 | first hit the unit circle , a lot of students | |
11:59 | try to memorize things . I try to memorize how | |
12:02 | many degrees it is . If I go here and | |
12:03 | there and all that , you don't ever have to | |
12:05 | do that . Okay , what you have to do | |
12:07 | is recognize that you can count by certain increments of | |
12:09 | 90 . If you want to know what this angle | |
12:11 | measure and is down here just say one chunk of | |
12:15 | 92 chunks of 93 chunks of 90 . Okay . | |
12:17 | Three times 90 . That's 270 four chunks of 90 | |
12:20 | is 3 60 . Notice that that's what we said | |
12:23 | here . Right ? But then we can go to | |
12:24 | 455 40 . How do we know that these are | |
12:27 | the angle measures ? Well , that's because if this | |
12:29 | was four times 90 this is five nineties and then | |
12:32 | six nineties and then seven nineties and then eight nineties | |
12:35 | eight times 9 to 72 . So this is this | |
12:38 | 360 degrees . If you go around again 3 60 | |
12:41 | plus 3 67 120 degrees . Okay . Now if | |
12:45 | you were to take a look at these larger numbers | |
12:47 | , you might not recognize these larger angle measures . | |
12:50 | But if you take the 4 50 if you subtract | |
12:53 | off 3 60 from that , Because if I can | |
12:56 | do it three , Then what you're gonna get is | |
12:59 | 90°. . If you go down here and you subtract | |
13:03 | off 3 60 because you're taking a big number minus | |
13:05 | 3 60 then what you're gonna get is 90 I'm | |
13:09 | sorry , not 90 , you're gonna get one , | |
13:12 | 1 80 . If you take this guy and subtract | |
13:15 | off , Then what you're gonna get is 270 . | |
13:19 | And if you take this guy and subtract off 360 | |
13:22 | , what you're gonna get is 360 . So what | |
13:24 | I'm trying to say is these really large angle measures | |
13:26 | . If you don't know exactly where there are in | |
13:28 | the unit circle , like if I look at 630 | |
13:30 | degrees , I don't know off the top of my | |
13:32 | head , where is it ? Is it over here | |
13:33 | ? Is that I don't know . So just take | |
13:35 | those large numbers back off one revolution of the circle | |
13:38 | and then you have to 70 and then you immediately | |
13:40 | know it's here . So really 630 degrees means it | |
13:44 | goes all the way around the unit circle , But | |
13:47 | then all the way , another three more chunks of | |
13:49 | 90° To get down here to to 70 . So | |
13:54 | one way that you could look at that is to | |
13:57 | say uh this is one chunk to chunks , three | |
14:01 | chunks , four chunks , five chunks , six chunks | |
14:03 | , seven chunks , eight chunks and so on . | |
14:06 | All the way around to wherever you're trying to be | |
14:08 | . All right now , the same process works with | |
14:10 | negative angles . I'm not gonna write all of the | |
14:12 | multiplication is on the board for negative angles . But | |
14:14 | if you start here , you know that positive angles | |
14:16 | go this way and you know that negative angles go | |
14:19 | this way . So this angle measure down here , | |
14:22 | of course we know it's 270 as measured in the | |
14:24 | positive direction . But this angle measure here is also | |
14:28 | equivalently negative 90° right ? And then negative 1 80 | |
14:33 | the negative to 70 and then negative 3 60 . | |
14:37 | So all of these angle measures that have positive angle | |
14:40 | measures , they also have equivalent negative angle measures as | |
14:43 | well . So what I'm saying is this angle measure | |
14:46 | of 270 is measured from the positive access is exactly | |
14:49 | the same thing as negative 90° because you're going down | |
14:52 | by -91 chunk of 90 in the negative direction . | |
14:55 | Two chunks of 90 in the negative direction would be | |
14:57 | negative 1 80 This will be negative to 70 . | |
15:00 | So negative 270° is exactly the same thing as positive | |
