01 - Angles and Angle Measure in Degrees - Part 1 - Types of Angles & What is an Angle?

well welcome back the title of this lesson is called angles and degree measure this is part one so we're embarking on a completely new set of topics here which are actually incredibly crucial to your future education and math going into calculus going into engineering and more advanced topics here in math because we're going to be talking about angles we'll want to be talking about trigonometric functions sine cosine tangent we'll be talking about a lot of triangle trigonometry we'll be graphing trigonometric functions and all of this stuff boils into and kind of feeds into talking about wave motion because wave motion can be described using angles and and trigonometry concepts like this which has applications in every branch of engineering that you're going to study so I'm not exaggerating when I say what we're learning here is absolutely critical now the beginning of this journey starts with talking about what is an angle okay so what we're gonna do now is go over that right now in general when you're talking about angles there are two main ways there are others but there are two main ways that we measure angles we measure them in degrees that's what we're gonna be talking about here and then later on we're gonna be talking about another measure of angles called radians so I know you might want to might have confusion or be thinking about radians in the back of your mind but put the Radian questions off to the backburner I want to teach you how to do all the stuff in degrees first because it makes more sense then we're going to jump back in and talk about Radian angle measure in just a little bit okay so let's talk about what we all know and that is that a circle probably learn this you know back in geometry or even in earlier math the math has what we call 360 degrees so the question mark is what is a degree right well really what it is is if you take a circle and you chop going around that circle into 360 equal pieces then one of those little pie wedges of a circle after you chop it up like that is what we call a degree now why is it 360 degrees why isn't it 380 degrees or why is it you know not 270 degrees well that goes into the history of math and map making and other things and why that is the case we're not going to talk about the why we're gonna talk about what it is but ultimately the 300 sixty degrees is just the measure that we have landed on here it doesn't make a lot of intuitive sense why it's 360 degrees later on we talk about radians that will make a lot more sense why Radian measure is defying the way that it is but for now we're only talking about degrees and so we're just going to take it as a fact that the people that you know figured out geometry and map making and surveying and things like that in the past set on 360 degrees for a very good historical reason we're not going to talk about the historical reason here but what it basically means is if ice if I have a circle like this if I start at one point in the circle like this and if I travel all the way around like this all the way around back to my starting position so my arrowhead is right here and what I mean by that is if I kind of take from the radius here of the circle and start at this location and sweep out all the way like a clock all the way back to where I started with this is what we call 360 full divisions of that going around the circle we call that a degree and obviously we can talk about Radian measure which splits the circle into its gonna end up being two pi radians you see pi is involved when we have radians so we're gonna save radians for later because I want to get through the fundamental concepts first all right so this is a full circle right but angles in general like in triangles are not a full circle so what we have here for in general is if we have an acute angle or something then we might have an angle that goes up like this it's not a complete circle because this angle measure that we're gonna ultimately be talking about is only a portion of a circle you see the entire circle goes all the way around to 360 degrees and then this amount that this thing is open if you can imagine like a circle kind of going like the boundary of the circle kind of going like this this is just a fraction of going all the way around the circle so it's not 360 full degrees in this particular case it might be something like 45 degrees let's say and the angle measure is measured from you know this line up to this line right so you need to kind of start getting used to measuring these angles opening and closing in the following way we have two sides of this angle side right here it's called the initial side and then over here when we get up to where we're kind of going this part is called the terminal side initial and terminal now in this part of math when we kind of get out of basic geometry and we're starting to talk about really weird angle measures that go all the way around the circle we generally have to start measuring the angle from some known location so from here on out we're going to be measuring angles from the positive x-axis we have to have a starting point so we measured from the positive x-axis what I mean by that is this imagine that this is an XY plane and the x axis actually continues on here so this is the x axis right and so you can imagine this part right here this would be the y axis