A LEVEL MATHS: CAST Diagram Explained WITH Exam Practice LESSON 8

Hey guys, just to let you know that you can catch these lessons live every Thursday from 6pm until 7pm UK or BST time And I'll just allow you to ask any questions that you want and get some answers to them. Without further ado, let's get on with the lesson Hey guys, welcome back to another A-Level free tuition lesson. Today we're going to be covering trigonometric equations and also Doing the cast diagram. So the cast diagram tends to be quite a popular method of solving it There are other ways as well. But just as a brief introduction, let's say I wanted you to solve the equation sine x equals 0.5 Now most of you would probably tell me that all you'd have to do is do x is equal to inverse sine of 0.5 and then you'd get your answer, right? So then you just go straight to your calculator So let me just bring out my calculator quickly And again, you just do inverse sine 0.5 And you'll be done. 30 degrees. Easy peasy, right? Now the thing is, if we actually look at the sine graph, right? So if we draw the graph down the middle Starts with 0 and it goes up and down And so on, right? And even on the other side it repeats as well Now if you're looking at the values of a half, so this is plus 1, this is minus 1 Values of a half are going to be here And if you look, it actually crosses at multiple points Right where the y-axis equals a half So the problem is, is that although this is a correct solution, it's only one solution So one solution of an infinite number of solutions Okay, so we could also have a negative answer as well And we could have an infinite number of positive answers and an infinite number of negative answers because these graphs go on forever The same is true for the cosine graph and the tan graph as well So basically we can't actually tell where or In a sense which answer is correct, so to speak So what you'll find is in A-level, what they decide to do is they decide to give you certain boundaries and limits For what you're going to do. So what they might say is that solve or find all solutions, so solve sine x equals 0.5 in the region of And generally speaking, it would be like 0, x to 360 So what you'll do is once again, we have that x equals 30 degrees Then what can we do with it? Well, we use something called the Karst diagram to find the other solutions Okay, Karst diagram is a fairly simple draw a little 4x4, 2x2 grid and then you label it C, A, S and T And what this allows you to do is work out all of the other values for sine So again, this can be kind of tricky But the way I like to teach it in the way that seems to work well for Other students is to kind of go through a bunch of questions and slowly but surely add more and more information While there are questions that are this simple Most questions actually are a bit more complicated than just doing this You can have negative signs, you know negative angles and you can also have extra things before the x and after the x2 But for this one, what would you do? Well, what you'd first do is you draw 30 degrees Which is our answer onto our graph now we always start from Clockwise and we move anti-clockwise. So basically this is 0 degrees This is 90 degrees This is 180 degrees This is 270 degrees and then background to 360 degrees Okay, so our 30 degrees would look something similar to this And what we need to ask ourselves now is whether or not our value is positive or not Well, this is a positive value So we need to find the other place where sine X is a positive number and sine X is positive in the A quadrant which stands for or And the S quadrant which stands for sine. So we draw 30 degrees there again, and you're always drawing this from the x-axis Not the y-axis So again, just so you know all of these values here all these letters Stand for different trig functions. So cosine the A stands for all which means trig That's right tan sine and cos are all positive in that first quadrant sine for the S and tangent for the T Okay, and the idea is that In each of these quadrants a certain number of trig functions are positive. So in the C quadrant only cosine is positive So if we had cos X equals 0.5, we'll be drawing another line in cos. If it's tan X, then we're drawing the tan sign And that's it So now the question is what is this actual angle? Given that we always measure angles anti-clockwise Well, it doesn't get to 180 degrees. It's 30 degrees less than that. So it would be 150 degrees. So our two answers are 30 and 150 degrees Okay Now because we're only going up to 360 I'm not going to go back around to the next one because if I want to go all the way around That would be 360 plus 30 degrees 390 degrees, which is outside of our range. So I don't include it So hopefully that was okay as an introduction I know it's quite overwhelming, but I believe that through practice and a good explanation Anyone can learn