 Let's talk about the idea of a reference angle. So what is a reference angle? So the reference angle of any angle theta which which is put in the standard position standard position here meaning of course that the Initial side coincides with the positive x-axis and the terminal side terminates well wherever it wants The reference angle of angle theta which we're in this lecture series We're going to note the reference angle as theta hat you draw a little hat symbol on top of the theta here This is not a universally accepted notation. Some people will do things like theta prime Or you could talk about like ref of theta or a bunch of other things. Some people don't even give a notation But just so you're aware in our lecture series if you ever see a theta hat that reference that will be the reference angle So the reference angle of angle theta is going to be the positive acute angle Form between the terminal side of theta and the x-axis So what I mean by that is if we have the terminal side right here and you have Excuse me the initial side right there and then the terminal side over here. So this is our angle theta Well, then the positive x-axis which would be this line right here What is the angle the positive acute angle form between the terminal side and the x-axis? You might get something over like this theta hat and so the reference angle in general is not the same angle as The angle theta itself although there are some exceptions, of course to that So for example, if you take a 30 degree angle notice that a 30 degree angle if you're in standard position Terminate in the first quadrant right here and which case you're then looking for the positive Acute angle form between the terminal side with the x-axis. That's going to be the same angle So in the first quadrant The in the first quadrant the reference angle is just the original angle There's no difference whatsoever. So in just make a little comment about this in q1 The reference angle is just the angle theta. They're one in the same thing. No nothing different there But if you move on to site think the second quadrant take a hundred and thirty five degree angle for for example This terminates here in the second quadrant you see that right there in the second quadrant the angle between the Terminal side in the positive axis is going to be formed right here And so if your angle start off with a hundred and thirty five degrees then the reference angle is gonna be 45 degrees How many more degrees do we need to complete the hundred eighty degrees to complete the line? That's gonna be 45 degrees So in particular if you're in the second quadrant you get that the reference angle theta is gonna equal a hundred and eighty degrees Minus theta that is in the second quadrant the reference angle is just the supplement of the given angle Okay Let's move on here to example C Example C is an example of an angle that will terminate in the third quadrant So if you take theta to be 240 degrees that terminates here in the third quadrant Well, how do you compute the reference angle the reference angle will be the angle between the x-axis With the terminal side so that's a positive acute angle so we can see something like this So for 240 degrees the reference angle will be 60 degrees How did we find out that angle right here? Well if you're in the third quadrant you have to figure out what portion is Past what angle measure is past 180 degrees? So in the third quadrant your reference angle is how far past 180 degrees are you so your theta hat here is gonna equal Theta minus 180 degrees like so and we could do some other examples of this Like what happens if you have an angle that terminated in the fourth quadrant something like this well If you're in the fourth quadrant the reference angle is gonna be how much more do you need to get to 360 degrees? So let's make a comment about that as well So in quadrant 4 your reference angle is gonna be how short of 360 degrees are you? So you take 360 degrees minus theta that would give you the reference angle right there And so as long as your angle is between Zero degrees and 360 degrees you can use these strategies these four formulas that put on the screen here to compute any reference angle But what if you're larger than 360 degrees? What if you're less than zero like what if you have a negative angle for example? consider example D here if if you take D to excuse me take theta to be negative 210 degrees then the reference angle is gonna be 30 degrees notice that negative 210 degrees you wrap backwards Negative degree measures mean a clockwise rotation. This would actually terminate in the second quadrant right here, but of course The reference angle is gonna be 30 degrees you go you go negative Well, excuse me. You went negative 30 degrees past the 180 degree mark So how do you get that? Well again one strategy is always just to switch it over to something that's in You know between zero and 306 degrees so if we added 360 degrees to this we end up with of course 150 degrees in which case that terminates in the second quadrant in which case you get 30 degrees from there. So geometrically it makes sense but from a computational point of view, it's almost just easier to always just put something between zero and 360 degrees Two other examples is take theta to be negative 140 degrees that terminates of course in the third quadrant the difference between negative 140 and To all get all the way up to this right here be another 40 degrees You can see that again geometrically very simply if you want to do it from a numerical point of view One strategy will just be add to it 360 degrees right so put it between the range zero and 360 if you did that of course That's going to give you 220 degrees like so for which how far past 180 degrees is 220 220 degrees minus 180 degrees is going to be 40 degrees Which is the reference angle? One last example F here. What if you take something larger than 360 degrees theta equals 540 that means we went around Once and then we went into their half spin So 540 degrees is a spin and a half and so it's going to terminate on the left x-axis right there What's the angle between it? Well, if you're at all confused maybe subtract 360 from it to help you out there That's going to give you 180 degrees and then remember or if you're since that this will stop at 180 degrees Are you the second quadrant of the third quadrant kind of both doesn't really matter You either take 180 minus 180 or 180 minus 180. Oh, it's the same thing You're going to get to the reference angle here is zero degrees because there's no gap between 450 and that so the reference angles again If you're between zero and 360 degrees you can use these formula right here to help you compute the reference angle And if you're outside the zero and 360 range, you're something co-terminal to that Just then compute a corresponding angle to that is between zero and 360 degrees And this will help you compute these reference angles Which are very critical in sugar on tree Which is something we'll talk about of course in the next video about the reference angle theorem