I'm a calculus Professor but often my students are not irritated by the calculus it's about the trigonometry that they use throughout calculus it feels like there's just so many things to memorize like you have all these special values and all these different crazy graphs and don't even get me started on the long list of trig identities it's just a lot and they can all feel a little bit random so in this video I want to share my favorite perspective on trigonometry and kind of review all the trigonometry that you might need need for calculus but hopefully in a way that means you don't have to memorize very much and just can understand a few basic concepts this video is sponsored by brilliant more about them at the end of the video I want to begin with the unit circle imagine I have a circle of radius one and I'm going to imagine a point on that Circle starts on the positive x-axis and rotates around in a counterclockwise way I will use Theta to denote the angle here and note that the fact that I'm starting from the positive xais and going counterclockwise this is just a convention that we're all going to agree on so that we can talk about the same things when it comes to our trig definitions and then I can create a right triangle by considering the vertical and horizontal components of my particular point and now I've done the setup to Define my trig functions I am going to define the horizontal and vertical components to be cosine of theta and S of theta respectively this is a definition of what I mean by cosine and S of of theta when I'm in the first quadrant now for a right triangle you might have previously used the word adjacent for being close to the angle opposite the opposite side away from the angle and hypotenuse for the long side and when doing this had seen the definitions that s is going to be the opposite over the hypotenuse and cosine is the adjacent over the hypotenuse that's great but for our purposes where I'm talking about a unit circle the hypotenuse is always one and this is going to mean that adjacent and opposite are cosine and S respectively and as we allow the angle Theta to increase the sides of the triangle can either point in the Positive Directions with respect to the X and Y AIS going out from the origin or along the negative directions and so s of theta and cos Theta are sometimes positive and sometimes negative now as for that angle of theta in calculus we pretty much always use radians for this and the basic idea here is that if if I imagine going the entire way around the circle this is going to be called 2 pi radians and this corresponds to the notion that the circumference of a circle is 2 pi * the radius so if it's a unit circle then it's just 2 pi * 1 or 2 pi and so the reason I like radian so much is that my circumference is 2 pi and the angle is 2 pi they exactly match or or if I just take a portion of it like like for example suppose I do a quarter of a circle this is Theta going to 2 Pi / 4 or 4 Pi / 2 and then the arc length from that quarter Circle also Pi / 2 there's this correspondence when you use radians and the consequence of this in calculus is that you avoid a bunch of weird stretching factors appearing if you try to do things in degrees you're going to have all these two pi divided 360° factors appearing it's really nice when you use radians but you do have to remember that if you're going to your calculator and typing anything in please make sure you've got the radian button selected on your calculator for anything that we're doing in calculus now one of the things that immediately we get from this definition of s and cosine is our first trig identity if I take a normal right triangle can just zoom on this one call it sides a b and c we have the Pythagorean theorem that a s + B2 equal c² so in our specific context where we have Co and S I get that cos squ plus sin squar is the hypotenuse squar or one this is the so-called Pythagorean trigonometric identity and we use this one all the time in calculus all right next up graphs remember how cosine of theta is thought of as the horizontal component and so if I just record that horizontal component as I rotate along then I just get the graph of cosine of theta playing it slowly when Theta is zero cosine of theta is going to be 1 but then as I rotate around to Pi / 2 that horizontal component shrinks down to zero rotating further to Pi it starts increasing now in the negative Direction so I'll say the graph goes down to - 1 rotating to 3 pi/ 2 it's going to go back to zero and then finally stretches back out to one as you get around to 2 pi similarly we can look at s of theta which is now plotting out the vertical component as the point rotates around the circle this is going to give the graph of s so the point is that you can always generate the S and cosine graphs and answer questions like cosine of 3i / 2 just by knowing the unit circle by the way you'll notice that the two graphs look really similar to each other just sort of shifted over so indeed this could be captured by another identity such as cosine of theta is just s of theta plus pi 2 it's it's kind of like the S of theta is playing catch up to the cosine of theta and just running a bit behind and again I don't memorize the formula like where exactly does the pi/ 2 Go is it plus or minus if you understand the unit circle and how to come up with the graphs you can figure