to use the class diagram and it becomes very very useful indeed So again, that was just a basic example. Let's go on to these questions So here we go. Solve for and then they give us a range. So that's a good first thing to look at And now it says cos 2x plus 15 equals 0.3 so The first thing you might notice is that we have inside the bracket 2x plus 15 and we're solving for X Now the problem with this is it's actually going to change the range of values we can accept. Okay We're going to end up halving what this is So we can actually go twice around. So what we need to do is to both sides of this We multiply it by 2 and add 15. So 0 times 2 is 0 plus 15 is 15 less than equal to 2x plus 15 180 times 2 gives me 360 plus 15 gives me 375 So for the initial part, we're just going to work out What the overall values are for this whole thing Then we're going to subtract 15 and divide them by 2 in order to work out what X would be so 2x plus 15 is equal to the inverse cos of 0.3 Which if we work out on our calculator 72.5 degrees and it says give you answer to one decimal place. So 72.5 Okay, now we go straight to the cast diagram Okay, we're going to ignore the 2x plus 15 for now and do that right at the end Okay, now we're going to find all of our values in the range of 15 to 375 So again Our first value and again, we can label this cast So again the angle that we get from working out on our calculator. We always put in the all column 72 Now our answer is indeed positive right 0.3 is positive The angle is positive. So we're going to go anti-clockwise and Since we're doing cos we're going to go into the C section We down here and again always draw your angle from the x-axis So we're looking for this value. That's all the way around the back, which is going to be 360 Minus 72.5 Which gives us 287.5 degrees. Now if I go around again and look at this angle I Can see straight away that I'm going to get a number that's outside my range. What I would get is 432.5 now if I look at my range the upper limit is 375 so I don't include this and obviously all future values will also be outside the range. I also don't go the other way into the negatives because my lowest value is a positive number so These are my two values for this so with X equals What we need to do is subtract 15 from these numbers and divide them by 2. So we have 72.5 Minus 15 and then divided by 2 Gives me an answer of 28.75 so 28.8 degrees and do the same for the other one we have 287.5 minus 15 divided by 2 gives me an answer of 136.3 degrees to one decimal place Hopefully that wasn't too bad so far I'm going to keep going Perfect. So again solve for between 0 and 180 the equation is this. So again, we have 3 theta minus 15 So on both sides of our equation, we need to times them by 3 and minus 15 0 times 3 is 0 minus 15 is minus 15 And then 180 times by 3 is 540 minus 15 gives us 525 so in this one, we're more than likely going to go around the Cast diagram multiple times. So it might get a bit messy. Hopefully you can see the idea so So that equals an answer of 44.4 degrees and it says two decimals so 44.43 degrees. So again, let's go and look at the cast diagram Cool so again in the all section we're going to write 44.3 degrees and Since we used the sine symbol and these are all positive. We're going to draw the other line in the s section Okay. So again, our next angle will be from here all the way to the line, which would be a 180 minus 44.3 0.43, I keep saying 0.3 135.57 and Again, since we haven't reached the upper limit of 360, I'm going to go around one more time So go all the way around to here that would be 360 plus 44.43 which gives me 404.43 And we're still not at the edge so we can go all the way around back into the s section again which would be So in this case, this is 360 add another 180 would give me 540. So it'd be 540 Minus 44.3 at 4.3 Gives me 495.57 Which is still in the range and we can't go again Now you may be wondering I have a negative in my range so I could actually go anti-clockwise But as you can see if I go anti-clockwise It's about like an angle all the way from here around to here That's going to be too big. It's going to be a lot bigger than minus 15 There'd be minus 180 minus 44.43, which would be a lot bigger than 15 So once again, if we want to work out what theta is we need to add 15 to all these numbers and divide by 3 So 44.43 add 15 And then divide it by 19.81 and again to two decimal places 50.19 So and the reason why I'm doing it in this particular way and there are other ways to do it Is that a lot of the times in this question students again know how to do the work They understand it, but they end up missing a value or two And the problem is is you drop entire marks and just missing one value So when you do it this way, it kind of guarantees you get every value in the range Technically speaking you can do it In one step But I find it tends to lead