05 - What is a Radian Angle? Convert Degrees to Radians & Radians to Degrees - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The title of this lesson | |
| 00:02 | is called . What is radiant angle measure ? Also | |
| 00:06 | if I had to subtitle it I would say converting | |
| 00:08 | between degrees , going to radiance and also radiance going | |
| 00:12 | two degrees . So up until this point we have | |
| 00:14 | learned all about trigonometry and how do you sign and | |
| 00:17 | co signing things ? The unit circle in degrees because | |
| 00:19 | we all know that there are 360 degrees in a | |
| 00:22 | circle . Now we need to switch gears and start | |
| 00:24 | to talk about radiant angle measure . Once we now | |
| 00:27 | understand what to do with the trigonometry we can then | |
| 00:30 | switch measurement systems into the radiant system . Now the | |
| 00:33 | first thing you might want to say or just ask | |
| 00:36 | is why do we have to systems ? Why don't | |
| 00:37 | we just work in degrees ? Well the truth is | |
| 00:40 | the radiant system is we're gonna find out in just | |
| 00:42 | a second is a very much more natural way to | |
| 00:44 | talk about angle measures . So when you're generally talking | |
| 00:47 | about triangles and you know surveying and just triangle trigonometry | |
| 00:52 | , usually we work in degrees . But once you | |
| 00:54 | move past that into I don't even want to say | |
| 00:57 | advanced concepts really . It's just when you move out | |
| 00:59 | of triangles , we generally start working in radiant measure | |
| 01:02 | because any kind of complex problem , anything more complex | |
| 01:04 | than a triangle , It actually makes a lot more | |
| 01:06 | sense to talk about the angles and radiance . So | |
| 01:09 | why is that ? What is the fundamental concept of | |
| 01:12 | a radiant anyway ? Alright , so again , first | |
| 01:14 | of all , keep in the back of your mind | |
| 01:15 | a circle has 360°. . However we talked about , | |
| 01:19 | why is it 360°? ? Why isn't it 380 degrees | |
| 01:23 | or 400 degrees ? What is special ? About 360 | |
| 01:27 | degrees . Well , the answer is you can go | |
| 01:28 | to the history of that degree system and there is | |
| 01:31 | a history behind it , of course . But the | |
| 01:33 | reality of it is being divided into 360 degrees . | |
| 01:36 | Doesn't have like a fundamental it's not a fundamental feature | |
| 01:40 | of our universe . It's just a human system that | |
| 01:42 | we made up . Radian measure , however , is | |
| 01:45 | much more fundamental , and that's why it's more natural | |
| 01:47 | to talk about radiant measure . So let's talk about | |
| 01:50 | a circle and let's talk about a unit circle , | |
| 01:53 | which we've been discussing , which has a radius of | |
| 01:55 | one . That's how we're going to introduce the concept | |
| 01:57 | of radiant . All right , So , what we're | |
| 01:59 | gonna talk about is a radius a circle with radius | |
| 02:04 | equal to one . Now , I'm gonna draw a | |
| 02:07 | quick little circle with radius one . It doesn't have | |
| 02:09 | to be exact , it doesn't have to be perfect | |
| 02:12 | , but we're gonna try to do our best . | |
| 02:13 | So here is roughly why is equal to one over | |
| 02:16 | here somewhere is X is equal to one down here | |
| 02:19 | . Why is equal to negative one ? And over | |
| 02:21 | here , right around here is when X is equal | |
| 02:23 | to negative one . So I'm just trying to sketch | |
| 02:26 | a circle with a radius of one . I'm not | |
| 02:28 | gonna do a very good job here . It's probably | |
| 02:30 | gonna look more like an ellipse , but we're gonna | |
| 02:32 | try our best to make it some kind of circle | |
| 02:35 | . It's not too bad . Obviously , it's not | |
| 02:36 | rounded perfectly , you know ? But the point is | |
| 02:39 | it's a circle of radius one . All right . | |
| 02:41 | How many degrees are in the circle If we start | |
| 02:43 | from the X axis and we measure all the way | |
| 02:45 | around we get to of course there's 91 82 70 | |
| 02:49 | we get to 360 degrees . All right , now | |
| 02:51 | , let's talk about the concept of radiant . Right | |
| 02:54 | , Let's say , since we know that this circle | |
| 02:57 | has a radius one , let's calculate the circumference the | |
| 03:00 | distance all the way around this circle . The circumference | |
| 03:04 | you might have remembered from fifth grade or something is | |
| 03:06 | two times pi times are two times pi times the | |
| 03:10 | radius . That's what the distance is . If you | |
| 03:13 | measure the radius in meters or centimeters , then the | |
| 03:15 | circumference will be measured in meters or centimeters , whatever | |
| 03:18 | system you're using . So , but now we have | |
| 03:21 | a unit circle . So that means the radius is | |
| 03:23 | one . So that means that the circumference is two | |
| 03:26 | times pi times one . That means the circumference of | |
| 03:29 | this circle , because the radius is one is equal | |
| 03:31 | to two pi . Right ? So let's say for | |
| 03:35 | instance that the that the radius of this circle was | |
| 03:40 | one centimeter . Right ? Let's just say So , | |
| 03:43 | then that would mean that the circumference all the way | |
| 03:45 | around was to pi centimeters . Right ? You have | |
| 03:49 | to be the same unit . If you measure the | |
| 03:50 | radius in meters then the circumference will be two pi | |
| 03:53 | meters . So what do I mean by two pi | |
| 03:55 | centimeters ? I mean pi is 3.14159 and an infinite | |
| 04:00 | number of non repeating decimals because pi is irrational . | |
| 04:03 | Okay , so it's not a decimal that stops at | |
| 04:06 | 3.14 but we can approximate it as 3.14 You multiply | |
| 04:10 | by two and then that's the actual number of centimeters | |
| 04:13 | . If I were to measure it with a kind | |
| 04:16 | of a flexible tape measure and measure all the way | |
| 04:19 | around , I would measure two times pi pi is | |
| 04:21 | about 3.14 so two times three is six , so | |
