we are now going to look at solving trig equations let me say a few more a few quick things before we actually look at some examples here's the first problem I'm going to do in a minute a very the most basic kind of trig equation to solve and by the way in solving trig equations the angles we're looking for are going to be in radians so for now none of these problems will be solved in terms of degrees but we will give the answers to the angles in terms of radians so this is a very the simplest kind of equation to solve and solving this we think of theta as our variable so solving this equation means I need to find the angle if I plug it in here the sign of the angle equals 12 now before we talk about this let me say first that there are going to be two kinds of solutions that you're going to be asked for in the homework in other words some homework [Music] problems the first problems you do they will ask for what's called what we call General Solutions General Solutions means every angle that makes this equation true so there might be more than one angle that when you take the sign it equals a half as a matter of fact if I were to just grab the unit circle and if I begin to look for which angles the sign is 1/2 hopefully by now you're pretty good at this if I look here at pi/ 6 the sign of p/ 6 is a half so Pi / 6 would be a solution to this [Music] equation I'll even write it here but that's not the only angle that would solve this equation how about if I go over here to 5 piun over 6 what is the S of 5 pi over 6 it's also 12 so as of now if I just take one quick trip around the unit circle there are actually two angles that are solutions to this equation but you know what we know from trig now there are two solutions on one trip around unit circle but what if I go around the unit circle I go 2 pi and then come back here so one trip around plus pi/ 6 is actually 12 piun over 6+ piun / 6 13 piun / 6 there's another solution to this equation and if you think about it theoretically I can now go over here and take 12 piun / 6 + 5 5 pi/ 6 and I would get another solution as a matter of fact theoretically I keep going keep going around and around and if I kept stopping here and here those would be Solutions that's the principle of co-terminal angles which you learned about earlier so what does that say theoretically how many solutions how many solutions are there to this equation how many angles can I plug in here and and this equation be true there's actually an infinite number of solutions so that's one thing we have to take into account so initially you're going to be given problems where you're asked to find the general solution which another word for that is like all solutions so we have to think about all the solutions we go round and round and round but the second kind of problem after we do some general solution problems I call them specific Solutions and what they do when they first give you the problem they will say to Li liit [Music] Your solution your angle usually usually they say from 0 to 2 pi so for this problem if if this was the equation to solve and then they said only give the solutions where Theta is between 0 and 2 pi then you would only have two solutions and you can ignore the rest of these CU all these solutions would be larger than 2 pi so there's going to be two kinds of questions you're going to be asked as we're solving these trig equations now what's going to happen is especially for these problems with specific Solutions sometimes like for this problem if we're ask for a specific solution the answer is pretty clear there's going to be some problems later on where we actually asked for a specific solution but in order to get the correct answers you have to still use the general solution as the first step and once you determine the general solution then you go determine the specific solution so we need to understand what this General solution means and then we can also understand specific specific Solutions so a general solution for this problem we've already sort of started right here's what you do I'm going [Music] to matter of fact I would just show you a homework problem so you can see how they word it solve the equation give a general formula for all the solutions and then they say list six Solutions so let's go ahead let me give you the general Solutions or what they call the general formula and then we will look at this problem to see in what form they want you to put the answers so let's explain General Solutions first and then we can look at how they want you to uh type in the answers we've already started talking about this what you do with the the general solution is [Music] you find all solutions [Music] Theta from 0 to 2 pi so one trip around the unit circle so for instance this equation the first angle we found was P / 6 so I'm going to write down this first solution p/ six now the whole key is we have to think of a clever and smart way to say okay yes p/ 6 is a solution but if we go around the unit circle once and stop back here that's another solution which we called 13 power 6 and we could go around the unit circle twice which would be 4 pi and then stop here and that would also be a solution how can we express the whole idea of going around and around the unit circle and here's how they do it you take your initial Solution on the first trip around the unit circle and you simply add 2 piun * k a k is called a counter where k [Music] equals -2 -1 0 one two dot dot dot in other words what this means is normally you start at zero and your first solution is you plug in zero for k okay just so it's clear let me for this first one sometimes I'll even make a little table here you start out with zero for K when I plug in Zer for K my solution 0 * 2 pi is 0 my solution is pi over 6 next I'm going to plug in one for K so p/ 6 + 2 piun we already did that that's that 13 piun / 6 and that's another solution and if I plugged in two for K I would have P 6 + 4 Pi I also have to include negative numbers so if I say let's set k equal to -1 so now I have piun / 6 plus a 2 piun so it's piun / 6 - 12 piun / 6 if you go to unit circle it's not on here but if I were to go to1 piun over 6 I would go like this and I would end up right here which once again is a solution to this equation so this is a short hand clever way to express all the solutions all the angles that end up right here from p/ 6 to one trip around and Power Six to two trips around it even takes into account if you go in the negative angle Direction so this is the basis for what we call the General solution now this so I said find all the angles from 0 to 2 pi this was the first one but there was the second one right