 and what we need to do we need to recreate our circle right what I'm gonna do is I'm gonna move this loop a little bit closer to the top because that way I don't have more you know too much jiggle room right so let's do this again we got one two three four five six seven eight nine one two three four five six seven eight nine and we got nine here we got one two three four five six seven eight nine okay so we'll go a little slower on this one so that's not a that's a pretty little circle there I think that's one of the best ones I've ever done so what we did was we drew a circle when you draw a lot of circles lately right and this is a circle in space and for us to analyze a circle what we need to do is be more exact with our measurements and what we end up doing is we end up throwing a grid cartesian coordinate system on the circle with zero zero being at the center of the circle and as soon as we do that we put a cartesian coordinate system on there that we've done over the last I guess six videos five or six videos is what it does it basically breaks the circle into four quadrants right so let's throw our grid on here and again I'm going to bring out my trusty little lever right or level not lever level right and let's throw our grid on here and we're going to put an x and y axis right so talked about this a lot right our this is our x axis and for a unit circle this is our going to be our cos theta and this is our y axis and for a unit circle we're basically going to call this sine theta right now what we've done by creating this grid we've broken this thing down into four quadrants right one well I guess one two three four right and that allows us to analyze the circle and we talked about you know when we're talking about the trig ratios and we're graphing them basically if we're standing if we're standing at the edge here right at this point here as we move around the circle our sine theta is basically mapping out what our y coordinate is going to be right and our cos theta is mapping out what our x coordinate is going to be so this is you know sort of where we're left off and you know some of the stuff we've talked about and a lot more right and the way we should think about this is we're using the circle right now to allow us to analyze the ideal cyclic functions right sine cosine and tangent the three basic trig functions but the circle can be anything we want it to be right we can think of the circles any type of system anything that we want to take a look at and for us to analyze the circle what we end up doing is breaking it down into smaller segments breaking it down into a piecemeal and taking a look at what happens in certain certain segments of the system of the circle right so for example if this thing was you know us doing financial reporting for a corporation the corporation does you know every year they do quarterly reports right so for us to go from here to here that would be one quarter right because we've broken a whole system a year's worth of finances into quarters right so each one of these nodes right would be one quarterly report well beyond that what we want to do we want to take a look at maybe what happens on a monthly basis right monthly basis would be us taking the circle and breaking it down into 12 even pieces right and that would be taking a look at it on a monthly basis right it could be a pizza the circle or a piece of pie where people come in into you know you're throwing a party you want to give everyone an equal piece you break it down into you know number of people that there are right you give everyone a their equal share right so what we're going to do right now is break the circle down into smaller pieces and the first thing we're going to do is we're going to take one of these quadrants and break the quadrant in half right that makes sense that's the first first step if we're gonna if we're going to try to break the circle down we're not going to go and break it down into a hundred pieces right away we're going to break this quadrant down into two pieces right so right now what we have is right now what we have is going around the circle full circle is 360 degrees it's also equal to two pie right each quadrant ended up being 90 degrees right because 360 divided by four is 90 so we have one quadrant that's 90 degrees what we're going to do right now in this quadrant is take this 90 degrees and cut it in half which means we're going to have two 45 segments okay so let's draw that let's draw that line in where we're cutting this thing and what we're going to do is make a triangle again and what that triangle is going to be is going to be one of our special triangles okay so 45 degrees is you know if I had a protractor I you know put my protractor down here measure 45 degrees and go across right but that's easily done with 45 because all I have to do is just hit the grid where my grid the quad the crosshairs are right for the axes and the y's so I'm just going to go straight down 45 degrees so let's go down here and I'm just going to do this in blue so it comes out nice and dark right so right now this angle here is 45 degrees right I'm