| 04:24 | just a little bit more than six centimeters around . | |
| 04:27 | If I measure the radius to be one centimeter , | |
| 04:29 | if I measure the radius to be one m , | |
| 04:31 | then it's gonna be just over six m around . | |
| 04:34 | If I measure the radius to be one light year | |
| 04:36 | , then it's going to be just over six lightyears | |
| 04:38 | around . That's how circumference works now . Here's the | |
| 04:41 | thing , since I know that this distance is just | |
| 04:44 | a little bit over 62 pi centimeters all the way | |
| 04:47 | around , then I can use the distance around the | |
| 04:50 | circle to tell me basically what the angle is . | |
| 04:54 | Because remember , the angle is just telling me in | |
| 04:57 | the previous system , we had 90 degrees up here | |
| 04:59 | and then we had 100 and 80 degrees here and | |
| 05:01 | then to 70 here and then we had uh 360 | |
| 05:05 | right there . But basically , since we know the | |
| 05:08 | circumference of the unit circle has two pi uh centimeters | |
| 05:12 | , then we can kind of like drop the unit | |
| 05:14 | measurement , we can drop the centimeters part and we | |
| 05:17 | can say that every circle has two pi radiance that | |
| 05:20 | go all the way around . A radiant is just | |
| 05:23 | like pretending the circle has a radius of one and | |
| 05:25 | then you have to pie of those that goes all | |
| 05:28 | the way around and see . That's why the concept | |
| 05:30 | of radiant measure makes a lot more sense because 360 | |
| 05:33 | degrees . It's like you pull it out of the | |
| 05:34 | air and you say , oh there's 360 of these | |
| 05:36 | things in a circle that's kind of like off the | |
| 05:39 | wall and comes from nowhere . I mean really there's | |
| 05:41 | history there , but I mean it isn't a fundamental | |
| 05:43 | thing , but this is a fundamental thing because if | |
| 05:45 | I take a circle has a radius of one , | |
| 05:48 | I know because of geometry , because of the fundamental | |
| 05:51 | geometry that there's going to be two times pi distance | |
| 05:54 | units around . So I just dropped the distance and | |
| 05:57 | say , okay , there's there's every circles identical . | |
| 05:59 | So there's two pi of these things called radiance . | |
| 06:02 | So you can think of a radiant as being the | |
| 06:04 | distance around the circle there if the radius is one | |
| 06:08 | . So what we say is we kind of dropped | |
| 06:11 | the distances here . As we're talking about unit circles | |
| 06:14 | , we say that there are two pi radiance around | |
| 06:22 | every circle all right around every circle . So , | |
| 06:28 | because it involves the number pi the rating and measures | |
| 06:31 | that you see for the angles that we're gonna be | |
| 06:33 | using , they're always gonna have pie in there . | |
| 06:35 | So , that's why they have pie in there . | |
| 06:37 | Because the definition of going all the way around the | |
| 06:40 | circle is two pi radiance . The reason is two | |
| 06:42 | pi ratings is because if you look at a unit | |
| 06:44 | circle , the distance around is two times pi uh | |
| 06:47 | in whatever unit you're talking about . All right . | |
| 06:49 | So for instance , if I wanted to draw a | |
| 06:52 | picture of it right ? I would say that if | |
| 06:57 | I normally if I say this is zero degrees 90 | |
| 06:59 | degrees . 180 to 70 back to 360 . But | |
| 07:02 | in this case I'm going to say I'm gonna go | |
| 07:04 | all the way around and measure back to where I | |
| 07:08 | started from Now . In the old system that would | |
| 07:10 | be 360 degrees . So I could say data is | |
| 07:13 | 360 degrees . But I can also then say that | |
| 07:17 | data is two times pi radiance . So you don't | |
| 07:21 | put a degree symbol when you put a degree symbol | |
| 07:24 | that means degrees . But when you have to some | |
| 07:26 | number of pie and you put our A . D | |
| 07:28 | . That's what you put , it tells people that | |
| 07:30 | you're talking about radiant measure . This 360 degrees is | |
| 07:34 | the same thing as two pi radiance . Now just | |
| 07:37 | like you can take 360 degrees and start chopping it | |
| 07:39 | up . Like if you chop it into four pieces | |
| 07:42 | you're gonna get 90 degree chunks , the quadrants of | |
| 07:45 | the unit circle . You can take the two pi | |
| 07:47 | ratings and you can chop it up into pieces too | |
| 07:50 | . So going all the way around the unit circle | |
| 07:51 | , you're gonna have fractions of pie right ? And | |
| 07:54 | we're going to learn and going to calculate those fractions | |
| 07:56 | of high in just a second . But what you | |
| 07:58 | need to realize is that basically these two things are | |
| 08:00 | the same thing . So probably the most important thing | |
| 08:03 | I can write in this lesson is an important conversion | |
| 08:07 | factor 360 . I can put the degree symbol but | |
| 08:10 | I'm gonna put D . E . G . To | |
| 08:12 | make sure it's all clear 360 degrees is equal by | |
| 08:15 | definition to be two times pi of this new unit | |
| 08:18 | called a radiant . And you can think about it | |
| 08:20 | as being the distance . If this is a unit | |
| 08:23 | circle here , every bit that I go two times | |
| 08:27 | pi distance units around that's one full revolution . Right | |
| 08:30 | ? So if I go half the distance over here | |
| 08:32 | then I go to pi divided by two radiance . | |
| 08:35 | Because if I only go halfway and we're gonna get | |
| 08:37 | to that in just a second . So you might | |
| 08:39 | say Well it doesn't just just work for like circles | |
| 08:43 | of of distance of Radius one . I mean what | |
| 08:46 | if I have circles bigger ? What if I'm trying | |
| 08:48 | to measure the angle of a you know some large | |
| 08:51 | part of the circle , that's not a unit circle | |
| 08:54 | . I mean the same thing works for degrees . | |
| 08:56 | I mean you know that let me draw an interior | |
| 08:58 | circle right here . So let's say the radius of | |