to errors among students. You should only really do it if you're hyper comfortable If you look at our range again 0 to 180 We've got four values. It's pretty impressive. It seems that we've got them all Okay, so just be really careful when it comes to that And again in a five mark question losing a mark is pretty devastating. It's 20% of the grade gone Leave it up there for a second Okay, so here we have our first negative, okay, and I'll show you exactly what happens with that so first off if we're doing theta plus 30 Then our range now becomes minus 150 degrees Also, please pay attention to the equal sign as well So sometimes the value is exactly 180 or minus 180 in this case minus 180 is accepted 180 is not, you will lose a mark for every additional value that is not on the mark scheme. So please be careful So theta plus 30 Just equal to the inverse term minus 2.5 which equals Minus 68 and this is one decimal place, so minus 68.2 Perfection. Okay, so now It's a negative number so we're actually going to go clockwise Instead of anti-clockwise. So this is where that will be and The other section so keep in mind our value is negative So you remember how in the previous few questions asked well, where else is it positive in this case? I'm looking where else with this value for tan be negative. Well, it's going to be in the sine pole because a and tan are both going to be positive and that's going to give me 68.2 So my other value or one of my other values it's going to be that angle there So again, there is sometimes confusion of whether or not we'd still measure it clockwise if you measure it clockwise You'll get a negative answer So you do 180 plus 68.2 and make it negative which in this case would be outside of our region, right? Because 68 plus 180 would be approximately 248.2 which is outside this range and outside this one too So we don't need to worry about that we're looking at the inside one 180 minus 68.2 And again, we're not doing minus minus 68.2 or anything like that. That would be the negative answer So you're 111.8 degrees Now ask myself if I go around again either way will I get an answer? Well if I go around this way So I read all the way around Until this line I would have 360 minus 68.2 Which again is a valid answer, right? So 360 minus 68.2 That is a valid answer, but it's outside our range. So yes, you would go around to the C But it's outside our range so we don't need to do it And if I go around again, obviously that's going to be outside our range So in short the only two answers we have are these two now, that's not our final answer to work out What theta is we need to subtract? 30 degrees from both of them. So we have That I'll just be minus 98.2. That's not too bad And that would be 81.8 degrees That'd be it Not too shabby So the cost diagram is extremely powerful and working out answers The other way to do is to memorize a bunch of rules of like adding and subtracting from 180 or from 360 Some people like that. I think it's a bit too much memorization in general There's a way you can do something without memorizing something. That's the best way to do it And yes, I recognize that you know, you have to learn like times tables and stuff, but you know So in a situation like this where we have That what do we do? Well, it's quite simple really we can solve it just like a normal equation, right? So generally speaking. I just want to cause by itself. I'm going to divide both sides by five and That is what I'm going to solve Okay The only reason why we don't do that for the bits inside is you can't really get rid of the stuff inside right without inverse Causing first which is where we have the multiple solutions bit. Okay, so we have to get rid of the number in front first But for this one we have before the bit in the middle we have to inverse cos first Which means we need to change our boundaries and all that stuff So X minus 40 so that makes our boundary minus 40 X minus 40 and 3 Okay, so X minus 40 would equal the inverse cos 0.4 which is 66.4 because again, actually, no, it's two displaces my bad 66.42 then bad once again, we're going to head over to the cost diagram, so Relatively straight so this is in the all section as always and We're looking at cos here. So our other value where it's positive is going to be here So that number there would be 360 minus 66.42 Which is 293.58 Now we have to ask ourselves. Well, what's our range? Well, we got to 320. So let's see if we can go again Well, if I go again, I get 360 plus 66. That's not gonna be in the range. So we're done So now with X all we need to do is have all of the plus 40. So just add 40 to all of these so 106 point four two degrees and 333.58 degrees Nice and easy, I think Hopefully that's all okay with that part Okay So this question once again, we do something very very similar to what we have before. However, Let's be very careful We're going to square root both