something like this so this going from the x axis to the y axis this is 90 degrees and about half way through there we stopped roughly halfway I'm a I'm not exact am i drawing here right halfway so this is half of 90 which makes this about 45 degrees and then of course you can continue past 90 and kind of go all the way around in which case you get back to 360 degrees so these are little angles that we've used and looked at in the past for instance in triangles but here when we get into the trigonometry and kind of more advanced learning we have even larger angles that we deal with and that can look a little weird at first so here we have this initial side instead of opening up and stopping somewhere here let's continue on and measure that angle all the way down let's just talk about an example coming down here so let's call this the terminal side now you might be tempted to talk about the angle measure in here and you of course can take a look at that angle right here but in general you want to start at the terminal of the initial side and when you want to go in a counterclockwise direction remember clocks go like this right so counterclockwise goes the other way so we want to take a look at the angle measure that goes through this so if over here is 90 degrees then over here would be 180 degrees that makes sense 180 degrees over here and this is just a little bit bigger than 180 degrees so in this example this is 225 degrees so it is perfectly valid to have an angle larger than 90 degrees and angle larger than 180 degrees now it's a little weird the first time you see that because usually in geometry you're only talking about little angles inside of a triangle and they're always these are often these little acute angles but in trigonometry and precalculus and calculus we have angles that go way beyond 90 degrees way beyond 180 degrees all the time so it is true that you do have a smaller angle in here we could define that by the way it would be 360 degrees minus 225 that would be what this angle is because going all the way around is 360 so if we subtract that away that would be that angle of course we could take a look at that but in general we're measuring counterclockwise from the initial to the terminal side and it's they could be large numbers like this which leads me to the next thing that I want to write down that's important okay so when you're measuring angles when you have something like clockwise measurement clockwise which means going the other way like this these are what we call negative angles that's the other weird thing when you get out of geometry and you start getting into trig and precalculus like this you can have negative angles and positive angles now in all of these example we start up here and we're going counterclockwise counterclockwise counterclockwise so the other direction like this is counterclockwise this is the direction we've been measuring for these guys right here so counter them put a C clockwise okay these guys are always going to be positive angles it's extremely important you understand that you know the basic ideas of algebra you learn what a positive number was that's going along the to the right going on the number line towards the right and negative values those are going the other direction pass zero on the number line same thing for Y positive is up negative is down you had to learn all that in the beginning so here we're talking about something similar it's extremely important for you to understand if I start at some initial side and I measure angles in the counterclockwise direction in this way I'm measuring by definition by convention positive angles that's why this is positive 45 and positive 225 but if I start at this initial side and I measure an angle below the axis like this and then I'm going the other direction in the clockwise direction I'm going to by definition define and be looking at negative angles so let me give you an example of that so if for instance you have the initial side right here so I'm going to call this initial remember they're always measured relative to the x-axis like this so the initial side and then I have an angle that comes down something like this but for whatever reason I do not take and look at this angle over here that's a positive angle instead I start here and I go and look at the angle below I start sweeping down below like this in a clockwise fashion this means this is a negative angle so it's negative 60 degrees in this case so when you see negative angles it just means angles that are measured below the x-axis and these angles of course you can measure negative angles all the way around right but if you start here and you go in the counterclockwise direction you're measuring positive angles so one direction you're measuring positive angles the other direction measuring negative angles you're never going to get away from this idea in Matt so you have to remember that okay so let's take a look at what this actual angle is we just said it's about negative 60 degrees that makes sense because if you imagine this vertical line being here this is about 90 degrees negative 90 degrees and so it's not quite there so it's a negative 60 degrees okay now what to describe something to you and that is the idea that we are measuring the angle from here to here and so it's a negative and this is if you get a protractor it's about negative 60 degrees but we're perfectly free to measure this