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CCSS.Math:

so looks let's explore the unit circle a little bit more depth so let's just start with some angle theta and for the sake of this video we'll assume everything is in radians so this angle right over here we would call this theta and now let's flip this I guess we could say the terminal ray of this angle let's flip it over the x and y axes so let's just make sure we've labeled our axes so let's flip it over the positive x axis so if you flip it over the positive x axis you just go straight down then you go the same distance on the other side you get to that point right over there and so you would get this ray you would get this ray that I'm attempting to draw in blue you would get that ray right over there now what is the angle between this ray and the positive x axis if you start at the positive x axis well just using our conventions that counterclockwise is from the x axis is a positive angle this is clockwise so instead of going theta above the x axis we're going theta below so we would call this by our convention an angle of negative theta now let's flip our original green ray let's flip it over the positive y axis so if you flip it over the positive Y axis we're going to go from there all the way to right over there and we can draw we can draw ourselves arrey so my best attempt at that is right over there and what would be the measure of this angle right over here what was the measure of that angle in radians well we know if we were to go all the way to the positive from the positive x-axis to the negative x-axis that would be pi radians because that's halfway around the circle so this angle since we know that that's Theta this is Theta right over here the angle that we want to figure out this is going to be all the way around it's going to be pi minus it's going to be PI minus theta notice pi minus theta plus theta these two are these two are supplementary and they add up to PI radians or 100 80 degrees now let's flip this one over the negative x-axis so if we flip this one over the negative x-axis you can get right over there and so you're gonna get an angle that looks like this that looks like this and now what is going to be the measure of this angle so if we go all the way around like that what is the measure of that angle well to go this far is PI and then you're going another theta this angle right over here is Theta so you're going pi plus another theta so this whole angle right over here this whole thing this whole thing is PI plus theta radians so PI plus theta let me just write that down so this is PI plus theta now now that we've figured out these kind of you know these have different symmetries about them let's think about how the sines and cosines of these different angles relate to each other so we already know that this coordinate right over here that is sine of theta sorry the x coordinate is cosine of theta the x coordinate is cosine of theta and the y coordinate is sine of theta or another way of thinking about it is this value on the x axis is cosine of theta and this value right over here on the y axis is sine of theta now let's think about this one down over here by the same convention this point this is the really the unit circle definition of our trig functions this point since our angle is negative theta now this point would be cosine of negative theta comma sine of sine of negative theta and we can apply the same thing over here this point right over here the x-coordinate is cosine of pi minus theta that's what this angle is when we go from the positive x axis this is cosine of pi minus theta and the y coordinate is or is the sine of pi minus theta and then we could go all the way around to this point I think you see where this is going this is cosine of I could say theta plus PI or PI plus theta that's right PI plus theta and sine of PI plus theta and now how do these all relate to each other well notice over here out here on the right hand side our x coordinates are the exact same value it's this value right over here so we know that cosine of theta must be equal to the cosine of negative theta so that's pretty interesting let's write that down cosine of theta is equal is equal to I mean this blue color is equal to the cosine of negative theta that's a pretty interesting result but what about their signs well here the sine of theta is this distance above the x-axis and here the sine of negative theta is the same distance below the x-axis so they're going to be the negatives of each other so we could say that we could say that sine of negative theta sine of negative theta is equal to is equal to the negative sine of theta equal to the negative sine of theta it's the opposite if you go the same amount above or below the x-axis you're going to get the negative value for the sine and we could do the same thing over here how does this one relate to that well these two are going to have the same sine values right the sine of this the y-coordinate is the same as the sine of that so we see that this must be equal to that let's write that down so we get sine of theta sine of theta is equal to sine of PI minus sine of PI minus theta now let's think about how to the cosines relate what was the st. mark they're going to be the opposites of each other where are the x coordinates are the same distance but on opposite sides of the origin so we get cosine of theta cosine of theta is equal to the negative of the cosine let me do that same color actually let me make sure my colors are right so we get cosine of beta is equal to the negative of the cosine of pi minus theta and now finally let's think about how this one relates well here our cosine value our x-coordinate is the negative and our sine value is also the negative we've kind of flipped over both axes so let's write that down over here we have sine sine of theta plus pi which is the same thing as PI plus theta is equal to the negative of the sine of theta and we see that this is sine of theta this is sine of PI plus theta or sine of theta plus pi and we get the cosine cosine of theta plus PI cosine of theta plus pi is going to be the negative of cosine of theta is equal to the negative of cosine of theta so even here and you can see you could keep going you could try to relate this one to that one or that one to that when you can find all sorts of interesting results so I encourage you to to really try to think this through you're on your own and think about how all of these are related to each other based on essentially symmetries or reflections around the X or Y axis