| 09:01 | this circle right here is one . Okay , But | |
| 09:04 | you know from the previous definitions that if I draw | |
| 09:07 | a circle outside of that that's larger than one , | |
| 09:10 | I can certainly measure degree measurements around this larger circle | |
| 09:14 | . If I draw an even larger circle around this | |
| 09:17 | guy , something really , really , really big like | |
| 09:19 | this . Even though this is like a radius of | |
| 09:22 | two , Maybe this thing's a radius of three or | |
| 09:23 | four . I can still say this is zero and | |
| 09:26 | I can still say this is 90 degrees . It | |
| 09:28 | doesn't matter how big the circle is . In other | |
| 09:30 | words , there's always 360 degrees in every circle . | |
| 09:33 | But for radiance we say the same thing , there's | |
| 09:35 | always two pi radiance , no matter how large the | |
| 09:38 | circle is . But the way you can think about | |
| 09:40 | it is if you look inside of your circle to | |
| 09:42 | a circle of radius one , you increments a number | |
| 09:45 | of radiance as basically you walk around the circumference of | |
| 09:48 | the circle . So again two pi is two times | |
| 09:50 | 3.1 or just a little bit over six . So | |
| 09:54 | it's gonna be basically six centimeter . If it's a | |
| 09:56 | one centimeter it'll be six centimetres around when you by | |
| 09:58 | the time you get around to the other side . | |
| 10:00 | So when you get over about halfway it's not going | |
| 10:03 | to be six , it'll be about just a little | |
| 10:05 | bit over three . Right ? Because six , just | |
| 10:08 | a little bit over six radiance is all the way | |
| 10:10 | around . So then halfway it's about around three radiance | |
| 10:12 | . So you're just measuring the circumference of that unit | |
| 10:15 | circle , That is the number of radiance that you're | |
| 10:18 | walking around this boundary . That's why radiance are more | |
| 10:21 | fundamental because they actually represent geometry . You're measuring the | |
| 10:25 | circumference of a unit circle . That's the number of | |
| 10:27 | radiance you have as you walk around . Whereas degree | |
| 10:30 | measure is just some random 360 number that was just | |
| 10:33 | comes about from history . Alright , so just like | |
| 10:37 | we said That 360° is two pi radiance . What | |
| 10:42 | if I only go halfway around , In other words | |
| 10:46 | , every circle has two pi radiant as you go | |
| 10:48 | all the way around . What if I don't go | |
| 10:51 | all the way around ? What if I only go | |
| 10:52 | halfway around ? What if I say I'm not gonna | |
| 10:55 | go all the way around , but I'm going to | |
| 10:56 | go all the way to here . Now , we | |
| 10:58 | know from previous discussions that this is 100 and 80 | |
| 11:01 | degree angle , We already know that , right ? | |
| 11:04 | But we know that there is 360 degrees in every | |
| 11:07 | circle and we know that 360 degrees is two pi | |
| 11:10 | radiance . So what would this radian measure be ? | |
| 11:12 | Well , if the entire way around the circle is | |
| 11:15 | two pi then if I take that to pi radiance | |
| 11:17 | of going all the way around and I just cut | |
| 11:19 | it in half , then this must be the measurement | |
| 11:21 | of half a circle . Right ? You can see | |
| 11:23 | the two's cancel . So pie radiance Pi radiance is | |
| 11:31 | equal to 180 degrees . In fact , I think | |
| 11:34 | I'm gonna write that again underneath . It's an important | |
| 11:37 | note . I'm gonna say conversion factors and you can | |
| 11:42 | use Either of these conversion factors that you want . | |
| 11:46 | They're both the same thing . You can say that | |
| 11:49 | 180° is equal to pi radiance , Right ? Or | |
| 11:55 | you can say that 360° Is equal to two pi | |
| 12:00 | radiance . You see what's happened here , This is | |
| 12:03 | the fundamental conversion factor . But if I multiply by | |
| 12:06 | two I'll get 3 60 I'll get to pie either | |
| 12:09 | one is the same . Um I honestly usually just | |
| 12:12 | say two pi radiance is 3 60 degrees . But | |
| 12:14 | I know lots of people and lots of books that | |
| 12:16 | just say 180 degrees as pie radiance . It's more | |
| 12:19 | fundamental for you to understand why it is the case | |
| 12:21 | though . The reason that these work is because a | |
| 12:24 | full circle is two pi radiance . So a half | |
| 12:26 | circle must be that amount of radiance divided by two | |
| 12:29 | . That's how you get to the pie radiance . | |
| 12:30 | But every time you walk some degree measurement around the | |
| 12:33 | circle it will be some fraction of of some multiple | |
| 12:37 | of pi , some fraction of pie because going all | |
| 12:39 | the way around is two times pi . All right | |
| 12:42 | , so let's take a second to do a couple | |
| 12:44 | of quick conversions and uh we see how much more | |
| 12:48 | space do I have . You have a lot of | |
| 12:51 | space . So let's go and say for my next | |
| 12:55 | for the next part here let's do a little conversion | |
| 12:58 | . Let's convert 30 degrees two radiance , 30 degrees | |
| 13:06 | the radiance . So this is how you do it | |
| 13:08 | . And I'm gonna show you how to convert units | |
| 13:11 | in a way that you may have seen before but | |
| 13:13 | maybe you haven't seen before . But the way I'm | |
| 13:14 | gonna show you how to convert between units in this | |
| 13:17 | case between degrees and radiance is the way that I | |
| 13:19 | convert units for every class for everything . I do | |
| 13:22 | . Chemistry , physics , engineering , electric circuits , | |
| 13:26 | mechanical systems . It doesn't matter . This method I'm | |
| 13:28 | about to show you of converting between things is way | |
| 13:32 | easier than just trying to wonder if I should multiply | |
| 13:35 | or divide . Okay so let me show you how | |
| 13:36 | to do this . What you do is you start | |
| 13:40 | with what you know you have 30 degrees , that's | |
| 13:42 | what you're trying to convert from . So you're right | |