15:04 | 90 . Uh , negative 1 80 is exactly the | |
15:07 | same as positive 1 80 . And so instead of | |
15:09 | memorizing all these things you need to learn to count | |
15:12 | . It's literally like learning third grade math again , | |
15:15 | Second grade math . When you learn to count , | |
15:16 | we have to learn accounting degrees . All right . | |
15:18 | So I think we've exhausted counting in 90 degree chunks | |
15:21 | . We're not going to do probably quite as much | |
15:24 | talking for the next ones , But now that we | |
15:26 | have the idea , we can certainly talk intelligently about | |
15:28 | what we're going to do next . So let me | |
15:30 | pick , try to pick a different color and let's | |
15:33 | figure out what this angle is . We just talked | |
15:36 | about it . If this is 90 than what's this | |
15:37 | ? This is 45°. . So if this is 45 | |
15:41 | degrees , it's going to be right here . That | |
15:44 | will be the 45 degree angle measure . So now | |
15:46 | , instead of counting by nineties , let's count in | |
15:49 | chunks of 45 degrees . So what I want to | |
15:51 | do over here is I want to say we're going | |
15:53 | to count bye chunks of 45 degree measure . That's | |
15:59 | what we're gonna do . All right , so what | |
16:01 | do we have here ? This pie wedge , this | |
16:03 | chunk right here . Forget about this line . This | |
16:05 | doesn't exist just this chunk from here to here . | |
16:07 | This whole segment here . This is a 45 degree | |
16:10 | object . So one chunk of 45 is right here | |
16:13 | and then 1 45 is here and then two chunks | |
16:16 | of 45 is here . That would be here . | |
16:18 | Three chunks of 45 would be here for chunks of | |
16:21 | 45 is here . Five chunks of 45 6 45 | |
16:24 | . 7 45 . 8 45 . I can keep | |
16:27 | going . 9 45 . 10 45 . 11 45 | |
16:29 | 12 45 13 45 14 45 15 45 16 45 | |
16:34 | . I can keep going forever . All right , | |
16:36 | but let's write down a few things . Let's say | |
16:38 | we count by 45 . We're gonna start with zero | |
16:41 | and then we're gonna have one chunk of 45 And | |
16:44 | then we're gonna have to chunks of 45 And then | |
16:47 | 3 45s And then 445 and then 545 and then | |
16:55 | six whoops , 6 45 and then 7 40 fives | |
17:02 | and then 8 45 . All right , So let's | |
17:05 | go around that far . What do these correspond to | |
17:09 | ? All right . So then this is going to | |
17:10 | correspond to zero . This is one times 45 is | |
17:13 | 45 degrees . What is two times 45 ? It's | |
17:15 | 90 degrees . What is three times 45 ? It's | |
17:19 | going to be 135°. . Let me space things out | |
17:23 | to try to speak to kind of make them correspond | |
17:26 | a little bit of four times 45 is going to | |
17:29 | be 180°. . Five times 45 is 225°, , 6 | |
17:36 | times 45 - 70 . Um And then we're gonna | |
17:41 | have seven times 45 is 3 15 and then eight | |
17:45 | times 45 . And you multiply that out . You | |
17:47 | actually get 360 degrees . Now you see these angle | |
17:51 | measures ? These are the angle measures that exist on | |
17:53 | the unit circle . Now of course I could just | |
17:55 | I could just write them down and say , Hey | |
17:57 | , remember them . But that's no fun . So | |
17:59 | what we have is this is 45 and this is | |
18:03 | two times 45 , which is 90 . All right | |
18:06 | ? So actually I think what I'm gonna do is | |
18:08 | I'm gonna erase this in red and we're gonna try | |
18:10 | to keep everything in the same colour . So here | |
18:12 | we have a 45 degree increments one chunk of 45 | |
18:16 | 2 chunks of 45 was 93 chunks of 45 . | |
18:19 | We just figured out was 1 35 . So this | |
18:21 | angle measure which is 45 from here is 1 35 | |
18:25 | . But then for chunks of 45 when you multiply | |
18:28 | that out comes out again to 1 80 then five | |
18:31 | chunks of 45 . This comes out to 225 and | |