going to close off my triangle I'm going to make this a 90 degree triangle what's leveled so what we've done right now is broken 90 degrees into 45 and 45 so we've taken one segment and broken it down into two pieces right now what happens with right angle triangles is this guy's a right angle right and for any triangle the sum of the angles in a triangle is 180 degrees so if this is 45 that's 90 degrees this angle up here ends up being 45 degrees and this is our first special triangle that we have to learn okay so this is 45 degrees as well okay now what we end up doing for our calculations we want to work in radians right for our degree measurements because radians are more useful and we can do a simple calculation to figure out what 45 degrees is in radians we can either use our formula that we had before which was 360 degrees over 2 pi has to equal whatever degrees we have here over radians right and since we're trying to figure out what 45 degrees is in radians we can put in 45 here and do our cross multiplication and we did a couple of calculations with this in a previous video right but there's an easier way or a quicker way to figure out what 45 degrees is in radians and this is a method that we're going to use to generate the table as we move around the circle for our two special triangles this is the first one 45 90 45 the other one is going to be 30 90 and 60 or 30 16 90 right 45 45 90 the the calculation that we're going to use to figure out to generate the table of what our coordinates are for those special triangles basically works like this to go half a circle is 180 degrees for us to get to 45 degrees we divide 180 by 4 since 180 is the same as pi in radians we have to be consistent so we're going to divide pi by 4 to get to 45 so 45 degrees is pi over 4 i hope that makes sense we'll delve a lot deeper into this right so 180 degrees divided by 4 gives you 45 degrees right well 180 degrees is the same thing as pi in radians right so 45 degrees is going to be pi divided by 4 we have to be consistent math is very symmetrical right so 45 degrees ends up being pi over 4 and that's us right now breaking this quadrant into two pieces so we've got 45 45 and we can continue this and break this circle or the system each quadrant into halves into two pieces so all of a sudden we've gone from having four pieces right we started off with a circle which was one piece no grid on there we put a grid on there breaks it down into four pieces right 90 90 90 90 gives us 360 now what we did was broke down the 90 into two even pieces right now we're going to have eight pieces right one two three four five six seven eight pretty simple eight times 45 360 right i hope so anyway so what we've done we've broken our circle down into eight pieces and what we do is we're going to start analyzing or taking a look at what our coordinate is when we get to here right when we get to 45 degrees around the circle when we get to 90 degrees when we get to 45 plus 90 is 135 when we get to 135 degrees 180 180 plus 45 225 270 315 right so we're gonna those are some of the special angles that we're going to look at that we generate from the special triangle okay now the other special triangle that we have we broke this thing down into two pieces from one piece right 90 degrees to two pieces well what we're going to do is break it down into three pieces right it makes sense we're going to work our way down right get break it down into tighter smaller and smaller pieces later on maybe we could break it down into four pieces four even pieces right or five even pieces or six even pieces all of those could be our special triangles depending on you know what type of system we're analyzing for us right now and in general in math 12 or basically high school mathematics the two special triangles we're going to take a look at is 45 45 90 and the other one if we break down 90 degrees into three even pieces what happens is we need 30 degrees here right because 90 divided by three is 30 so what we end up doing is taking a look at a 30 degree triangle here and we have a 90 here so 30 plus 90 is 120 some of the angles in a triangle have to equal 180 so this angle up here ends up being 60 degrees so the other special triangle we have is 30 60 90 and what I'm going to do so we're not busy in this quadrant I'm going to break this quadrant down into three even pieces okay and we're going to do approximate for this one because again it's it's not as easy as the 45 because 45 all I'm doing is going down the crosshairs of the grits right easy to make sure it's 45 degrees for this one I'm going to approximate it if I had a protractor I lay down my protractor and measure off 30 degrees and I would have our you know the special triangles right so this is going to be let's see so that should be about the pieces so let's assume this is three even pieces this quadrant right that means each one of these angles is 30 degrees so I just broke this into two pieces we can just imagine me breaking this down into three pieces right and each angle is going to be 30 degrees same deal right I can do the same thing down here and we will be