| 13:43 | 30 degrees and you could put the degree symbol but | |
| 13:47 | I liked putting D E G just so it's easier | |
| 13:49 | to read . Then you draw a horizontal line under | |
| 13:52 | this and a vertical line next to it . And | |
| 13:54 | now I have to put a conversion factor in place | |
| 13:57 | . Right ? So I I know that the conversion | |
| 13:59 | factor can be written as 100 and 80 degrees is | |
| 14:01 | equal to pi or 3 60 is equal to two | |
| 14:04 | pi . I know that either one is going to | |
| 14:06 | work so let's go and use the first one . | |
| 14:08 | And the way you do it is you you arrange | |
| 14:11 | the conversion factor in such a way that the units | |
| 14:13 | cancel . So the way you want to write it | |
| 14:15 | is 100 and 80 degrees equals pi radiance . And | |
| 14:22 | I write it like this way because now I have | |
| 14:25 | degrees on the top and degrees on the bottom . | |
| 14:27 | And if you remember things on the top cancels with | |
| 14:29 | things that are on the bottom , just like any | |
| 14:32 | kind of division , any kind of simplification of X | |
| 14:35 | . When X is on the top and X is | |
| 14:36 | on the bottom , you cancel them , three is | |
| 14:38 | on the top and three is on the bottom you | |
| 14:39 | cancel , 16 is on the top , 16 is | |
| 14:41 | on the bottom , you cancel , degrees are on | |
| 14:43 | the top , degrees are on the bottom , they | |
| 14:45 | cancel . So the only unit left here is radiance | |
| 14:48 | . So the way you do this is you grab | |
| 14:50 | a calculator and you take 30 which is the number | |
| 14:53 | you have . You multiply by pi . Anything on | |
| 14:56 | the top gets multiplied and anything on the bottom gets | |
| 14:59 | divided . So if you take 30 and multiply by | |
| 15:03 | pi and then divide by 1 80 . The way | |
| 15:06 | it's gonna work out is you're going to have 30 | |
| 15:08 | times pi divided by 180 . Multiply the tops divide | |
| 15:13 | the bottom . So grab a calculator uh or actually | |
| 15:16 | don't even have to grab a calculator because we see | |
| 15:18 | that this is 30 and this is 1 80 on | |
| 15:20 | the top and bottom . So we can divide the | |
| 15:22 | top by 30 and get one , we can divide | |
| 15:25 | uh the 1 80 by 30 and get a sick | |
| 15:28 | . So Three times six is 18 . So you | |
| 15:30 | have a one on the top and the six on | |
| 15:31 | the bottom , what you end up with is just | |
| 15:33 | a pie on the top and the six on the | |
| 15:36 | bottom . And the unit that's left is the only | |
| 15:38 | unit that didn't cancel . Now you would not want | |
| 15:41 | to do this with the conversion factor flipped over because | |
| 15:45 | if degrees were on the top and radiance are on | |
| 15:47 | the bottom , nothing would cancel . So you always | |
| 15:49 | arrange the conversion factor so that the units cancel . | |
| 15:52 | You know , a lot of students have problems with | |
| 15:54 | units because they start thinking well do I multiply by | |
| 15:57 | pi and divide by 1 80 ? Or do I | |
| 15:59 | divide by 1 80 ? Multiplied by pi ? Which | |
| 16:01 | way it makes sense ? And you start trying to | |
| 16:03 | think it , think about it , you don't have | |
| 16:04 | to think about it with unit conversions . Start with | |
| 16:07 | what you know and then the conversion factor is this | |
| 16:09 | that's what's given to you . You arrange it in | |
| 16:11 | such a way that the units cancel . The only | |
| 16:13 | unit left is radiance . So then you multiply this | |
| 16:16 | divide by 1 80 . Of course we don't need | |
| 16:18 | a calculator , we just simplify the fraction and you | |
| 16:20 | get pi over six radiance . Remember I told you | |
| 16:23 | every distance around the unit circle is going to be | |
| 16:25 | some fraction of pie in this case it's pi over | |
| 16:28 | six . It kind of makes sense when you think | |
| 16:30 | about it because all the way around the unit circle | |
| 16:32 | is two pi radiance 30 degrees . It's a tiny | |
| 16:36 | little fraction of that . So it makes sense that | |
| 16:38 | it's pi over six . A really small fraction of | |
| 16:40 | pie To go only 30°. . Okay , now what | |
| 16:44 | we're gonna do is we're gonna use this information , | |
| 16:46 | we have now figured out that 30 degrees is equal | |
| 16:49 | to pi over six radiance . So what I want | |
| 16:51 | to do is go to our unit circle . I | |
| 16:54 | have left this unit circle . We have already done | |
| 16:57 | problems with this unit circle . Nothing has changed . | |
| 16:59 | All I've done is I've moved the degree markings inside | |
| 17:02 | to make room . What we've said is that 30 | |
| 17:05 | degrees is exactly the same thing as pi over six | |
| 17:09 | radiance . So I'm not gonna write our A . | |
| 17:11 | D . You were going to know that when you | |
| 17:14 | see an angle with a pie in there , you | |
| 17:15 | know it's a radiant measure . So pi over six | |
| 17:18 | radiance is exactly the same thing as 30 degrees . | |
| 17:21 | So keep that in the back of your mind and | |
| 17:23 | let's move on to the to the next topic . | |
| 17:26 | Now , one more thing I want to tell you | |
| 17:28 | before I go is that some students are like well | |
| 17:31 | what if I didn't use this conversion factor ? What | |
| 17:33 | if I use the other one ? Let's do the | |
| 17:35 | same thing again . 30°. . And let's just say | |
| 17:39 | for some reason you chose to use this one that | |
| 17:42 | 360° Is equal to two pi radiant , two pi | |
| 17:48 | radiant . Again , the degrees are arranged so that | |
| 17:51 | they canceled . All right . So what are you | |
| 17:54 | going to get on the top ? You're gonna have | |
| 17:56 | uh 30 times two pies , you have 30 times | |
| 18:00 | two pi uh radiance On the top and on the | |
| 18:05 | bottom you'll have 360°. . Well , you don't have | |
| 18:10 | degrees , degrees have canceled . So you just have | |
| 18:12 | 360 . Okay , Do you see what's going on | |
| 18:15 | here ? I can basically say the two . I | |