18:35 | then six chunks of 45 is gonna come out to | |
18:38 | to 70 and then seven chunks of 45 is going | |
18:41 | to come out to 315 degrees . And then eight | |
18:45 | chunks of 45 is going to come out to 3 | |
18:47 | 60 grab a calculator , one times 45 2 times | |
18:50 | 45 . 3 times 45 . That's what you're gonna | |
18:51 | get now . In a similar way . I can | |
18:54 | go in the negative direction . This angle measure is | |
18:56 | measured as 3 15 in the positive direction 315 degrees | |
19:00 | now . But this angle measure , this location is | |
19:03 | exactly the same as negative 45 degrees , negative 45 | |
19:07 | degrees is the same thing as positive 3 15 , | |
19:10 | -90° is the same exact thing as this And negative | |
19:14 | 135° measured from here , -135 is the same as | |
19:19 | positive to 25 . All right , So you see | |
19:22 | the symmetry of things . So , a lot of | |
19:24 | students are like , well , should I count positive | |
19:25 | , should account negative . How do I do it | |
19:27 | ? What you really need to know is that these | |
19:29 | diagonals here that are on the diagonals ? Those are | |
19:31 | just increments of 45 degrees . And if you forget | |
19:34 | that 3 15 is over here , just remember , | |
19:36 | you're gonna have to remember some of these numbers . | |
19:38 | You know this is 2 70 45 more is going | |
19:40 | to be here here . 45 more is 3 60 | |
19:43 | so on . And then of course I can go | |
19:44 | up when this was eight times 45 9 and 10 | |
19:48 | and 11 times 45 12 times 45 so on . | |
19:51 | I can go all the way around and do that | |
19:53 | . So I can measure my angles like this . | |
19:56 | All right . So now we have counted by 45 | |
19:59 | degree chunks and we've counted by 90 degree chunks . | |
20:01 | Now I want to spend a few minutes talking about | |
20:04 | counting by 30 degree chunks and then we'll talk about | |
20:07 | 60 degree chunks . So on this unit circle , | |
20:10 | this angle measure we already said is a 30 degree | |
20:12 | angle measure from the axis . So this is a | |
20:15 | 30 degree chunk . So in your mind you need | |
20:17 | to remember that this is a 30° chunk . So | |
20:20 | this angle measure is the same as this angle measure | |
20:23 | which is the same as this angle measure and is | |
20:25 | the same as the single measure . So these are | |
20:27 | all chunks of 30° as you go around , right | |
20:30 | ? So you could say this is 30° and then | |
20:35 | another chunk of 30 degrees would be 30 plus 30 | |
20:38 | is 60 degrees . And then another chunk of 30 | |
20:40 | degrees would be 90 degrees . And then you can | |
20:43 | continue walking around . But instead of doing it doing | |
20:45 | it on the unit circle over there , let's go | |
20:47 | over here and say we're going to counts bye 30 | |
20:52 | degree chunks . We can count by 33 chunks . | |
20:55 | So we're gonna have 10 and then one times 30 | |
20:59 | and then two times 30 And then three times 30 | |
21:03 | and then four times 30 . And then five times | |
21:07 | 30 . And then six times 30 . Right now | |
21:12 | we have to keep uh actually no we're fine there | |
21:15 | . So six times three . Um Yeah . Actually | |
21:19 | let's go down here . Let's go down and say | |
21:20 | seven times 30 eight times 30 . Whoops . nine | |
21:27 | times 30 . I know this is a little boring | |
21:29 | but it's gonna pay off . Pay off 10 times | |
21:31 | 30 . 11 times 30 And 12 times 30 . | |
21:36 | All right . What do these correspond to in terms | |
21:39 | of angle measure ? You just do the multiplication . | |
21:41 | So what this is going to correspond to is zero | |
21:43 | . This is going to correspond to 30 . This | |
21:45 | is going to be 60 , this is going to | |
21:47 | be three times three is 94 . Times three is | |
21:49 | 120 five times 3 is 150 six . Times 3 | |
