Transcript for:
Understanding Trigonometric Identities and Equations

trigonometric identities and equations pure 2 chapter 6 wonderful resource by dr frost slightly adapted by myself so in this chapter we're going to be looking at exact trig values and the unit circle and introducing the cast diagram we're going to be using identities sine of cos equals tan sine squared plus cos squared equals one we're going to be solving trig equations and we're also going to be solving them in quadratic form so it's really exciting so the unit circle here we have the unit circle and from the unit circle if we just look at the values of theta in the range of 0 to 90. we know that sine theta and cos theta are the lengths on a right angle triangle and that is because we know if we go back to sarcoma so know that sine of x equals opposite over hypotenuse now in this case if we're using the unit circle our hypotenuse is one so our sine of x equals our opposite because opposite over one is just opposite so here we have sine of x and then our cos of x is adjacent over hypotenuse when our hypotenuse is one that means our cos of x equals our adjacent now what would be the gradient of the bold line now if we think about gradient we're doing difference in y divided by difference in x so our difference in y is sine x over cos x so i don't know why i've put a here i think it's because i was thinking of adjacent and sine x over cos x this is our gradient what we also know though is actually our sine is our opposite and our cos is our adjacent so this is the same as our opposite over adjacent now opposite over adjacent is in fact tan so this is where we lead to the first identity that tan equals opposite over adjacent which is also sine x over cos x so tan of x equals sine x over cos x so it's really good that we can use this right angle triangle between zero is less than theta is less than 90 to work this out and the unit circle can help us with this what we can do though is use the unit circle to actually help us in creating a cast diagram and what we're going to be doing with our angles is measuring them anti-clockwise so this is how we're going to start the cast diagram so the cast diagram is a fantastic diagram that's used and it uses the graph so we know for sine the sine graph looks like this and we can see we know this is 360 degrees 270 degrees 180 degrees 90 degrees zero and our minimum is minus one and our maximum is one and then for the cos graph cos cuts so it cuts the sine graph and we know this is a translation of the sine graph 360 degrees 270 degrees um 180 degrees 90 degrees and zero maximum one minimum minus one then we also have our tan graph as well which we're going to come to in a second so for these graphs for sine and cos and now we have the tan graph now the tan graph forget my drawing it's not always the best here we have 19. 180 and then we have 360 degrees might just sort that out a little bit see we've got our asymptotes as well at 217. and we have no maximum or minimum so what we're going to do now is we're going to look at whether we have positive values of y so if we look between 0 and 90. so if i was to just have a look between 0 and 19 we know here this is 90 degrees and if we imagine this is 0 and this is a 90 and we're going this way round okay so between 0 and 90. all of our graphs sine cos and tan are positive so we're going to call this a for all if we move on between 90 and 180 because we know here on our degrees between 0 and 180 this is a straight line and we know that angles on a straight line add to 180 and we also have another right angle here essentially so two right angles which do we have that's positive the only graph that's positive is sine all of the rest are negative so this is sine is positive here now if you imagine we're then going to 270 if i had another 90 degrees if i'm looking at this angle here between 180 and 217 the only positive graph is turn and then lastly if i was to make a full circle and go back to 360. if i have a look at where's positive between 270 and 360 it is cos so here you can see the cast diagram c a s t and a stands for where they're all positive all the graphs between 0 and 90 degrees all of them have positive y values then between 90 and 180 degrees it's only sine then between 118 to 70 degrees it's tan and then between 270 and 360 degrees it's only cos so this cast diagram is going to help us to find angles because even though we can use symmetry for example we know if 30 degrees is a half as far as to go across i'd get 30 degrees if i was to carry on going using symmetry i know that this angle must be 30 degrees so this is 150 degrees and it's nice and easy so we know sine of a half equals 30 degrees and 150 degrees and it would keep going if we went over the interval between zero is less than or equal to x is something equal to 360. so this is nice and easy but when we have a transformation this is when it gets a bit more difficult and this is where