| 18:17 | can do whatever I want . I can multiply this | |
| 18:20 | two times three uh is gonna be that's gonna be | |
| 18:22 | 60 degrees and I can do the exact same math | |
| 18:25 | there . Or I can just cancel and say two | |
| 18:27 | divided by two is 13 60 divided by two is | |
| 18:29 | 1 80 . And you see what I've got is | |
| 18:32 | the same thing 30 pi over 1 80 which is | |
| 18:36 | exactly what we got before . So it's going to | |
| 18:38 | reduce the pi over six radiance . So no matter | |
| 18:41 | what conversion factor you use , 180 being pie radiance | |
| 18:45 | or 360 degrees being two pi radiance . It doesn't | |
| 18:48 | matter either one is going to give you the same | |
| 18:50 | answer because of the same , there's the same ratio | |
| 18:52 | , the same conversion factor . So let's go and | |
| 18:56 | work through the rest of the most important angles . | |
| 18:59 | We already did 30°. . Now let's convert 45° to | |
| 19:03 | radiance and let's see what we come up with . | |
| 19:05 | So we say 45° , We draw a horizontal line | |
| 19:11 | , draw a vertical line and we have to use | |
| 19:12 | a conversion factor . The conversion factor we're gonna use | |
| 19:14 | is 180° is equal to pi radiance . We want | |
| 19:18 | to arrange it as 180° on the bottom so the | |
| 19:20 | degree units will cancel and the pie radiance on the | |
| 19:24 | top because that's the unit we want to have in | |
| 19:26 | our final answer . So again we see that the | |
| 19:29 | degrees cancels with the degrees . So what you're going | |
| 19:33 | to have is 45 times pi Times Pi divided by | |
| 19:40 | the 180 . And that unit is going to be | |
| 19:43 | radiant measure . All right . Now , if you | |
| 19:46 | think about it , if you grab a calculator and | |
| 19:48 | just kind of verify with me , you can divide | |
| 19:50 | the top by 45 and get a one and you | |
| 19:52 | can take 180 and also divide by 45 and get | |
| 19:55 | four . So basically I'm dividing the top of the | |
| 19:57 | bottom of the fraction by 45 . And so what | |
| 20:00 | you get at the end of the day only pies | |
| 20:02 | left on the top four is left on the bottom | |
| 20:05 | radiance . So what we have learned is that 45° | |
| 20:08 | is pi over four radiance so that we can go | |
| 20:11 | over here to our unit circle and say all right | |
| 20:14 | now we've learned that 45 degrees is now pi over | |
| 20:17 | four . So you're right this is pi over four | |
| 20:19 | radiance in this location . So essentially the unit circle | |
| 20:22 | is going to stay the same . Nothing changes as | |
| 20:24 | far as the signs and the coastlines of the angles | |
| 20:27 | . But we can use degree measures if we want | |
| 20:30 | but soon we'll be using radiant measures to to the | |
| 20:32 | signs and the coastlines but the exact same unit circle | |
| 20:35 | applies . It's not anything different Thai over four is | |
| 20:38 | 45°. . All right let's continue on . We'll continue | |
| 20:43 | on doing the most important ones here . What about | |
| 20:45 | 60°. . Now let me ask you just one question | |
| 20:49 | really quickly before we actually convert it . Okay so | |
| 20:52 | this is 30 degrees you know that double of 30 | |
| 20:55 | or 60 . So if we want to convert 60 | |
| 20:58 | and we know that double of 30 or 60 degrees | |
| 21:00 | what do you think the radiant measure is gonna be | |
| 21:02 | ? It's gonna be double of what that radiant measure | |
| 21:05 | is . It's just an angle measure in a different | |
| 21:07 | unit . So if this is pi over six radiance | |
| 21:10 | this angle measure and ratings should just be double of | |
| 21:12 | that . Let's see if it actually works out that | |
| 21:14 | way . Right ? What we do is we say | |
| 21:17 | we start out with 60 degrees D . E . | |
| 21:19 | G . Horizontal line , vertical bar . We say | |
| 21:23 | that pie radiance Is equal to 180° 180°. . Um | |
| 21:31 | and actually you know what just for variety let's do | |
| 21:34 | it a little differently . I told you that we | |
| 21:36 | can do pie radiance is 100 and uh 180°. . | |
| 21:41 | But I also said that we can use the other | |
| 21:42 | conversion factor . So just to mix it up a | |
| 21:44 | little bit let's make it two pi is equal to | |
| 21:47 | 360° because it's the same exact conversion factor . Uh | |
| 21:51 | let's just see how this works out . So what | |
| 21:53 | we're gonna have then is we will have uh 60 | |
| 21:58 | Times The two Pi . Yeah All divided by the | |
| 22:01 | 360 . Everything on the bottom gets divided the degrees | |
| 22:06 | , cancels with the degree . So the only unit | |
| 22:08 | left we have is radiant . So all I have | |
| 22:09 | to do is do this multiplication . Okay , so | |
| 22:12 | you can do it however you want . But if | |
| 22:14 | I tell you that , Hey , I have a | |
| 22:16 | two on the bottom . So let me just divide | |
| 22:18 | two divided by two . And then I'll divide this | |
| 22:20 | by to give me 180 . You can see what's | |
| 22:23 | going to happen on the top . It's going to | |
| 22:25 | be 60 Pi over 180 radiance . But then I | |
| 22:31 | realized I can simplify this further , Right ? Because | |
| 22:35 | I can divide by 60 giving me one and 180 | |
| 22:39 | divided by 60 is going to be three . I'm | |
| 22:42 | just dividing top and bottom by 60 . So what | |
| 22:44 | I'm going to get is pi over three radiance And | |
| 22:49 | that was equal to 60° pi over three radiance . | |
| 22:52 | All right , so let's go over here and write | |
| 22:56 | it down . And we have pi over three radiant | |
| 23:03 | . So , the question was , is this pi | |
| 23:05 | over three actually , double pi over six . We'll | |
| 23:08 | think about it . It should be double . Right | |
| 23:11 | ? So , I could say pi over six times | |
| 23:14 | two . That would be doubling it . Right . | |
| 23:16 | So it would be to pi over six . And | |
| 23:20 | if I simplify this fraction , what do I get | |
| 23:22 | pi over three ? So , it is double . | |
