now let's talk about quotient identities you need to know that tangent Theta is equal to sin / cosine andent Theta is equal to cosine over s now keep in mind tangent is also y ID X and coent is X over y and used in the reciprocal identities coent is 1 / tangent so those are some formulas you need to know so these two are known as the quotient identities when you hear the word quotient think of division you're dividing two things so let's say if sin Theta is 4 / 5 and cosine Theta is 3 over 5 what is the value of tangent Theta and coent Theta so tangent is simply s / cosine so it's 4 over 5 / 3 over 5 now you can write that like this 4 over 5 / 3 over 5 and then use Keep Change Flip you keep the first fraction change division to multiplication and flip the second fraction and you're going to get 4 over3 as your answer another way in which you can get the same answer which is a technique I like to use is when you get a complex fraction like this eliminate the denominator by multiplying the top and bottom by five if you do that notice that you'll be left over with 4 over three and I think it's much faster to do it that way now once you have tangent the best way to find Cent you could use cosine over sign to get the answer but just realize that cang is a reciprocal of tangent so you just got to flip 43 and it becomes 3 4 that's the fastest way to get Coan once you have now let's say that sin Theta is equal to 5 / 13 and cosine Theta is 12 / 13 find the values of tangent and coent tangent Theta is going to be equal to s / cosine so s is 5 13 cosine is 12 over 13 so let's multiply the top and the Bottom by 13 so these will cancel and those will cancel and so this is going to be 5 over 12 and then coent it's simply 1 / tangent which is is 1 over 52 and that becomes 12 over 5 so that's the value of tangent I mean coent and tangent now keep in mind you can always do it this way you can say coent is cosine of a sign and cosine is 12 over 13 s is 5 over 13 and you know the 13s will cancel and this will also give you 12 / 5 so that's coent and this particular problem now let's say if we want to find a value of tangent Pi / 4 how can we do so so we need to go back to the unit circle and we need to find the values that correspond to pi over 4 so at pi over 4 we have the point < tk2 over 2 comma < tk2 over2 Now tangent we know is s / cosine and we know that s corresponds to the Y value and cosine corresponds to the x value so therefore tangent is simply YX in this case Y and X are both the same so tangent Pi 4 is going to be < tk2 divid two / itself which they cancel and give you one so that's the value of tangent Pi 4 let's try another one what is the value of tangent 2 Pi / 3 so feel free to pause the video and work on this example 2 pi over 3 is in quadrant 2 and a reference angle of 2 pi over 3 is pi 3 which is in quadrant one now the point that corresponds Tok / 3 is 12 comma < tk3 / 2 and in quadrant 2 x is negative but Y is positive now that we have that we can evaluate tangent so we know tangent is going to be S over cosine which is simply y / X it might be easier just to use that so here's the yv value and here's the x value so that's going to be < tk3 / 2 over /2 so let's multiply the top and the Bottom by two so then this will become < tk3 / 1 so the final answer is < tk3 so that's the value of tangent 2 pi 3 now what about coent 4 Pi / 3 try that one so 4 pi over 3 is located in quadrant 3 and a reference angle for that is pi 3 so we know the points for pi over 3 the x value is 12 and the Y value is < tk3 / 2 so then it corresponds to this point where the x value is -2 and the Y value is < TK 3 / 2 X and Y are both negative in quadry now coent is cosine / s and we know that cosine is associated with the x value and S is associated with the Y value soent is basically x / y so so here's the x value and this is the Y value so we got to put -2 on top /3 over two so let's multiply the top and the Bottom by two to get rid of the twos so what we now have is -1 / < tk3 which is POS 1 over posi < tk3 the two negative signs will cancel now the last thing we need to do is rationalize so let's multiply the top and the bottom byun3 so this becomes theun 3 / 3 and so that's cent of 4 pi over 3 it's positive < tk3 over 3 try this one tangent of 210 so if you have a negative angle just go ahead and make it positive by adding 360 if it's in radians add 2 pi to it to make it positive so this is going to correlate to 150 150 is in quadrant 2 so this is 150 which is which also relates to a -210 now the reference angle is going to be 180 minus the angle in quadrant 2 which is 150 and so that's 30 so once you have the reference angle you know what point it corresponds to so an angle of 30 which is the same as pi/ 6 that corresponds to the point < tk3 over 2 comma 12 and so 150 and -210 will correspond to the point negative3 over2 positive 12 in quadrant 2 remember X is negative Y is positive as you travel towards the left X is negative but as you go up you have a positive yv value now let's go ahead and evaluate tangent we know that tangent is y / X Y is 12 x is < tk3 over2 so this is what we have and we can cancel the twos you can multiply the top and bottom by two if you want but this is faster so it becomes POS 1 / byun3 so we need to rationalize and the final answer is < tk3 / 3 now what is tangent of 0 de and tangent of 90 go ahead and find those two so here's zero here's 90 at Z we know the point is 1 comma 0 and at 90 the point is 01 so tangent is y / X so Y in this case is 1 and X is zero so it's 1 over actually I'm using the wrong point I need to use 0 degrees which is here so this is the Y value that we need to use and this is the x value we're looking for tangent of 0 degrees so the Y value at 0 degrees is 0 the x value is one 0id 1 is 0 so tangent 0 is z now tangent 90 we need to use these values so here's the Y value here is the x value so it's YX which is going to be 1/ Z and anytime you have a zero in the denominator of a fraction the value is undefined So Tan 90 is undefined but tangent Z is z go ahead and try cotangent of 180 and also cotangent of 3 pi ID two so we know 180 is the negative xais 3 pi/ 2 which is the same as 270 that's the negative y AIS and this corresponds to a point of 01 and this point is 1 0 now coent is x / y tangent is YX and so need to use the point that corresponds to 180 so this is X this is y therefore this is going to be -1 /0 which means coent of 180 is undefined now coent of 3 pi/ 2 it's going to be X over y just as before and this time x is zero but Y is 1 soent at 3 pi/2 is z so anytime you need to find T tent coent or secant or cosecant of an angle that's either 0 90 180 270 these are angles that are not in any particular quadrant they're on the X or Y axis so whenever you have one of those four functions tan Coan secant or cosecant the answer as you've seen is either zero or undefined