the cast diagram can help us so we're going to understand how to use the cast diagram on the next slide so very quickly a mini exercise have a look at your values and the gradient and see if you can understand what's going on so between zero is less than theta is less than 90 degrees we know that cos is positive sine is positive and tan is positive when theta is 90 degrees if i was to draw that so we have our x and our y say 90 degrees this is my 90 degree angle so here's my angle there's my theta what do we know we know that cos is zero our sine value is one and our tan is undefined because we know that's where the asymptote is because it doesn't have a gradient between 90 is less than or equal to theta and less than equal to 180. we're having a look at this the angle is going to be roughly around here it's going to be in between 90 and 180 degrees now in this section we know that our cos is negative and our sine is positive and our tan is negative moving on when theta is 180 degrees we know it's going to come here because this is 180 degrees and our cos is at minus one sine is zero and our tan is zero between 180 and 270 is about here we know our angle is here it's in between 180 and 270 degrees i'm going to have negative gradient a negative gradient and a positive um sorry not great negative values and that would lead to our time being positive then theta equals 270 we know this is going to give me this angle here all the way around to 270 degrees now we know our cos is going to be zero sine is one and again our tan is going to be undefined because if you think about it as well our our tan is sine one divided by zero which is going to give us undefined and then between 217 360 our angle is going to be roughly in this quadrant here we go we know this angle is going to be roughly between 270 and 360 and we have a positive value of cos a negative value of sine and a negative value of tan and again here it's drawn slightly better if you need it so the unit circle explains the behavior of the trig graphs beyond 90 degrees and that's why we use them so what we're going to do is we're going to have we're going to have a go at a few questions now using the cast diagram there are also some angle laws that we could use but today we're going to focus on the cast diagram so these are just a few trig angle lulls which we spoke about about symmetry and when we were doing igcse we we've already seen this because this is what we had to do in our trig objective and we've had a look at the symmetry and we know that sine and cos repeat every 360 and we know that sine that causes a transformation of sine this is all well and good but how could we apply apply the cast diagram to what we know so without using a calculator we are going to do that for the next set of questions okay so to start with we've got tan of 225. so first to draw my cast diagram tan of 225 if i think about where 225 is i know if this is 0 90 180 270 it's got to be in between here so roughly 225 know i'm going to reflect it here so now i can think 180 plus what was 225 that was 45 so that means this angle is 45 degrees now we all know what tan of 45 degrees is and that is one so tan of 225 is the same as tan 45 which is in fact one tan of 210 again if i was to change this now to 210 180 plus what's 2 tenths 30 degrees that means this angle is 30 degrees so tan of 210 is the same as tan of 30 degrees which is in fact one over root three so this is my term okay if i have a look at my sign now i have sine of 150 150 is roughly here and if i think how did i get 180 take away what's 150 30 degrees so that means this angle is also 30 degrees with sine we always reflect it in this vertical line with tan we carry on the diagonal and with cos we reflect it in the horizontal line because if we think about our diagram c a s t we're looking for where it's positive so sine of 150 is the same as sine of 30. i don't know why i put 10 sorry sine of 30 which we all know is a half now i'm going to have cos cos of 300 300 degrees is roughly here if i think if i go from zero all the way around it's going to give me 300 degrees and with cos i'm going to reflect this in my horizontal line so 360 to what is going to give me 300 and that is 60 degrees so i know that's 60 degrees so we're going to have cos of 60 which we know is a half now sine of minus 45. if i was to draw minus 45 here it is 45 again i'm going to reflect this in the vertical line 45 degrees sine of negative 45 is the same as cry all the way around 180 225 which doesn't help us actually but it's just good for us to do but i do know what sine of 45 has are so that's actually the same as minus sine of 45 which is minus 1 over root 2. cos of 750 now if i think about this if this is 0 and i go around 90 180 270 360. i can keep going 450 540 630 720 i'm now at 720 degrees so 750 would mean my angle would now be here and this would be 30 degrees i would still reflect it here and now i could start at zero so