| 23:24 | So this is the radiant measure of 30 degrees . | |
| 23:27 | It's five or six . If I double it to | |
| 23:28 | get up here , it's pi over three . Okay | |
| 23:30 | , we're gonna come back to that in just a | |
| 23:32 | second because there's more that I want to talk about | |
| 23:35 | when it comes to this , but when it comes | |
| 23:36 | to that . All right . The next probably the | |
| 23:38 | most important angle measure that we're gonna be using is | |
| 23:41 | 90°. . So , let's turn 90° into a radiant | |
| 23:47 | measure . So let's go ahead and use that 180 | |
| 23:51 | degrees is equal to pi radiance again . We arrange | |
| 23:56 | it like this and not upside down because the degrees | |
| 23:59 | have to be on the top and the bottom to | |
| 24:00 | cancel so that we're left with radiance . So what | |
| 24:04 | we'll have is 90 times pi on the top , | |
| 24:07 | divided by 180 radiance . And then we try to | |
| 24:10 | simplify this fraction And of course we can do it | |
| 24:14 | 90 , divided by 90 is 180 , divided by | |
| 24:16 | 90 is too . So what we actually end up | |
| 24:18 | with is that 90 degrees is equal to pi over | |
| 24:24 | two radiance , pi over two radiance . So we | |
| 24:29 | come over here to our unit circle and we're just | |
| 24:31 | going to fill it in and say this is now | |
| 24:33 | equal to pi over two ratings is equal to 90°. | |
| 24:37 | . All right now , I want to stop for | |
| 24:39 | just a second because we are going to do more | |
| 24:40 | conversions in just just a minute . Okay . But | |
| 24:42 | before we go any further , it's important to stop | |
| 24:45 | and look at what we've actually figured out . We've | |
| 24:47 | said that every circle in existence in degree measure has | |
| 24:50 | 360°. . And you've learned to chop that 360° , | |
| 24:55 | You know , into 91 , 70 . And even | |
| 24:57 | to chop it further 30 , 60 and so on | |
| 25:00 | . Now we're using another system of measurement based on | |
| 25:03 | the circumference of a unit circle . And we're saying | |
| 25:06 | that there's two pi of these radiance that are in | |
| 25:09 | a circle because if a circle has a radius of | |
| 25:11 | one , its circumference is just two times pi right | |
| 25:14 | , so the farther around the circle you go , | |
| 25:17 | the more radiant you kind of are crossing on your | |
| 25:19 | way around to two pi of those radiance . Right | |
| 25:22 | then we said what would 30 degrees be in radian | |
| 25:25 | measure ? We converted it , we got pi over | |
| 25:26 | 6 45 . We converted it to pi over 4 | |
| 25:29 | 60 . We converted it to pi over three and | |
| 25:31 | 90 . We converted it to pi over two . | |
| 25:34 | So you kinda should commit this to memory . You | |
| 25:36 | need to commit these two memory . The other ones | |
| 25:38 | around the unit circle . I'm going to show you | |
| 25:40 | how to count in such a way that you won't | |
| 25:42 | have to memorize them . But the ones in quadrant | |
| 25:45 | one you should memorize . Here's an easy way to | |
| 25:47 | remember it . 45 degrees . Is the one with | |
| 25:50 | pie having a four on the bottom , that's easy | |
| 25:53 | to remember . 45 degrees is the only one that | |
| 25:55 | has a four on the bottom . So pi over | |
| 25:57 | four you should think oh that's 45 degrees because of | |
| 25:59 | four is on the bottom . Okay , 30 degrees | |
| 26:02 | you would think it would have a three on the | |
| 26:04 | bottom but actually it's backwards . So the way you | |
| 26:06 | remember it as you say 30 degrees , it's not | |
| 26:08 | pi over three . Its the other its pi over | |
| 26:11 | six and the same thing with this , the 60 | |
| 26:13 | degrees you would think of six would be on the | |
| 26:14 | bottom . It's really a three . So the way | |
| 26:16 | you remember these is the smaller number is the pi | |
| 26:19 | over six . So it has to be For 30°. | |
| 26:21 | . It's the other number six over here and for | |
| 26:23 | 60° it's the other number three under there . So | |
| 26:27 | it's a little bit cumbersome to remember . Pi over | |
| 26:28 | six has the smaller angle and pi over three is | |
| 26:31 | the bigger angle . But after a while you'll start | |
| 26:34 | to remember that and then finally pi over two is | |
| 26:37 | up here at 90 degrees . That's just one that | |
| 26:39 | you'll just have to remember and you'll remember pretty quickly | |
| 26:41 | that it's pi over two . All right . And | |
| 26:43 | honestly you can remember that is pi over two partly | |
| 26:45 | because when you look back at the one of the | |
| 26:49 | first conversions we did we said that 180° worked out | |
| 26:53 | to be pie radiance because all the way around was | |
| 26:56 | to pie . So halfway around is pie . And | |
| 26:59 | if I cut it in half again this has to | |
| 27:01 | be pi over two . So if this is pie | |
| 27:03 | it makes sense that this up here has to be | |
| 27:04 | half of that which is pi over two . So | |
| 27:07 | I want you to try to remember um remember these | |
| 27:12 | . And at the end of the lesson I'm gonna | |
| 27:15 | show you how counting by by radiant measure how counting | |
| 27:19 | works . But before I'm gonna kind of count and | |
| 27:21 | show you how it all works like that . I | |
| 27:23 | want to go and do a few more conversions going | |
| 27:24 | the other direction . So far we have taken degrees | |
| 27:29 | , convert to radiance degrees to radiance degrees to radiance | |
| 27:33 | . Now I want to go the other way and | |
| 27:34 | we want to take radiance and convert two degrees . | |
| 27:37 | So let's say we have three pi over two radiance | |
| 27:42 | and we want to go to degrees . How do | |
| 27:44 | we calculate this ? So let me show you how | |
| 27:46 | to do it . The thing you start with is | |
| 27:48 | what you write down first . The way you write | |
| 27:50 | it is like this three pi over two long bar | |
| 27:54 | and you put the unit right on top so three | |
| 27:57 | pi two on the bottom radiant over here . You | |