on one trip around you in a circle we also said that 5 pi/ 6 was a solution so now there's a second General solution so you do the same thing except now I write down 5 pi/ 6 and then I add 2 pi K again and just so you know for now on when I do this kind of notation I'm not going to write down every time this last part where k equal -2 1 012 just because it's tedious and when you see this notation you sort of realize that this is also included so basically here are your two general Solutions so once again the idea is you take one trip around the unit circle you figure out which angles would be a solution and you take each of those angles and you add 2 pi K and that's a general solution and then if there's another angle you take and add 2 pi K and there's a second General solution so this is a way to write the solution to write every solution actually every possible angle that would make this equation true so now let's actually look at when you do this kind of problem in the homework what they're going to want you to do so the first thing they say is solve the equation and then they say give a general formula so we've done that for all the solutions and then list six Solutions so let's see the kind of thing they want you to do so the first thing is write the general formula based on the smaller angle see what they're going to do is since there are for this problem since there are two general solutions they ask for the smaller angle General solution and then they ask for the larger angle General solution so for us we would put in for the smaller angle since P 6 is a smaller angle and then for the larger [Music] angle so these are the two general Solutions and then they the second part of the question list six Solutions then they go in and they want you to list the first six solutions that are greater than or equal to zero and here's what you want to do you want to go ahead and I will show you what I did a minute ago so here are my two two solutions my two general Solutions now since they want the first six Solutions I started to here let me go and finish it now by first six and they actually say here let's read your instructions so you see why the answer list the first six solutions that are greater than or equal to zero so they're only asking for positive angles the first six positive angles so I already started what you do is this little table here I already started it so now we start we want to find the first six Solutions so you always start with k equals 0 when I plug in k equals 0 into this equation I got Pi / 6 now when I plug in k k = 0 to my second General solution I get 5 piun / 6 so when k equals z these are my two solutions these are going to be the first two of six now I bump up K to one when I plug one into my first teral solution 2 piun + < / 6 is 13 piun / 6 when I plug in one into my second General solution 5 piun / 6 + 2 piun is 17 Pi / 6 now I have four solutions for now I can ignore this because we only want positive solution so now since they want six I need two more so now I plug in two for K so piun / 6 + 4 piun well 4 4 Pi is the same thing as 24 piun / 6 so the first general solution would give me 25 pi/ 6 if I plug in K into my second General solution 24 piun / 6 should be 29 Pi / 6 and these are going to be the six Solutions so right here in this box you would put all six of these angles separated by a comma so this is the kind of problem you're going to have when they give you a trig equation and they ask for the general formula so this is how you do sign to cosine now let me talk about tangent for a minute because I think you will have to have a tangent problem look at this one they ask [Music] you to give the general formula for this trig [Music] equation let me expain belief briefly when you're getting a general solution for s and cosine you always add the 2 pi K however if you're ever solving a trig equation and you're asked for the general solution for tangent just because the way tangent works you don't add 2 pi K you just add pi over k so for instance the problem I was just looking at let's talk about it briefly it said tangent of theta equal < TK [Music] 3 over 3 now with tangent of theta I need to go find go to the unit circle and with the unit circle I don't even have to go all the way around for tangent I can just go look at the top half of the unit circle and see what are the what the solutions are now tangent can be sort of tricky because the way I do it is once again you think of tangent is s over cosine so I I need to get some kind of fraction like this so you do this enough and you realize pretty quickly that [Music] um here's something here's some algebra that we need to understand there's another way to write this [Music] fraction and this is going to help to be able to solve these problems 1 over < TK 3 is the same thing Asun 3 over 3 this also becomes important later on we're going to have problems if you go to the unit circle and you notice > 2 over2 is pretty common for S and cosine it's important to know that this also can be Rewritten as 1 over < TK two to get from here to here what you do is if you multiply the top and the Bottom by square < TK of three then you get this and this one if you multiply the top and the Bottom by square < TK of two you get this but these are really the same so when I'm looking at this you know what I'm really looking for I'm looking for some kind of sign and cosine that gives -1 overare < TK 3 and you'll see why I say this because now I'm looking to put a sign over cosine and I want to have a square of three in the bottom I do that because I'm going to either have this angle or this this angle since I want the sare of 3 in the bottom what I'm going to do is I want the cosine to have thek 3 over2 let me just do it let's look at the tangent of pi/ 6 the tangent of pi/ 6 is the S of PK / 6 over the cine of pi/ 6 if I go to my unit circle here's the cosine here's the sign so the sign is 12 the cosine is < TK 3/ 2 you simplify this you get 1 over Square < TK 3 now that's the number I want however the twist is actually I need it to be negative but what I do is I realize if I go to the reference angle over here if I were to do the tangent of 5 pi/ 6 the number would be the same except I would have a negative so in other words the solution to this is actually going to be 5 piun over 6 now I said for tangent once you find this one solution that's all you need because what you're going to do is instead of adding 2 pi K when you're looking at tangents you just add Pi K and then if they ask for six [Music] Solutions you just put in six vales of K all right so that's that's the idea of solving equations and finding the general solution