actually cos of 750 is the same as cos of 30 degrees which is root three over two and then lastly with cos i've got cos of 120 degrees if i go from zero all the way around this is about 120 this would mean this is 60 degrees again i reflect in the horizontal line 60 degrees so this is the same as 120 go all the way around so i'm gonna have 120 degrees or this is going to give me cos of 240 degrees again nothing's really helping me right now but cos of 120 as you can see normally we have our cos here so this would mean we could also flip it and do the negative this is negative cos of 60 degrees which equals negative a half okay test your understanding give these questions a go it says without a calculator but just use the cast diagram and see what other answers you can get and you can use your calculator if you feel the need okay so here are the answers is this what you got i hope it is and that we're ready to move on okay so trigonometric identity so returning to our unit circle when our hypotenuse is one okay our sine is our opposite over hypotenuse isn't it okay so this gave us the sign then as we were doing our spaghetti okay you remember we did we got the spaghetti did our unit circle and then we measured all the angles and then our sign would be this one because we started zero and then when we got to one go up to nineteen and we remember we came back down and went across and this side is our cos now using our trigonometric identities so sine is opposite over hypotenuse cos is adjacent over hypotenuse what is my tan yeah tan is opposite which this is our opposite the angle over our adjacent so tan is sine over cos or as i like to say silly cow okay silly cow fc silly cow so our tan is sine over cos this is the first identity now this will hold true for example tan of 50 is sine of 50 divided by cos of 50. this will hold true this is an identity now we have a right angle triangle who's our favorite man pythagoras absolutely okay and pythagoras says that a squared plus b squared equals c squared so we're going to come on to our second identity so um it doesn't matter which way you do it but sine squared theta plus cos squared theta equals one squared now i'm actually going to just rewrite this i skipped the gun okay it's cos bracket bracket squared squared brackets okay remember to put your brackets in yeah but sine theta squared can be written as sine squared theta plus cos squared theta equals one squared which is one so there is our second identity now i can't remember if you need to know these because i don't have a formula sheet on me but what you find is that you learn them okay so what we do with these identities is we try and prove other identities for example it says prove that 1 minus tan theta sine theta cos theta equals cos squared beta okay you have to prove this now the only way to prove this is to use what you know the two things you know you should put in a magical box okay silly cow and sine squared plus cos squared equals one now you can manipulate these for example tan x equals sine over cos well you could also say that tan x multiplied by cos x equals sine x couldn't you because all i've done is multiply my tan and my cos here okay this is just manipulation so if i multiply both sides by cos the same with speed equals distance over time time multiplied by speed equals distance you know i can manipulate this however i want it's still the same as long as we manipulate it correctly obviously so how do we prove this normally we start on the most complicated side okay and we write down are we going to do the left hand side or the right hand side left so i'm going to write l h s this is an abbreviation and what does it stand for yeah left hand side you know like other abbreviations that you know so left hand side equals minus tan theta sine theta cos theta okay now you can manipulate do whatever you want we could probably do this question in a few different ways some of us are going to get us to the right hand side quicker than others okay now it is ingrained in my brain that if i see a tan i immediately think silly cow okay so instead of writing tan what am i gonna substitute in yeah so this equals one minus bracket now whenever i substitute i write a bracket sine theta over cos theta now we remember with proof multiplied by sine theta cos theta with proof we have to show every step that's the whole point we're proving it step by step so this equals 1 minus sine theta times sine theta times cos theta over cos theta doesn't it now immediately we have a factor of cos theta on the numerator and the denominator they can cancel out and this equals one minus sine squared theta because sine theta times sine theta is sine squared theta yeah five times five is five squared okay now as i said we can manipulate now we know that sine squared x plus cos squared x equals 1. so if i minus sine squared from both sides i get cos squared x equals 1 minus sine squared x do i know here i have 1 minus sine