| 27:59 | draw your vertical line and here you don't have to | |
| 28:02 | worry about . Oh should I multiply shit . I | |
| 28:03 | divide . It's just all set up for you because | |
| 28:06 | the way you write it is you say that pie | |
| 28:08 | radiance Is 180°. . The reason you're writing it this | |
| 28:15 | way is because you have to make sure that the | |
| 28:17 | radiance are the thing that's canceling you want degrees left | |
| 28:20 | over . You're trying to go to degrees . So | |
| 28:22 | you cancel the radiance and you're left with this . | |
| 28:24 | You wouldn't flip it over like we had done before | |
| 28:26 | because if degrees were on the bottom , nothing would | |
| 28:28 | cancel at all . Alright . Also notice that . | |
| 28:32 | Well let me go and just finish the multiplication . | |
| 28:34 | It's gonna be three times pi times 1 80 . | |
| 28:38 | I'll write it like this three pi times 1 80 | |
| 28:40 | on the bottom . You have a two times a | |
| 28:42 | pie . All right . But now we can and | |
| 28:45 | then of course the unit is degrees . But notice | |
| 28:49 | that there's a pie on the top of the pile | |
| 28:51 | on the bottom . So they go away and you | |
| 28:53 | have a two and a 1 80 . So two | |
| 28:55 | divided by two is 11 80 divided by two uh | |
| 28:58 | is 90 . So really , what you have on | |
| 29:01 | the top is three times 90 . So three times | |
| 29:03 | nine is 227 . So three times 90 is 272 | |
| 29:08 | 170 degrees . So , what we figured out is | |
| 29:13 | three pi over two is 270 degrees . So let's | |
| 29:17 | go over here to the unit circle and let's mark | |
| 29:20 | this guy right here at 270 degrees . Three pi | |
| 29:23 | over to remember how I told you that every I | |
| 29:27 | don't like the way this three is written . So | |
| 29:29 | let me try to clean that up a little bit | |
| 29:30 | . Uh Every radiant measure around this unit circle is | |
| 29:34 | going to be a fraction of pie , every one | |
| 29:36 | of them . If you don't see a fraction of | |
| 29:38 | pie somewhere in a radiant measurement , then something's wrong | |
| 29:41 | . You've already done something wrong . So three pi | |
| 29:44 | over two . Uh and we've already figured out from | |
| 29:46 | before that 100 and 80 degrees was pie . So | |
| 29:50 | I can actually fill , fill this one in as | |
| 29:53 | well . I can say 180 degrees as pie radiance | |
| 29:56 | . And over here at zero , we can say | |
| 29:59 | that there's zero radiance here and we already know that | |
| 30:02 | there's two pi radiance and a whole circle . So | |
| 30:04 | instead of going to 3 60 when you go all | |
| 30:06 | the way around , you get to two pi radiance | |
| 30:09 | . So all the way around is two pi . | |
| 30:11 | You start at zero . Going to pi over six | |
| 30:13 | pi over four pi . Three pi over two . | |
| 30:15 | We're gonna fill the rest of these out in a | |
| 30:17 | future lesson but you have pie over here and three | |
| 30:19 | pi over two over there and I want to do | |
| 30:21 | one more conversion before I wrap the lesson up with | |
| 30:25 | showing you how to count properly . Uh in these | |
| 30:28 | uh in this unit system here which is gonna be | |
| 30:30 | really easy to understand to what if I want um | |
| 30:34 | five pi over four radiance and I want to go | |
| 30:40 | to degrees . Okay you set it up the same | |
| 30:43 | way you say five pi long bar over four . | |
| 30:48 | The unit is radiance and now you have to arrange | |
| 30:51 | your conversion factor , you have to have radiance on | |
| 30:53 | the bottom . So you say there are pie radiance | |
| 30:57 | In 180 degrees . Alright . The reason we arrange | |
| 31:03 | it like this is because ratings is on the top | |
| 31:05 | ratings on the bottom . So we got the units | |
| 31:07 | correct ? Now we multiply through . What do we | |
| 31:09 | have ? We have five times pi times 180 Then | |
| 31:16 | we have on the bottom four times pi . Just | |
| 31:20 | right . And then the unit is degrees . All | |
| 31:24 | right . So what do we have ? We have | |
| 31:26 | a pie on the top , cancels with a pile | |
| 31:28 | on the bottom . Uh And then what we see | |
| 31:30 | is four divided by four is 180 divided by four | |
| 31:34 | is 45 . When I divide this by four I | |
| 31:37 | get 45 and then five times 45 if you think | |
| 31:41 | about it is 225 . So really everything cancelled the | |
| 31:45 | pies canceled five times 45 is to 25 degrees . | |
| 31:49 | Okay . 225 degrees . So we figured out is | |
| 31:52 | five pi over four is 225 degrees . So 225 | |
| 31:58 | degrees is here . So it is five pi over | |
| 32:02 | four . So by now and we're not going to | |
| 32:05 | do everything in this lesson . But by now you | |
| 32:08 | can see that all around the unit circle is gonna | |
| 32:10 | be these weird fractions of pie and you might be | |
| 32:12 | tempted to memorize them . Like I told you at | |
| 32:15 | the beginning , don't memorize like a signing coastline of | |
| 32:18 | all these angles around you . You don't need to | |
| 32:20 | memorize things . You just need to really focus on | |
| 32:21 | quadrant . one same thing is true of here . | |
| 32:24 | Remember I did a lesson in the past on how | |
| 32:26 | to count by chunks of degree measures all the way | |
| 32:29 | around . I told you it was gonna be critical | |
| 32:31 | that you understand that for radiant measure . Let me | |
| 32:33 | get into it here . I'm gonna do a little | |
| 32:35 | bit of it now and we'll do some more of | |
| 32:36 | it in a later section . But it's irresistible to | |
| 32:39 | to uh to do it here as well . So | |
| 32:42 | let's start by counting by the easiest thing Pi over | |
| 32:45 | two . Right , so this is pi over two | |
| 32:48 | radiant measures . It is a chunk of of of | |
| 32:51 | an angle an angle measure . That's pi over two | |
| 32:54 | radiance . So we're gonna count in units of pi | |
| 32:57 | over two . So this is pi over two . | |
| 33:00 | And then what would happen if we counted this ? | |