21:54 | is 180 . I'm gonna put my little degree symbols | |
21:57 | here like this . All right , seven times 3 | |
22:02 | . It's 210 . 8 times three is 24 , | |
22:07 | nine times 3 is 27 , 10 times three is | |
22:10 | 300 , 11 times three is 330 . And then | |
22:14 | here you have 12 times 3 36 . So 360 | |
22:16 | . You see what's going on here ? We're basically | |
22:18 | calculating the degree measures as we walk around . All | |
22:21 | right . So all of these corresponds to here have | |
22:24 | zero . Then you have 30 60 90 . The | |
22:26 | next one after that is 120 degrees . So this | |
22:29 | is another 30 degree chunk which comes out to 120 | |
22:33 | degrees . And then here you skip over this because | |
22:35 | this is another 30 degree chunk and it comes out | |
22:38 | to 150 degrees and this is another 30 degree chunks | |
22:42 | . So you add 30 you get 1 80 then | |
22:44 | this is another 30 degree chunk , and you get | |
22:46 | 210 . And then this is another 30° chunk , | |
22:50 | right ? And then you end up with 240 , | |
22:53 | is checking myself here , 240 . Just add 30 | |
22:56 | to this and then at 30 to get this , | |
22:58 | you get to 70 at 30 of this , you | |
23:00 | get 300 And then add 30 to get this . | |
23:04 | And you get 330 at 32 this and you get | |
23:09 | 3 60 . So you see you can count by | |
23:12 | 45 degree chunks and get all the way back to | |
23:14 | 3 60 . You can count by 90 degree chunks | |
23:16 | and get all the way back around the 3 60 | |
23:18 | . You can count by 30 degree chunks . You | |
23:20 | have more chunks but you're still going to count all | |
23:22 | the way back around the 360 . And the same | |
23:24 | thing happens in the negative direction . This angle is | |
23:27 | a positive 330° as measured from the positive direction . | |
23:32 | But that's exactly the same thing as a negative 30 | |
23:35 | degree angle . This is exactly the same as a | |
23:37 | negative 60 degree angle . Uh And this is a | |
23:41 | negative 90 degree angle . So account by negative 30 | |
23:43 | negative 60 negative 90 negative 90 . We already said | |
23:46 | the same as positive to 70 . So we have | |
23:49 | now labeled essentially everything on the unit circle . But | |
23:52 | notice we never counted by sixties . So I want | |
23:56 | to spend a second and I want to count bye | |
24:01 | 60 degree chunks and actually the work is already done | |
24:03 | for us . So let's just go ahead and say | |
24:06 | we start with 01 times chunk of 62 chunks of | |
24:10 | 60 three chunks of 60 . Uh Then we have | |
24:15 | four chunks of 60 and we have five chunks of | |
24:19 | 60 . And then we have six chunks of 60 | |
24:22 | . 1 of these correspond to , you might have | |
24:25 | guessed this corresponds to zero degrees 60 degrees , two | |
24:30 | times six is 12 , so 120 degrees three times | |
24:33 | 6 18 . So 180 degrees four times 6 , | |
24:37 | 24 . So 240 degrees five times 6 30 so | |
24:41 | 300 degrees and then six times 6 36 or 360 | |
24:45 | degrees . So you see when we count by 60 | |
24:48 | degree chunks , we get 0 61 21 80 and | |
24:50 | so on . Now . Think about it in terms | |
24:52 | of 60 degree chunks . Forget about the 30 degree | |
24:55 | line . Forget about the 45 degree line . This | |
24:57 | is a chunk of 60 degrees . This is a | |
25:00 | 60 degree chunk . Another chunk when you add it | |
25:03 | to this is going to be this chunk you're gonna | |
25:05 | land over here . So you're counting by 60 60 | |
25:08 | . Then you land over here . Uh This is | |
25:11 | 60 then you land over here on 1 20 . | |
25:13 | Another 60 degree chunk is down here , you land | |
25:16 | on 1 80 . So that's the numbers were getting | |
25:18 | 0 61 21 80 you have zero then 60 then | |
25:23 | 1 20 then 1 80 . What's another 60 degree | |
25:25 | chunk ? It's going to be down here at 2 | |