angle in the other direction going the other way it's perfectly fine to define angles measured to different directions okay when you measure going below the x-axis it's negative measurement of angles when you measure one the other way it's a positive so how would you figure out what this positive angle is if I were to start here and measure in the positive direction okay how would I do that well you know that this that this line right here is the zero spot right and you know that you can go all the way around the circle and get back to 360 degrees right and you also know that this angle is negative 60 degrees so what I can do if I want to define this angle I'm gonna call it theta I can say that it's the angle I started with negative 60 and I can add 360 degrees to it because any angle this is important any angle that I add 360 degrees to it just brings me back to the exact same location because remember if I start anywhere on this circle like if I start here and add 360 degrees to that position I'm gonna end up exactly where I started if I start down here and add 360 degrees I'm going to end up exactly where I started if I have an angle of negative 60 degrees and I add 360 degrees to it I'm gonna get exactly where I am this arrow will be the same but the angle measure will actually be a positive number what will this angle B so negative 360 plus 60 you might have guessed is 300 positive 300 degrees so this angle here can actually be measure two ways I can measure it as a positive 300 degrees makes sense right because this is 90 degrees this is 180 this is 270 degrees going down and just a little bit past that is gonna be 300 degrees so that's 300 degrees but it's exactly the same angle measure as negative 60 degrees so to kind of bring that home let me shorten this this little leg right here just because the actual length of it doesn't matter and I'm gonna put kind of a nice conclusion down under here that's really important what we're basically saying is that 300 positive 300 degrees is the same angle that this little symbol means angle the same angle as a negative 60 degree angle in other words if I go to the grocery store and tell somebody hey I have a 300 degree angle or if I tell somebody hey I have a negative 60 degree angle I'm talking about exactly the same angle right just like five tenths is a fraction and it means exactly the same thing is one half but that also means exact Klee the same thing is 50 100 s but that means the same thing as 500 1000 so you see the pattern here or for 8 or 3 6 I can go on and on all day they all mean 1/2 so angle is the same way I can have multiple ways of representing the position of this terminal line right here someone positive someone negative angles in fact I'll take this opportunity to tell you then of course I started here I added 360 degrees to it might not this angle what if I add 360 degrees again I could add 360 degrees to the 300 degree angle what am I going to get 660 degrees so 360 plus the 300 is 660 so this angle also can be negative 60 degrees it could be 300 degrees or if I add 360 again it could be 660 degrees I can add another 360 and another 360 you know I can keep doing this forever and I can come up with many many many an infinite number of nut of angles measurements that actually correspond to exactly the same thing okay you have to kind of get used to that when we talk about this alright so we talked about angles how they're measured positive angles negative angles now we need to talk about quadrants quadrants of the XY plane all right because you're gonna be asked to be a giving or talking about the quadrants where these angles kind of exist in the XY plane okay so we'll talk about a couple right here so this is the x-axis this is the y-axis right so let's take a look at an angle that exists let's say right here now every angle has to have an initial side and a terminal side the initial side from now on unless I tell you otherwise the initial side is always going to be the x-axis the initial side is the x-axis and the initial side is the x-axis even here I kind of didn't draw it but the initial side was the x-axis right there so the angle that defines this guy right here starts from here and ends right here so you could say that that angle goes up something like this I can draw it really close if I want to or I could draw it far away it doesn't matter as long as I start somewhere and in somewhere that angle can be defined like this what is this angle well I'm going to tell you that this angle is 6 degrees okay now you have to go back and think about quadrants we talked about quadrants a long time ago this is quadrant number one this one over here is quadrant number two this one over here is quadrant three and then this one over here is quadrant four so it goes the same measurement the same direction of positive angle measures goes quadrant one quadrant two quadrant three quadrant four like this so if I asked you what quadrant is this angle in then it would be a simple answer you would say well the quality a terminal line here this is the initial and this is the terminal the terminal part of the angle is in quadrant one so we say this is quadrant one angle because you're gonna have some problems that'll basically say let me know what quadrant this angle is in all you do is look and see where the line the terminal line of the angle Falls that's the quadrant it's in okay but we've said that all angles can be