these out each time on your own these are all sort of nice values of theta because as we'll see you can just compute cosine and sign of them and get really clear exact answers you don't have to put it into a calculator and get an approximation often people memorize a couple special triangles here and I want to show you how to come up with it so so let's take a look at this particular triangle and what I notice is the hypotenuse has length one as always but I don't know what the other side is going to be let's just call it something generic like a pi over4 is cutting things exactly in half so I have to have this isoceles triangle here by Pythagoras A2 + a s is equal to 1 so 2 a^2 is equal to 1 and if I solve this gives the value of a equal to 1/ < tk2 and mostly because that's less cumbersome to say out loud I actually threw this trick of similar triangles where I can always multiply a triangle by a stretching Factor so if I multiply this triangle by every side by the factor of < tk2 it gives me 1 one < tk2 so this is how I memorize this this is the 1 one < tk2 triangle and that's fine but be careful if you actually were to put it on the unit circle the unit circle has hypotenuse one and so I want to label the two different sides of it 1/ < tk2 1 over < tk2 if I wanted to compute something like for example s of pi over 4 thus I'd get 1/ < tk2 now for reasons that I've never understood a lot of high school teachers will look at this and say you should rationalize the denominator and by this they mean they don't want that that root two that irrational on the bottom so multiply the top and the Bottom by < tk2 give the value of < tk2 / 2 okay uh I think this is a little bit ridiculous I don't see any reason why you should rationalize your denominators I think it's actually in many cases in calculus more transparent to leave it precisely like this but you do you okay we should also do pi over 6 and pi over 3 uh quickly here I've got the pi over 6 triangle note that because one of the angles is pi over 6 the other one is pi over 2 that's the right angle then that means that in that top corner it has to be the value of pi over 3 in order for the sum of the interior angles to add up to Pi or 180° so the the pi over 6 and pi over 3 triangle will sort of do at the same time again you can memorize a special triangle but if you ever forget there is a little trick here uh in this case what we're going to do is put the mirror image of this down and what this means is now we've got pi over 3 pi over 3 Pi over3 appearing in each of the three corners this is an equilateral triangle so the right side of it has to be of length one split in half this means half and a half so now that I have this little trick I can get rid of the bottom and now just by Pythagoras I'm going to get the of < tk3 / 2 going on the bottom and again since this maybe looks a little bit converson you could stretch this all out multiply everything by two and I call this the one two root3 triangle it rolls off your tongue very nicely by the way in in this video I'm trying to prepare you for what is commonly taught in calculus it's a very different question about should it be taught this way so don't treat this video as me saying what I think calculus should be like maybe just saying I'm trying to prepare you for how it is now what can be a little bit tricky here is if I do some other weird value like how about 11 Pi 6 so I kind of want to show you how to do all of those other ones and basically these are multiples of either pi over 3 pi over 4 or pi over 6 so this one I've chosen 11 pi over 6 and and what I like to do first if I'm working by hand is try to figure out like which of the four quadrants is it sitting in in this case 11 pi over 6 is just pi over 6 less than 2 pi that is 11 pi over 6 counterclockwise is the same thing as pi over 6 clockwise no matter which one you've chosen here you're always going to be able to come and put in that special triangle with the pi/ 6 at whatever location you've got at and this means we already know it values it's going to be 1 and a half and a < tk3 / 2 so let me suppose I was going to ask I don't know how about uh s of 11 Pi / 6 well s was the vertical components and according to this triangle that's the value of 1/2 but notice crucially my point here is in negative values of Y so I get from the special triangle the value of 1/2 but I have to adjudicate is it positive or negative and because I'm down here beneath the x-axis the answer to this is now one2 so you can use that kind of trick read to figure out at least for the special values multiples of Pi 3 Pi 4 and Pi 6 what you're supposed to be doing okay so we've mostly talked about s and cosine but there's other trig functions as you likely know tangent coent secant and cosecant and basically if you think about a triangle having the opposite adjacent and the hypotenuse there's six ways you can take a ratio of one of those to one of the other ones and each of those six ways is given a name these sort of foundational principles unfortun you have to sort of memorize the name of which one Associates to which there's not really a shortcut there however for the other four tan cotangent secant and cant I always like to have in my mind how they relate to the original sign and cosine in calculus this is really really useful as we start taking derivatives of one of the other functions so