sides. So now we have tan X equals plus minus root 3 Okay, so we have to do the cost diagram twice Okay, so what we're going to do here is we have tan X Equals positive root 3 and tan X all the way over here We're getting a bunch of different answers. Then we have to put them all together at the end. Okay, so it is still one equation we're solving it's just that we have an option for plus and an option for minus so X here equals 60 degrees. I probably should have just known that at the top of my head didn't think and our range is 0 to 360 So let's go Again when they give you these options where it's 2, like a plus and a minus, generally speaking They don't make you do anything else. Otherwise it's a bit bit too much work so 60 and Since this is a positive square root 3, the other value will be here So that would be 180 plus 60 which is 240 degrees and if I go again, it'll be 360 plus 60 which is 420 which is bigger than 360. So we're done. Those are our two values. Now with minus root 3 We get minus 60. So you can probably guess what our answers are going to be but we don't need to because we're very intelligent Sorry So one of the other answers is going to be negative 240, but that's outside of our range. So again, we go negative 60. That's here Because again, oh god Remember if we go clockwise, it's a negative angle. So since we have negative 60 to draw negative 60 we go clockwise Okay, the other place where tan is negative is in the sine section Now if I read this Anti-clockwise, I get a positive number. If I read it anti-clockwise, I get a negative number. Now if you look at our boundaries There are no negative answers because the smallest number we can have is zero So we have 180 minus 60 which is 120 degrees Now if I go round again the positive way Round to here. I have another value. It's 360 minus 60, which is 300 degrees So whenever you draw like a line on this graph Think about going clockwise and anti-clockwise Now if you can see by going clockwise you're going to get a negative and you're not looking for negatives You don't have to draw it, but still think about it in your head Okay, because I know for a fact every time there's a question like this this answer here is missed Because people don't realize that although you wrote a negative angle you can get a positive angle going the opposite way Most students forget that so please please please make sure that you're being very very careful Okay, any time you draw a line you can get two answers a negative angle and a positive angle So just keep that in mind and then of course you can go around again to get more answers But the idea is that any time you draw a line there's immediately two answers to get so The solution for the whole thing is going to be four values 60 240 120 It's going to be very very careful Sometimes there's also a part a where So sometimes there's a part a value. Okay, and this is where part a value a part a part where you have to Manipulate trig equations now manipulating trigonometric equations is extremely common in a level Okay, a lot of a2 content focuses on manipulating trig equations as well for integration differentiation Solving trig expressions just rewriting them in different forms, etc. So We have to show that this Can be written as this now what I like to do in can be written in the form Equations is I like to compare the two equations Okay in this first equation. I have a sine squared in this one I don't have a sine squared so I need to get rid of sine squared the only way I know of doing that is by using the expression sine squared plus cos squared equals 1 So if I want to just replace sine squared I get sine squared equals 1 minus cos squared So let's sub that in shall we? And again, we're going to keep looking back at this to see if we're getting close to the answer Well now if I look at the answer I've gotten rid of the sine squared and I know I can see I'm going to get 2 cos squared Which is in the answer and I have the 7 cos x Plus 3 come from plus 5 minus 2 So we're on the right tracks So I do apologize for my voice basically, I'm recovering from a flu and Although I feel completely fine. Unfortunately, my throat gets very dry very quickly and It makes talking very painful and if we move this over to the other side This becomes positive 2 cos squared x There's nothing that affects the 7 cos x and then we have 5 minus 2 which is 3 Which if we have a look is our answer here 3 marks all done. You get a mark for the substitution A mark for expanding the brackets and a mark for rearranging Pretty standard marking scheme there Perfect Now this is hence solved Now I've told guys before, as part of our guys, students before That whenever you see hence in a question what it means is Always use previous part So you'll find this in maths, you'll find in physics, you'll find in chemistry, biology Everything. If you see the word hence, it means using the