squared theta so this i instead of writing 1 minus sine squared theta i could in fact write cos squared theta which equals the right hand side therefore proved so 1 minus sine squared theta is in fact cos squared theta using this rule i know that sine squared plus cos squared equals one this is the same as cos squared equals one minus sine squared so instead of writing one minus sine squared i'm going to write cos squared okay here are more examples if you want to go ahead and do them you can without me and then i will um i'll do these in a second step one write down the two rules okay tan is silly cow tan feet equals sine theta over cos theta and we know our favorite man is pythagoras sine squared theta plus cos squared theta equals one sorry for the crumpling on the video so i see a tan theta what am i going to write cow but before i do any of this we forgot it says prove it says prove what do i write left hand side the only reason i'm writing the left-hand side it looks more complicated yeah left-hand side equals tan theta plus one over tan theta now tan theta is sine theta over cos theta 1 over tan theta is 1 over sine theta over cos theta which is in fact so i'm going to get sine theta over cos theta plus cos theta over sine theta because we know that one over sine over cos is cos over sine okay then when i add fractions i need to make sure they have the same denominator don't i yeah so i'm going to multiply this whole side by sine multiply this whole side by cos or you could cross multiply i don't mind if i multiply this side by sine over sine remember because whatever i do to the denominator i have to do to the numerator and i'm going to multiply sorry wrong way round i'm going to multiply this side by cos theta and this side by sign because remember cos over cos is essentially multiplying it by one so we're not changing the actual value because anything multiplied by one is still the value it's really important that we understand that so i'm going to have sine theta times sine theta is sine squared theta sine theta times cos theta is sine theta cos theta plus cos theta times cos theta is cos squared theta over sine theta cos theta do they now have the same denominator absolutely so i'm going to have sine squared theta plus cos squared theta over sine theta cos theta now can somebody tell me what is sine squared plus cos squared one so this is actually one over sine theta cos theta which is the right hand side therefore proved okay now i need to make something clear to you when we are proving you should write this if you write these at the top of the question make sure you write the word using and you have those two identities there so when you do this question i would suggest you would write it before you write the left-hand side using and you've got the formulas there yeah so that when you go through it they know where your substitution okay now the next one sometimes might catch you out but it's really easy okay you know that sine squared plus cos squared equals one if you see a sine squared you want to try and manipulate this we've got five though which is a bit annoying but both of them are a factor of five so step one i can take out a factor of five from both to get one minus sine squared theta now where how could i get 1 minus sine squared theta well using that formula i know that sine squared theta plus cos squared theta equals 1 and cos squared theta equals 1 minus sine squared theta okay now if i have to rearrange a formula to then help with our simplification you should write that down too then you could put therefore equals five lots of cos squared theta because we know that one minus sine squared theta is cos squared theta yeah okay test your understanding they look complicated it's really not break it down discuss with someone off you go so remember we should always start with using tan theta equals sine theta over cos theta and sine squared theta plus cos squared theta equals one if you have to rearrange them you have to write it down we always have to start with this left hand side equals and we should write the equation again tan x cos x over the square root of one minus cos squared x then now we can start substituting we know that tan is silica sine x over cos x multiplied by cos x over the square root we know that one minus cos squared is sine squared x which then equals we can now we know that the causes are going to cancel so we're going to have sine x over sine x which equals one which equals the right hand side therefore proved so just quickly sine x cos x if i multiply that out it's going to be over cos x and then we can cancel them because we have a common factor on the numerator and denominator so step one make sure you are writing use it tan theta equals sine theta over cos theta and sine squared plus cos squared equals one then it's really important to write left hand side equals cos to the power of four theta minus sine to the power of four theta over cos squared theta