tangent for example in my mind is s divided cosine and so on this is the way I typically try to have it memorized in my brain then if you have to figure out I don't know tan of 11 Pi 6 you just do s of 11 pi over 6 and you do cosine of 11 Pi 6 you go from there I can also come up with the graph of other trig functions like for example tangent here I've put in the graph of cosine and the graph of s and me just overlay the graph of tangent tangent is this one that I've done here in the pink and if you ever forget how to come up with the graph of tangent here's the idea we know that tangent is s divided by cosine so whenever cosine is zero which you'll notice happens at pi/ 2 3 piun / 2 pi over 2 and so forth any places where that denominator is zero it has to have a vertical ASM toote so we have these spikes that are vertical ASM tootes of pi/ 2 negative pi/ 2 and so forth then in the numerator which is s well s which is the blue graph is z at zero and pi and 2 pi and negative pi and so forth and so our graph of tangent has to be 0 through Pi 0 Pi 2 pi and so forth so knowing those anchors where the vertical ASM tootes are where the zeros are going to be it's just a matter of connecting like do they go up to positive Infinity do they go down to negative infinity and say in 0 to Pi / 2 we know that it has to go to positive Infinity because s / Co in this region the graph of cosine is positive the graph of s is positive so the tangent has to be positive well so I know that in this region is going to positive Infinity whereas for example between Pi / 2 and negative Pi / 2 my sign is positive my cosine is negative so s divided Co better be negative and that's why tangent starts at negative infinity and goes up to zero you can do the same kind of analysis to come up with the graphs of secant cosecant whatever you like and then now that we have the other trig functions we can look back for example remember the old Pythagorean identity sin squ plus Co squ equal to 1 that we saw earlier in the video well I can get the analoges of this for the other trig functions for example I could divide both sides by sin squared this would give me 1 plus well cos over s is cotangent and 1/ sin squ is going to be cosecant squar and I get a second Pythagorean identity this time relating cang cosecant or if instead I prefer to divide by cosine I get tan sare + 1al SEC squ now what happens a lot in calculus is that there's these three different pairings s and cosine cotangent and cosecant and tangent and secant and they really often work together and so what I do a lot of the time is take whatever complicated Expressions I have they're combining lots of things and try to make them all be signs and Coes or all cotangent cosecants or all tangent and secants these sort of three pairs will keep on working together now you then might ask well hold on don't don't those other trig functions mean something like geometrically and indeed they do I have my standard picture here with a particular Theta but but what I can do is actually draw a bigger triangle so this one on the base it's going to have length one I'm going all the way out and then I go straight up and then I have this sort of now longer hypotenuse bigger triangle what is cosine of this bigger Triangle Well it's one divided by the hypotenuse in other words the hypotenuse is 1 over cosine which was our definition of secant so so secant has a a geometric meaning on the unit circle if I go to the other side try to take sign right remember sign is opposite over hypotenuse we just decided the hypotenuse was going to be secant of theta see of theta is one over cosine so moving it to the other side I've got s divided cosine that's tangent so the opposite here is going to be tangent now I'll actually be honest this sort of geometric meaning isn't something you're going to be using all the time in calculus it's okay if you forget this geometric meaning prec precisely maybe the larger point is just to note that all these trig functions do have a nice geometric meaning and from time to time that can be useful all right now we got to talk identity there are so many identities out there it can be really tumbers some so I want to show you a nice path forward to learning the major trig identies so I've actually put up only six out of the much larger selection that I think are most commonly found and used in calculus now more important than memorizing CU indeed you can always look it up and unless your Kus Professor is one that really wants you to memorize these for your test what's more important for solving homework problems is like knowing the general purpose behind the identities and and being recognizing that there's one that you should go and look up so for example the Su identities is an expression for any time that you're trying to take the sign of a sum or the cosine of a sum and then the double angle ones are actually just a special case of that they're useful and they come up in a lot of you know standard calculus problems but really all they're doing is saying well what if I say that the two angles the Alpha and the beta are both equal to Theta and I get these results and so anytime I see a sin 2 Theta or Co 2 Theta I I know there's a formula here and I can go and look them up if I happen to have forgotten this point they'd be drilled in my mind and so I haven't forgotten them and then the one that actually