measured positively or they could be negative measures how would you measure this angle in the negative fashion if I were to measure this angle in a negative fashion then I get the right color let's use this one if I measure this angle in a negative fashion I would have to start again at the x-axis but I would have to go the other direction all the way back to here like this and this angle would not be a positive angle it will be a negative angle because it's measured down going below in a clockwise fashion like that how can I figure out what this angle measure is well this is negative 90 degrees this is negative 180 degrees this is negative 270 degrees and this is just a little bit more than that and since you know that this is 60 and you know that this is another 90 you could probably guess what this angle is this angle is negative 300 degrees negative 300 degrees how do you know it's negative 300 because this is negative 90 negative 180 negative 270 and this has to be 30 more because this is 60 and 30 and 60s 90 hasn't been 30 more from 270 so it has to be a negative 300 so another way to say that I mean we kind of talked through it and that's all great how do we figure this out mathematically well if I take the angle that I think that it is negative 300 right and I add 360 degrees to it I think it's negative 300 I'm gonna add 360 degrees to it so I'm gonna spin all the way back around to where I started then what I'm gonna get is positive 60 degrees so another way of saying this is I know that this angle measure is also negative 300 because if it is negative 300 then that tells me where this is and if I add 360 degrees to it and I'm gonna spin all the way back around to where I start and I'm gonna get a positive 60 degree angle so this angle is 60 degrees positive or negative 300 degrees negative those are just two of infinite number of ways to write this angle because again if I know that this angle can be expressed to 60 I can add 360 again what am I gonna get 360 plus 360 plus 360 that would be another 360 plus 60 degrees would be another angle measure and then I can add another 360 and another 360 or whatever in other words I can add and subtract 360 degrees two angles all I want and continue to get new angle measurements that are telling me the exact same information where is the terminal end of that angle that's all that's all I care about okay but the point is is no matter if this is a positive 60 degree angle measure or if it's a negative 300 degree measure the terminal side here is still in quadrant one so it's a quadrant one angle oh that's all it matters to these kinds of problems okay so to bring that point home a little bit more let's do another one let's say I'm gonna have an angle measure that's a little bit easier to understand let's say x + y let's go ahead and draw another angle measure in here and let's draw it over something like this and let's say that this angle starting here measured in a positive direction is 120 and by the way I keep using this angle theta angle theta used it here angle theta I've probably wrote it down somewhere over here this variable this Greek letter theta right it just means angle so in generally we use X&Y and algebra we start getting into angles and trigonometry and precalculus we use theta for angles we also have other variables we use for angles but theta is the one that's used the most so if you see theta it just means heads-up there's an angle measurement coming all right so this is 120 degrees measured from here that makes sense because this is positive 90 and just 30 more is going to be of me 120 so what quadrant is this angle and this is the second quadrant okay if I take this and I add or subtract 360 degrees to it I will get different angle measures some positive some negative but it's still expressing where this is in quadrant two so it's a second quadrant angle okay let's take a look at another one this one's actually a little bit more interesting what if x and y like this what if my angle is what I want to draw it here down here what if the angle is not exactly over here or over here or over here over here but what if the angle is basically right on top of one of the axis like this so the angle measure going all the way around here this is 90 this is 180 this is 270 so this angle is 270 degrees what quadrant doesn't fall into well it's not the first quadrant it's not the second quadrant and it's not in the third or in the fourth quadrant it's right in the middle if one of the angle measures falls right on one of these lines right here it's not in any of those quadrants it's just called a quad rental angle quadrantal angle okay so the bottom line is you'll be asked some questions here's an angle tell me what quadrant it's in and you just have to look and see where it falls if the angle falls over here it's in the third quadrant if the angle falls over here it's in the fourth quadrant if the angles over here it's the second quadrant like this one angles over here are in the first quadrant if the angle falls on this line or on this line or on this line or on this line in other words 0 90 180 270 or even back around to 360 degrees then it's not in any of those quadrants we just say it's a quadrantal angle that's all that we say alright now we have gotten at least about halfway through this lesson we've learned the most important concepts but we have one more thing to talk about we have to talk about coterminal