okay so what we notice here is really simple actually we have a difference of two squares in our numerator so we're going to write this as cos squared theta minus sine squared theta cos squared theta plus sine squared theta all of this is going to be over cos squared theta what else do we know we know that sine squared plus cos squared equals you know so this equals cos squared theta minus sine squared theta multiplied all by one over cos squared theta i'm then going to have cos squared theta minus sine squared theta all over cos squared theta i can split my fraction up to have cos squared theta over cos squared theta minus sine squared theta over cos squared theta we know that cos squared over cos squared is one sine over cos silly cow is tan squared theta and this equals the right hand side therefore proved okay here is an easy one for you off you go maybe you want to start with the right hand side this time okay now you're okay to start with the right-hand side you're okay to start with the left-hand side but you always go with what you've got more to play with okay step one write your rules okay using tan theta equals silly cow sine theta over cos theta and cos squared theta plus sine squared theta equals one now on this side i'm actually going to do right hand side equals first because i've got more to work with on the right hand side one over cos squared theta minus 1. another thing to know is if you want tan squared theta think about what do you need to get there i know that tan squared theta can be made by doing sine squared theta over cos squared theta okay using this rule so think about trying to achieve this okay so right hand side now this one's a bit of a sneaky one okay but what do i know i know that the number one equals cos squared plus sine squared so i'm going to write cos squared theta plus sine squared theta over cos squared theta so i've substituted that in then i'm going to have minus one i'm now going to split my fraction cos squared theta over cos squared theta plus sine squared theta over cos squared theta minus one cos over cos is one plus tan sine squared over cos squared is tan squared minus one we know the one and the one are going to cancel so it's just tan squared theta which equals left-hand side therefore proved so before we move on last lesson we looked at the rules now we know they're not called rules they're called trig identities okay and the first one is tan theta equals sine theta over cos theta silly cow and then our favorite man pythagoras we worked out that sine squared theta plus cos squared theta equals one given that cos theta equals minus three over five and theta is reflex okay first of all what is a reflex angle greater than 180 but less than 360. okay so we're all happy with what our reflex angle is brilliant we need to know this later okay and it says find the value of sun sorry if i didn't wear sidekick find the value of side so what we we have cos we want sign which equation are we going to use we don't want tan sine squared plus cos squared equals one but i'm going to rearrange this so we know that sine squared theta plus cos squared theta equals one so sine squared theta equals one minus cos squared theta and sine theta equals the square root of one minus cos squared theta okay this is simple rearranging here you don't need to remember what sine theta equals you just always start and we should be really confident rearranging so we know how to find sine theta so what we're going to do is we're going to substitute so we now know then that sine theta equals the square root of 1 minus cos theta squared so that's minus 3 over five squared and that is because it tells us that cos theta is minus three over five and this is cos theta squared everyone happy with this substitution perfect so this means that sine theta equals 1 minus 9 over 25 so the square root of 16 over 25 because someone just checked on the calculator i'm pretty sure i'm right so sine theta equals plus or minus four over five okay now this is where it's important it says theta is reflex so we know what sine theta equals we have two though don't we we have plus or minus four over five okay so how do we know which one is going to give us a reflex angle if i go back to the cast diagram okay c-a-s-t when we have a positive value okay our sign is only going to be in between here and here and this is zero degrees all the way to 180 degrees is this going to give us a reflex angle so we need the value when when it's negative don't we yeah because sine is positive between 0 and 180 so the final answer is therefore sine theta equals negative 4 over 5 because this will give us a reflex angle what i mean by that if anyone does anybody want me to go into more detail with this explanation no okay so we're just going to do a little side note let's say i did sine theta equals wait there are doing a different colour let's say sine theta equals four over five what does theta equal everyone inverse sine of 