25:27 | 40 . That's what we get right here . What's | |
25:29 | another 60 degree chunk here ? It's gonna be sweeping | |
25:32 | through here to 300 then the 3 63 100 then | |
25:35 | 23 60 . So again counting by sixties 60 then | |
25:39 | 1 20 then 1 80 Then 2 40 and 303 | |
25:45 | 60 . And that can continue on Incrementally , I | |
25:48 | stopped at six times 60 , but this would be | |
25:51 | seven times 60 and then eight times 60 and the | |
25:54 | nine times 60 . This was six times 60 . | |
25:57 | This is seven times 60 . This is eight times | |
25:59 | 60 . This is nine times 60 and then I | |
26:02 | can keep going around and the negative angles is the | |
26:04 | same thing . This is a negative angle measure of | |
26:07 | 60 degrees , so this is negative 60 right here | |
26:10 | , -60 is exactly the same thing as positive 300 | |
26:14 | And over here , this would be negative 120 is | |
26:18 | exactly the same thing as positive 240 . So why | |
26:21 | am I doing all of this ? I mean , | |
26:23 | really all of it is just multiplication . I mean | |
26:26 | nothing we've done is more than arithmetic and multiplication . | |
26:29 | Why are we taking the time ? Because when we | |
26:32 | ditch the degree measures , which we're going to do | |
26:35 | pretty soon , you'll no longer have the comfort of | |
26:37 | the experience of knowing . Oh , that's about 90 | |
26:39 | degrees . Oh , that's about 270 degrees . Because | |
26:43 | we don't have a good numbers in our mind when | |
26:45 | it comes to radiant measure , which is coming very | |
26:47 | soon . So in radian measure , it's going to | |
26:49 | be critical that you count around the unit circle in | |
26:52 | the proper increments of the radian measure to figure out | |
26:55 | what angle you're on . So I'm showing you that | |
26:57 | by counting by thirties counting by sixties counting by 45 | |
27:00 | counting by nineties . You can land on any part | |
27:03 | of this unit circle in the positive direction or in | |
27:06 | the negative direction and figure out what quadrant urine and | |
27:10 | where you're at . Because later on when we learn | |
27:12 | about sign and co sign , it's critical for you | |
27:16 | to know what quadrant you're in . It's extremely important | |
27:19 | to know that I'm somewhere over here in this quadrant | |
27:22 | at 315 degrees or that I'm over here in this | |
27:26 | quadrant at 210 and that's 30 degrees from from the | |
27:29 | X axis over here . It's important for you to | |
27:31 | know where you're at and how many degrees you are | |
27:33 | away from the different axes . So we're gonna solve | |
27:36 | some problems in the next lesson on on counting around | |
27:40 | the unit circle and getting a lot more practice with | |
27:42 | it for now . I want you to kind of | |
27:43 | watch this a couple of times and make sure you | |
27:45 | understand what I'm saying . And then also try to | |
27:48 | commit to memory these numbers around the outside . Uh | |
27:51 | It sometimes gets confusing and you sometimes forget but try | |
27:54 | to remember , I know we can all remember the | |
27:56 | first quadrant , but in the second quadrant 1 21 | |
27:58 | 35 1 51 82 10 . These are the important | |
28:02 | numbers in degrees because it's going to be numbers that | |
28:05 | will be using over and over and over again . | |
28:07 | Follow me on to the next lesson . Once you | |
28:08 | understand the concept here and then we're going to crack | |
28:11 | the crack , the very important topic of what is | |
28:14 | the actual meaning of sine and cosine . Forget about | |
28:17 | equations , what is the meaning of it ? We | |
28:19 | want to understand what it is so that we can | |
28:21 | calculate things and understand what we're doing . So follow | |
28:23 | me on the next lesson and we will conquer that | |
28:26 | right now . |
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