angle as a fancy sounding name but I've actually already taught you kind of what Kotex terminal angles really are coterminal angle just means that I can add and subtract 360 degrees to any angle I want and I'm going to get another angle that is exactly kind of in the same position and so those angles are kind of equivalent as far as like where they're at they have the same terminal point so we call them coterminal that's all it is and so let's talk about that it's a definition you'll need to know coterminal coterminal angle these are two angles where terminal side coincide all right and we've kind of done a lot of these over here so it's not going to be a huge surprise what we have here but let's take it one step at a time we're going to draw four pictures I'm gonna try to keep them all on the same board and try not to keep them too too large so I can make sure that you can see them all so let's say we have the first angle goes off like this and the angle here is 150 degrees okay first of all what quadrant is that angle at well this is quadrant one this is quadrant two so this is a second quadrant angle it's a positive angle measure because we're measuring counterclockwise like this this is 90 this is 180 so we're not quite to 180 so this is 150 it makes sense okay now what's gonna happen if we add 360 degrees to this angle we just said if this angle measure is here we can add 360 by just sweeping around a full circle from that in landing exactly where we start it's going to express the same location of that terminal side that's why it's called coterminal but the angle measure will be bigger than 150 because we're adding 360 to it so this is what it looks like okay the coterminal side is exactly in the same position so I'm going to do my best to draw more or less the same exact thing but here is that here's the weird thing what we do is we start at the the same initial side we go all the way to 150 but then we go right through and continue another 360 degrees landing on that terminal side so if you think about it from here to here is the 150 but then we add remember the line continued on around and it's kind of in a spiral the way we drew it another 360 degrees so what is this angle measure this is five hundred and ten degrees how do I know it's five hundred and ten because it's 150 plus 360 degrees and I'm gonna put a parenthesis one here I'm just adding one circle so it's one times 360 and so what I get is five ten now you might say why would I ever measure the angle is 510 degrees why wouldn't I just say 150 great question you're almost always gonna say 150 almost always we're kind of building the foundation here we we know that we're not gonna express 510 two degree angles very often but you need to know that it is exactly the same thing because sometimes when you're solving equations you get large angles but then you don't want it it's like a fraction you want to simplify it to the to the kind of the smallest number and you can and so you would need to know if I got 510 I'm just gonna subtract 360 from it and I'm gonna get the smaller angle that still expresses the exact same thing okay now we don't have to stop there we can of course add 360 degrees again if we want to and of course we want to because we're learning right so we're gonna take and say well the terminal side of this guy is the same physical location but now we're not going to go around one time we're gonna go around two times so what we do is we say well here's the 150 and then we go around one full circle again then we go a one full circle again back here so we had 150 then we went another full circle and added 360 then we have another full circle added another 360 what is that angle measure that is 870 degrees how do I know because they could take 150 degrees the initial angle and I can add two at two circles which is two times 360 so if you do this on a calculator 360 times two plus the 150 you're going to get 870 right so what we're trying to say is 870 degrees positive is the same thing as five hundred and ten degrees positive is the same thing as 150 degrees positive it always expresses exactly the same angle you're gonna have to get used to adding or subtracting 360 the angles because you end up having to do it a lot as you study this kind of math right now we've added 362 times and that was all wonderful but we can also subtract 360 degrees there's no there's absolutely no reason why we are Inuk we're not allowed to do that so we're going to show an example of that here same exact terminal angle as before but now instead of adding 360 when you add an angle you go in the positive direction of that angle measure okay here what I'm going to do is I'm going to say let's instead of taking a look at this as a hundred and fifty degrees positive let's measure the negative angle going down below here okay what would that angle measure be if I start here and I go this direction how would I get to that angle what would that thing will actually be well I know that this is 150 degrees positive right and instead of adding 360 what I'll do is I'll go the other direction and I'll subtract 360 so I'll subtract 360 degrees from it in fact just to make it a hundred percent clear what's going to happen here I'm going to subtract one full circle one times 360 and when you get that do that you get negative 210 degrees so this angle is a negative angle 210 degrees it's negative again because it goes down below