4 over 5 53.1 degrees okay so we know that's here roughly isn't it 53.1 degrees now we know with sine we reflect so here's going to be our next one which is going to be 180 minus 53 which is 126.9 okay at any of these angles reflex no but if i was to take sine of theta equals negative 4 over 5. what's my answer please it's going to give me minus 53 which is down here 53.1 and then we reflect we do the opposite so we know this is 53.1 okay so if i was to go round if i start at zero and go round the first one i'm going to hit is 180 plus 53.1 which is 233.1 this is a reflex angle and this is why if we were to take the positive we're not going to get our first reflex our first reflex occurs when sine of theta equals negative four over five so that's why it's important that we really understand the cast diagram okay part two it says given that sine theta equals two over five and theta is obtuse okay what does obtuse mean more than 90 but less than 180 absolutely find the exact value of cos theta so again i have sign i want cos so i'm going to use um the sine squared plus cos squared equals 1. i'm going to rearrange it so i'm going to get cos theta equals the square root of 1 minus sine squared theta so i'm going to have cos theta equals the square root of 1 minus 2 over 5 squared 4 over 9. so cos theta equals 5 over 9. root five over three is that right have i gone completely wrong no sorry everyone oh what a fool i didn't square five okay wait there four over twenty-five what's one minus four over twenty-five twenty-one over twenty-five so root 21 over 5 plus or minus okay now how is this gonna which one is going to give me the obtuse now when in doubt draw it out okay so we could try let's do cos theta equals plus equals root 21 over five what does theta equal 23 now if i was to draw my cast diagram 23.6 is roughly here i'm going to reflect it here okay is this going to give me an obtuse angle no because obtuse is in this quadrant here between 90 and 180. so i don't even need to work out the next angle i just know then that theta equals sorry cos theta equals negative root 21 over 5. okay so now we're going to look at solving trigonometric equations okay so they're going to become really really useful now now it's really simple we're just going to ease into it so for example it says solve a theta equals a half in this interval between 0 and 360. now we've already previously looked at this so this is just a bit of a quick um recap so sine theta equals a half so when sine theta equals a half theta equals so i'm going to do inverse sine of a half and i'm going to get 30 degrees now remember it wants it in between the interval of 0 and 360. so i'm going to draw my cast diagram okay i know my first one is 30 degrees it's positive so i'm going to reflect it in this is sign so we're going to reflect it i'm going to have it in the other sign okay and remember this is 0 19 180 270 and 360. so i start from zero and i'm going to go all the way to 360 and see where it intersects if i start at zero my first intersection is at 30 which i've already got written down carry on my second intersection is here this is 180 minus 30 which is 150 degrees carry on going round until i get to 360 there's no more intersections i stop because that is where my interval is so to solve sine theta equals a half in this interval gives me two solutions and the two solutions are 30 and 150. so these these trig laws here and these rules that dox frost has used on this powerpoint is definitely something you can do and this is how i was taught i was never taught using the cast diagram i just learned using symmetry in these rules but when we get to pure free and pure for it's a lot easier using the cast diagram so forget about this we're going to be using cast okay just for the first bit and then if you can figure out and you understand a different way that's fine but we're going to be using cast it's going to help us when we get to some of the harder questions okay the second one so solve tan theta equals 10 between this interval of minus 180 theta is less than 180 okay now first of all to solve anything this says five tan theta equals ten i don't want i want tan theta equals so what am i going to do divide both sides by 5. it's really simple so i'm going to get tan theta equals 10 over 5 which is 2. now i need to find what theta equals now when i inverse tan of 2 i'm going to get 63.4 degrees now rule of thumb is three significant figures okay sometimes one decimal place but roughly it will tell you in the question if it doesn't just go with three significant figures now when i draw my cast diagram remember this is zero this is 90. this is 180 this is 270 this is 360. okay our first step is we just label our angle and our angle here is 63.4 which is roughly here this is 63.4 degrees this angle now we know tan it goes c a s t so we're going to reflect well it's not really a reflection it is but in the line y equals minus x and this is our