| 33:01 | It would be two times at two times pi over | |
| 33:04 | two . What is two times pi over two ? | |
| 33:07 | Well the twos would cancel , it's gonna give me | |
| 33:09 | pie radiance . That's why this is pie radiance . | |
| 33:12 | So this is pi over two . This is two | |
| 33:15 | pi over twos . Then this is three pi over | |
| 33:18 | two's . Notice the angle that we actually have here | |
| 33:20 | is three pi over two . Then we count again | |
| 33:22 | . So here we go . Let's go again . | |
| 33:23 | One pi over 22 pi R two's . Three pi | |
| 33:26 | over two . So this would be four pi over | |
| 33:27 | two . What is four pi over two ? You | |
| 33:31 | divide here . What do you get to pie ? | |
| 33:33 | So you see these radiant measurements that are around there | |
| 33:37 | . The simplified fractions . You can count by radiance | |
| 33:40 | all the way around and just simplify the fractions and | |
| 33:42 | get whatever is written on the unit circle . So | |
| 33:44 | let's say I forget this is 3.2 . I don't | |
| 33:46 | remember but I remember everything in quadrant one . I | |
| 33:49 | know this is pie over the pie or two . | |
| 33:51 | So this is two pi over two . So this | |
| 33:53 | is three point or two so I don't have to | |
| 33:54 | remember that . I just count . Okay so let's | |
| 33:58 | do something else . Let's count by pi over fours | |
| 34:01 | . So if this is pi over four this angle | |
| 34:04 | measure then this would be to pi over four . | |
| 34:06 | So let's check that to pi over four . What | |
| 34:08 | would two pi ? Uh Two times pi before over | |
| 34:13 | four . B . Simplify this fraction . I'm gonna | |
| 34:15 | get pi over two . So you see I remember | |
| 34:17 | this but I also can count by pira force . | |
| 34:19 | Here's pi over four . There's two times pi over | |
| 34:22 | four which reduces to this . This is three times | |
| 34:24 | pi over four . And I'm gonna write this down | |
| 34:26 | later but it is going to be three times pi | |
| 34:28 | over four . This is four times pi over four | |
| 34:30 | . What is four times four pi over four . | |
| 34:34 | What do you get pie ? So this pie notice | |
| 34:37 | when we counted by pi over two . Is it | |
| 34:39 | reduced to pi ? If we count by pira force | |
| 34:41 | pi over four to pi over 43 pi over 44 | |
| 34:43 | pi . Before it also reduced the pie then we | |
| 34:46 | continue five pi over four . That's what's written here | |
| 34:49 | . This will be six pi over four . If | |
| 34:51 | you do six pi over four let's do it right | |
| 34:53 | here . Six pi over four . If you divide | |
| 34:57 | by two you'll get three pi over two . That's | |
| 35:00 | three pi over two . So this is six point | |
| 35:03 | before this will be seven pie before that's what I'll | |
| 35:05 | write it as in a minute in a future lesson | |
| 35:07 | . This will be a pie before what is eight | |
| 35:10 | pi over 48 or four is too so it's two | |
| 35:12 | pi So really to get to any place you want | |
| 35:15 | on the unit circle . Just if you're trying to | |
| 35:17 | count by a certain angle , just count around and | |
| 35:20 | wherever you land , then you simplify the fraction that's | |
| 35:23 | going to be the radiant measure there . So we | |
| 35:26 | can continue . This is pi over six to pi | |
| 35:29 | over six . That reduces to five or 33 pi | |
| 35:31 | over six . That's going to reduce the 5 to | |
| 35:33 | 45 or 655 or 665 or 665 or six reduces | |
| 35:39 | to pi 75 or 685 or six nine Pira 6 | |
| 35:43 | . If you do nine pi over six , it's | |
| 35:45 | going to reduce to three pi over two , 10 | |
| 35:48 | pi over 6 , 11 pi over six and 12 | |
| 35:51 | pira 6 , 12 pi over six again reduces to | |
| 35:54 | two pi one more time . We'll do it with | |
| 35:56 | a different one . Let's go with increments of pi | |
| 35:58 | over three . Here's pi over three . All right | |
| 36:00 | , which is a 60 degree angle . Here's pi | |
| 36:02 | over three to pi over three is going to be | |
| 36:04 | here . Right , then three pi over three is | |
| 36:07 | going to be here . That reduces to pi And | |
| 36:10 | then four pi over three would be here and then | |
| 36:12 | five pi over three would be here . And then | |
| 36:15 | six pi over three would be here . What ? | |
| 36:17 | Six pi over 3 . 6/3 is to two pi | |
| 36:20 | . So you see you can pick any radiant measure | |
| 36:22 | you want in quadrant one and just count around the | |
| 36:24 | unit circle in increments of that unit measure , wherever | |
| 36:28 | you land . Simplify the fraction . Then you're gonna | |
| 36:29 | have the radian measure at that point . So all | |
| 36:32 | of these angles that are gonna I'm gonna end up | |
| 36:33 | filling this chart in a future lesson . Right ? | |
| 36:36 | All of those measures that I'm going to write down | |
| 36:38 | that you see on your unit circle . You don't | |
| 36:40 | memorize them . You just start counting from here and | |
| 36:42 | you land somewhere . That's what usually happens if you | |
| 36:45 | know that . This is three pi over two . | |
| 36:46 | It's okay pi over 22 pi over 23 prior to | |
| 36:48 | this is 5.4 . It's 154 to 5435445455 Before you | |
| 36:54 | don't memorize those . I haven't memorized them . You | |
| 36:56 | get there by counting . So make sure you can | |
| 36:59 | convert everything that you know how to convert from degrees | |
| 37:01 | and radiance and radiance two degrees . So all of | |
| 37:04 | these problems yourself , make sure you kind of understand | |
| 37:06 | the concept here with the counting by radiant increments . | |
| 37:08 | We'll do some more practice problems in the next lesson | |
| 37:10 | and then we will finally discuss the full glory of | |
| 37:13 | the unit circle in radiant measure . |
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05 - What is a Radian Angle? Convert Degrees to Radians & Radians to Degrees - Part 1 is a free educational video by Math and Science.
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