the x-axis as a measurement in a clockwise fashion and it makes sense this is negative 90 this is negative 180 and this is just 30 degrees up from that from negative 180 going in that direction to give me this and I know that this other direction was 150 and so if I add the 150 plus the 210 I'm gonna get 360 so there's many ways of looking at it another way of looking at it is if you know this angle is 150 degrees like this instead of adding 360 to get a bigger number I'll start at 150 and I'll subtract 360 and when I subtract an angle I'm gonna have to go the other way remember negative angles are measured going clockwise and positive angles are measured counterclockwise so if I add 360 I'm gonna go this way but if I subtract 360 I'm gonna go in the negative direction down here and so when I subtract 360 that's why the ankle turns out to be negative because it's measured below the x-axis like that but wait there's more I can actually do one more time I don't need to just subtract 360 one time I can do it again okay and I'm getting to a punchline we're almost done but this kind of stuff is really important this foundational stuff so here's the terminal side right here and we went down and we said it was negative 210 degrees so I can say well I can go to negative 210 degrees but I don't have to stop there I'll just go another negative 360 degrees and stop where I be on the terminal angle here again so again I could say well I started at 150 degrees positive and instead instead of subtracting one circle of 360 to get this I'll just subtract two full circles of 360 degrees and what am I going to get 150 minus two times 360 I'll get negative 570 degrees so this angle measure is negative 570 degrees so let's take a minute to really understand what we've done here what we're basically saying is that when you look at angles and triangles that can geometry a more basic math or all these little bitty angles that are inside triangles acute angles mostly okay but when you get and grow out of that and start talking about trigonometry and precalculus and calculus the angles become much larger and in general you start at a initial position and you define the angle opening positive when you measure this way up to some final terminal position or you can measure the other direction going negative angles to the same terminal position whatever position the angle is in I can add 360 and get a larger angle I can add 360 again to get yet another larger angle but these three numbers represent the same physical location of the terminal side the angle which means these are coterminal angles likewise I can start at the same location instead of adding 360 going this way I can start this location and I can subtract which means I go the other way subtract 360 going here when I do that subtraction I get a negative angle which also again represents the same physical thing but I don't have to stop there I can subtract another 360 if I want to so I can start here and I can say well okay I'm going to subtract 360 and then I'm going to subtract 360 again from this angle again subtracting two full circles I get negative five seventy so I can add as many 360 degree cycles as I want and get the same coterminal angle or I can subtract as many 360 degree cycles as I want and get exactly the same coterminal angle this angle is exactly the same thing as this angle is the same as this this this they're all the same that's all I'm trying to say so to summarize the whole thing what we can really say is that all angles coterminal with 150 degrees can be written as follows 150 plus in times 360 degrees where n can be negative 2 let me do it this way dot dot dot negative 2 negative 1 0 1 2 dot dot dot so what it means as all integers in mathematical terms what it's saying is you start with some kind of baseline angle and I can add is meaning 360 degrees to it as I want if I add 0 360s to it then I just have my original angle if I add 1 360 to it then that we did those on the board if I add 2 & 3 & 4 & 5 we're gonna get bigger and bigger numbers or I could subtract 360 going the other way negative 1 negative 2 negative 3 and so on so there's an infinite number of ways I can express this angle measure in trigonometry and precalculus and calculus okay in general we don't care about all the other ones but it is important for you know they exist it's important for you know how this works so that you can solve more advanced problems so what we're going to do is wrap it up go into the next couple of lessons we're gonna solve real problems and we're gonna get a lot of practice it's very important for you to solve all of these yourself make sure you have confidence follow me into the next lesson we're going to continue on with angles and degree measure in trig precalculus and algebra

Channel: Math and Science

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Keywords: angles, angle measure, degrees, angle degrees, angles in math, trigonometry, precalculus, calculus, what is an angle, math angle, right angle, obtuse angle, acute angle, 360 degrees, angles in math definition, definition of angle, radians, radian measure, radian angle measurement, radian angles and quadrants, angle quadrants trigonometry, angle quadrant, quadrantal angles, complementary angles, supplementary angles, types of angles, degrees to radians, radians to degrees, math

Id: hy95VyPet-M

Length: 33min 15sec (1995 seconds)

Published: Mon Jun 08 2020