doing the same thing down there and down here right so each one of these angles is 30 degrees right now I'm not going to bother putting the angle here because it's going to get messy and what I'm going to do I'm going to close off my triangle here right generate a triangle here so this ends up being this angle here is 90 degrees so 30 plus 90 is 120 subtract that from 180 you get 60 now we're going to take a look at these guys and find out what they are in radians and what we could do again is use our formula that we've got right we could use this to do our cross multiplication put 30 degrees here do cross multiplication find out what 30 degrees is in radians and we could do the same thing with 60 degrees right but yeah let's use this method because we're going to end up using this method a lot more to generate the table and analyze our coordinates what happens to us as we move around the circle right so we're going to do the same thing from here to go to here half a circle is 180 degrees so we ask ourselves what do we do to 180 to get the 30 and the answer is divide by six right so 180 divided by six gives us 30 so 180 degrees divided by six gives us 30 degrees right 18 divided by six is three and you add a zero right well to be consistent to be symmetrical 180 is pi so I'm going to divide pi by six so 30 degrees is pi over six 60 degrees what do we do to 180 degrees to get the 60 what do we do to 180 to get the 60 we divide by three right so 180 divided by three is 60 degrees well that's the same thing here pi divided by three that's equivalent to 60 degrees so 60 degrees is pi over three and this same thing works here if I closed off this triangle this is 30 here and 30 here so that makes it 60 if I close off this triangle I'm going to have 90 degrees here so 60 90 that gives me 150 so this guy has to be 30 so I'm not going to draw this triangle here because it's going to be too busy okay so what happens is another 60 30 90 or 30 60 90 is going to break this thing into our three pieces that we want right and the beauty of triangles is 30 60 90 is the same thing as 60 30 90 right the ratios stay the same if they're basically similar triangles right just flipped a little bit so this right now we took a full circle one circle we broke it down into four pieces we took those four pieces we broke each four piece down to two pieces so all of a sudden our circles got eight pieces and we can take our circle and break each quadrant down into three pieces so if we do the whole circle with 30 60 90 triangle with this special triangle what we end up having is 12 pieces right three three three three so four times three is 12 pieces so every stop that we make is one-twelfth of the way if we're doing doing stops here right for the whole circle or we're going to do a combination of each right and that's the way you should really think about it is and that's the way we're going to lay down our table that we're going to lay down for us to analyze what happens with our coordinate system with sine theta cos theta tan theta as we move around the circle okay so basically if we you know put 45 degrees and 30 and 60 degrees this one into one quadrant what we end up having is this it's going to look like this this guy's going to go straight down so this thing right now is breaking this quadrant down into two pieces and what i'm going to do i'm going to do 30 degrees as well break this quadrant down into three pieces and it's going to look something like this anyway approximately because i don't have my projector here one two three four and a bit so this thing's going to look and again it's really important to keep in mind that we could have decided to break each quadrant down into four even pieces and that would be a special triangle of five even pieces or six even pieces whatever whatever we need to analyze whatever system we want to analyze right because right now we're taking looking at a generic system which is basically a circle right it's the most simplistic system that we can come up with um to analyze cyclic functions or any type of system right uh we're just going to start off from you know having one and then having you know if you wanted to you could have two right just draw an x-axis and that would have been hemispheres right two two sides of a circle and then we do a y-axis we're breaking four pieces break into eight pieces break into 12 pieces whatever whatever whatever right and what we're going to do is in the future video generate the table to find out what our sine cos and tan is going to be for every stop we make and we're going to put 30 degrees here at these locations so what we're going to do right now is figure out what sine cos and tan are for this special triangle 45 90 45 and for this triangle 30 60 90 which is the same thing as 60 30 90 right so let's take this down and keep this in mind this is pretty important so we're going to come back to this and dial a little deeper into this okay when we start generating a table for this thing but right now what we're going to do is draw our special triangles here and um take a look at what our trig ratios are for these special triangles