seems to be the most useful in calculus is these power reduction identities uh really all they are is taking the double angle identity for cosine rearranging it and then maybe using it uh Pythagoras in the case of the co squ but they're useful in calculus because sometimes when you're trying to do some weird integral and you've got a higher power like Co squar now you can reduce it to a lower power something to do with cosine 2 Theta and that's often a little bit easier so the point is that you can get a lot of these new identities from the old identities you don't have to remember very much but we still have the question where does the first identity th those those sum identities actually come from and there's actually a really nice little geometric proof of this coming from the unit circle the idea is let me imagine I rotated in amount Alpha and then let me imagine I then further rotated in a mount beta so my question now is asking if I wanted to know for example sign I want to know what happens for S of alpha plus beta where have I end it up if I do alpha 1 and beta second and now I just sort of want you to tilt your head a little bit I have formed this triangle here by dropping a perect pendicular on the triangle that's got this angle of beta and it's just the same kind of analysis we've done before like the adjacent was going to be coign the the opposite is sign when our hypotenuse is one so so this is a similar triangle to what we've seen before just sort of rotated around a little bit okay now I'm going to go and look at the triangle on the bottom here again it's a new one we haven't quite seen before instead of going all the way out to the edge of the circle I'm just going to the spot that intersected that perpendicular from the first one I got a new triangle here now if I want to look at the opposite side of this triangle well this actually I can say is just going to be well it's s of the angle Alpha but there's a stretching Factor since the hypotenuse is no longer length one the hypotenuse is is cosine of beta then that gives me my stretching Factor so it's sin Alpha Co beta and indeed if I compute s it could be sin Alpha Co beta divid sin Alpha and sin Alpha would be sin Alpha okay now here's the real insight this triangle that I've highlighted here is a similar triangle that means all proportions are the same to this one up here and most importantly what that means is that the angle Alpha is the same so I can do the same kind of thing if I want to come along here well this is the adjacent for that weird uh you know turned around triangle and so what do I have it's going to be the stretching factor of s beta to represent the hypotenuse and then is the cosine of the angle Alpha putting all this together if I look at just what the vertical is on both sides this is going to give me my double angle formula that sin Alpha plus beta is sin Alpha cos beta plus cos Alpha sin beta there I have my formula proven now again the geometric derivation here isn't necessarily the thing you're using all the time in Calculus what's more important is to note that all of these different identities let's let's put the list back up again all these different identities here don't come from nowhere like there's a very nice geometric meaning for the first one I'll leave it as an exercise for you all to figure out cosine of alpha plus beta if you saw the S one you just got to label different parts of that to be able to get to the cosine one and then the other identities were just a little bit of algebra coming from there so they're they're not completely magical they have these really sort of nice geometric meanings but nevertheless if there's one list you sort of want to sort of be familiar with this this is at least my short list okay so that was a lot of trigonometry facts but hopefully I've given you sort of a speedrun review or maybe sort of reframing of how you thought about trigonometry and hopefully touched on most probably not all but most of the important facts about trigonometry that you're going to be able to use in calculus but I don't want you stop there because to really internalize all of this now is the time for you to do some practice and for that I want to introduce the sponsor of today's video which is brilliant.org brilliant helps you be a better Problem Solver and a better thinker with courses on everything from well of course trigonometry like this video but across mathematics data science programming a guy whatever you're curious about as a professor I'm always talking about the importance of student centered learning this is where you are actively trying out problems interacting with the mathematics and science and ultimately getting feedback on your progress brilliant does a great job of breaking Big Ideas into really nice digestable chunks so that you can feel confident on every little step along the way and when you regularly Flex those mental muscles by actually doing mathematics a little bit each day the progress you can make over time is really phenomenal to try everything they have for free for a full 30 days go to brilliant.org bazit that's me or click the link down in the description Additionally you can get 20% off an annual premium subscription with that said and done I hope you enjoyed this video if you have any questions please do leave them down in the comments and we'll do some more math in the next video