previous part so Immediately, without even thinking, instead of this, I'm just going to write this Okay, so we write that again And just so we don't have to keep scrolling up and down, I want to copy down the range too so 0 to 360 Okay So why have they done that? Well, if we have a look at this cos squared x and this cos x It looks like a quadratic equation. Okay, so I'm basically solving a quadratic equation Where a is the number in front of this So we can even say it's the y equals cos x and solve it like that That makes it easier for you, so you can rewrite the whole thing as 2y squared 7y 3, if that makes it easier for you But if you want to factorise this, well You can see we're going to have a 2 cos x, obviously In a closed bracket, open bracket, we're going to have a cos x here Again, you can use the full method of a, c is equal to 6, 3 times 2, b is equal to 7, so two numbers that times to 6 add to 7 is 6 and 1, do it like that So if we want 6, then we're going to have a 3 here and a 1 here because 2 times 3 would give us the 6 Okay, so now we have either 2 cos x plus 1 equals 0 Or cos x plus 3 equals 0 So we have cos x equals negative a half or cos x equals minus 3 so Some of you eagle-eyed people may see That this is a math error Generally speaking So no solutions For those of you that didn't spot that, the reason for that is you cannot So you can't Inverse sine or inverse cos numbers that are bigger So numbers that are bigger than the modulus of 1, right? So what I mean by that is If it's bigger than minus 1 or plus 1 you can't actually inverse cos it Because they're a ratio of the opposite and hypotenuse, right? So If opposite over hypotenuse is bigger than 1 means the opposite side is bigger than the hypotenuse Which is impossible Okay, so you can't inverse sine or inverse cos numbers bigger than 1 or lower than minus 1 Okay, and again, the reason for that is because otherwise the adjacent slash opposite side would be bigger than the hypotenuse So we're only focusing on this Which, you know, helps us out a little bit So x equal to the inverse cos Negative a half again whoops 120 degrees, that's nice So this is quite an interesting situation Normally or in the previous questions that we've just done When the value here is negative, our angle is negative, but in this case it's positive. So what do we do? Well, we're still going into the all section, but if you notice it's 120 degrees This is 90 so it's actually going to be something Like that where this value is 120 degrees This value here would now be 60 degrees Okay so Where else is cos negative where it's going to be in the tan section? And if we were to go all the way around we have This would be 60 So we'd have 180 plus 60 which is 240 And if we go around again, obviously we get a number bigger than 360 Which is outside our range So the those two are only values there In terms of the class diagram, that's basically as difficult as it gets The key things to remember just to kind of go over Is so if we do a little summary now Is let's find this Okay, so we do a little summary What what don't know what happened there? Okay Okay negative angles are clockwise positive angles are anti Okay Now this both refers to drawing and reading the values So if you ever want to read a negative value you go And sorry clockwise if you ever want to write a negative angle you go clockwise as well Okay, so reading and drawing the next thing that I think would be important to remember is every line has positive And I know i'm not using my contraction now and negative angles with it right Which means if I ever draw a line on that thing I can go clockwise to it or anti-clockwise to it and get valid solutions Okay So really important for when I like the common place that I would say to be wary of is so be wary of negative angle limits So one of the questions So keep in mind this will come up in your year 12 mock and in your year 13 final exam The part where I was a really really tricky question where I said find the solutions in minus 360 degrees to zero degrees So many students lost marks now The interesting thing about this is students tend to say that this is easy Right, but if it's easy, then you wouldn't make mistakes if it gives you something like this Right, you're meant to go clockwise for all of the values They just broke down completely So make sure you're Practicing with negative limits positive limits limits bigger than 360 Etc. Right. So if this limit was minus 720, would you be able to do it? Let's see. Okay means you have to go around twice. Okay, so just be very wary of that This I would say is a pretty good summary of it Of everything you need to know so again If you have any questions feel free to comment and like make sure that you uh Join in the lessons live to make sure that you kind of get help when as and when you need it Okay, but other than that i'll let you guys get on with your evening