now let's talk about even and odd functions let's go over the even functions first cosine is an even function cosine ofx is equal to cosine of x and the reciprocal of cosine which is secant is also an even function secant ofx is equal to secant x so those are the even functions the other for our odd trigonometric functions so for example sin ofx is sinx notice that the sign changes and cosecant which is a reciprocal of s is also an a function cosecant Negative X is negative cose I was about to write cosine negative cosecant positive X tangent is also an odd function tangx X is tangent X and also cent ofx isang of X so those are the even and odd trigonometric functions now how do we apply this information so let's say if we want to find a value of cosine of -45° well we know that cosine ofx is simply positive cosine of positive X so therefore cosine -45 is equivalent to just cosine of 45 now using a unit circle we know that 45 corresponds to the point < tk2 / 2 comma < tk2 over2 so cosine 45 is < tk2 over2 and cosine 45 is the same thing < tk2 over2 so these are the same now what about s of < over 3 we know that s ofx is sin x so therefore S ofk 3 is sin 3 so at an angle of Pi 3 this corresponds to the point out of the unit circle which is uh 12 < tk3 / 2 and S is associated with the Y value so s of pi over 3 is positive < tk3 /2 but with a negative sign in front sinun 3 is going to be negative3 over2 and Pi 3 it's basically an angle in Quadrant 4 so this is piun over 3 and we know that sign is negative in the fourth quadrant and so does this is an agreement with the answer that we have negative3 2 now what about tangent of -3 pi over 4 let's try that one so tangent ofx is an odd function so this is Nega tangent of X so therefore this is going to be tangent of 3i 4 now at Pi of 4 we have the point < tk2 over2 comma < tk2 over2 Now at 3 Pi 4 which is in quadrant 2 we have a similar point but only X is negative Y is positive so if you recall tangent is going to be y / X in this case at 3 pi/ 4 we have < tk2 over2 / byun2 over2 so the square < tkos will cancel the twos will cancel and so we're left with < tk2 over 2 is just one but this one is negative so overall is going to going to be 1 so tangent 3un / 4 is1 but * is going to equal a positive value so tangent of Nega 2 pi 4 is positive 1 now let's make sense of it so here's 3 pi over 4 it's in quadrant 2 and in quadrant 2 we know that X is positive Y is negative so because tangent is y/x when you take a negative value and divide by a positive value you get a negative result that's why tangent 3 pi/ 4 was a negative answer now what about tangent of -3 pi/ 4 so to get3 pi4 we need to go in the other direction which is here and notice that3 pi/4 is in quadrant 3 and in quadrant 3 X and Y are both negative so tangent which is YX if you take a negative number and divide it by a negative number you're going to get a positive value and so that's why tangent 3i 4 was 1 but tangent 3i 4 was posi 1 which makes tangent an odd function now try this one cosecant of5 pi/ 6 now cosecant is reciprocal of s so therefore cosecant is going to be an odd function so this is equal to negative cosecant of positive 5 piun over 6 now 5 piun 6 can be found in quadrant 2 the reference angle of 5 pi/ 6 is Pi / 6 and we know the point for pi/ 6 it's < tk3 over2 comma 12 so 5 piun over 6 which is in quadrant 2 we only have to change the x coordinate now cosecant is 1/ S and S 5i 6 corresponds to the y value which is 12 so sin 5K 6 is positive2 and 1 /2 is 2 but overall it's -2 so therefore cosecant 5 piun 6 is -25 piun 6 is in quadrant 3 and in quadrant 3 the Y value is negative and so that's why cosecant 5 pi over 6 is going to be a a negative answer overall because if you actually plot this this will take it to quadrant 3 and in quadrant 3 Y is negative let's try secant of negative 11 pi/ 6 so feel free to pause the video and work on that example now secant is an even function because it's a reciprocal of cosine which is also even so secant x is the same as secant x so basically we're looking for a secant of positive 11 pi over 6 11 pi/ 6 is in Quadrant 4 and pi over 6 is in quadrant 1 in quadrants 1 and four cosine is positive and secant which is 1/ cosine is also positive there therefore our final answer should be positive now at pi/ 6 which is 30° we have this point so 11 pi over 6 only the yv value changes X is negative in quadrants 2 and three but X is positive in 1 and four and Y is negative in quadrant four right now this is in quadrant four Now using the reciprocal identity we know that secant is one cosine and cosine corresponds to the x value so at 11 Pi 6 cosine is root3 / 2 and 1 / root3 over 2 is going to be 2 over < tk3 and we know what to do here we need to rationalize so therefore the final answer is 2 < tk3 over 3 so that's see get1 piun 6 using the even and odd properties of trigonometric functions now let's try one last problem find the value of tangent of Pi so tangent is an odd function tangent Pi is Nega tangent of positive Pi now it turns out that positive pi and negative pi are the same this is positive pi and this is negative pi they both lead to the same point which is -1 comma 0 so let's evaluate negative tan Pi tangent is y/x y is z x is Nega 1 but once you have a zero on top the whole thing is going to be zero so Tang Pi or tan Pi which is YX the whole thing is just going to be zero but they're the same they equal to each other for