04 - What is the Unit Circle? Angle Measure in Degrees, Reference Angles & More. - Free Educational videos for Students in K-12 | Lumos Learning

04 - What is the Unit Circle? Angle Measure in Degrees, Reference Angles & More. - Free Educational videos for Students in k-12


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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