angle and this angle is still the same size 63.4 okay so step one we put our angle in we do the correct reflection okay so our cast diagram is now ready to use step two we look at the interval now the interval says between minus 180 and 180 okay so with the cast diagram i'm just going to start doing the positive so is everyone happy i could break this up we could do between minus 180 so less than or equal to theta is less than or equal to zero and i'm going to do zero is less than or equal to theta is less than 180 degrees is everyone happy i'm going to split these two up okay so i'm going to show you how we're going to find this one's the easiest between 0 and 180 between 0 and 180 is this whole section where does it intersect it intersects here so we know our first angle is 63.4 are there any other intersections between 0 and 180 no okay now normally so if we go clockwise 0 91 82 70 we're going around positive if we go anticlockwise we're going to go around this is zero this is minus 90. okay this is minus 180 this is minus 270 and this becomes minus 360. sorry not if we go clockwise if we go anti-clockwise it's positive if we go clockwise it's negative okay so now i want the interval between zero minus 180 and zero so where am i looking between zero minus nineteen minus one eighty okay i have found an intersection here how do i find this number this number is in fact minus 180 plus 63.4 which is minus 116.6 degrees and there are no other solutions solve sine theta equals negative a half in the interval 0 is less than theta is less than 360. so for our interval we know our cast diagram c a s t we know 0 90 180 270 360. so this is we're only going in this interval between 0 and 360. i'm just doing one holy okay so sine theta equals negative a half so that means theta equals minus 30 degrees okay so it's giving us a negative 30 degrees so step one draw this on the cast diagram negative 30 is going to be roughly here isn't it this is 30 degrees yeah now normally we reflect in this line if it's sign isn't it we're here we reflect it over so i'm still going to reflect on this line and here is my 30 degrees okay so it wants it in the interval so now this is set up i can start going round i start at zero is there any intersections between zero and 90 no 90 and 100 no uh sorry 180 180 to 270 yes i found my first intersection what is this value this is 180 plus 30 which is 210 degrees so there's my first intersection carry on going round found my second intersection this is 360 minus 30 which is 330 degrees carry on going round i stop when i get to 360. so i have three values for theta right now but this is where we need to read the question properly the question says it only wants it between 0 and 360. so therefore the final answer even though that minus 30 we used it to help us it's not included in the final answer because it does not want any negative values of theta it only wants feta between the interval of zero and 360 degrees so our final answer is 210 degrees and 330 degrees okay so next up solve sine of theta equals root three cos theta in the interval theta equal theta is less than equal to zero no it's not it's greater than or equal to zero and less than or equal to 360. okay so remember everyone um i want a number on one side and i want my trigger trig functions on the other side don't i so from here i'm going to have sine theta equals root 3 cos theta the most practical thing to do is divide both sides by cos theta so i'm going to have sine theta over cos theta equals root 3. hold on a minute what have we got silly cow yeah 10 because we know that tan theta equals sine theta over cos theta so i'm going to substitute in and say tan theta equals root 3. now i have one trig function i can find my answer so that means theta equals 60 degrees okay step one set up the cast diagram it wants it between um 0 and 360. so if i do my 60 degrees it's roughly here tan c-a-s-t so this is 60 okay set it up done now i'm going to go around and see what's going on so if i go from here to here this is my first angle this is 60 degrees which will we've already written down if i carried on if i start at zero and go all the way and this is the angle i want to find it's 180 plus 60 which is 240 degrees and then if i carry on all the way around it doesn't intersect again and i stop i stop when i get to 360 because that is where my interval is in between right test your understanding give it a go let's say five to ten minutes okay so solve two cos theta equals root three so that means that cos theta equals root three over two so inverse root 3 over 2 is going to give me theta equals 30 degrees so when i draw my cast diagram i'm going to have 30 degrees i know this is 30 degrees so if i go all the way around 330. there we go there's my two solutions and that's how quick you can do it for the second one i get root 3 sine theta equals cos theta so root 3 sine theta equals cos theta so that means root 3 equals cos theta over sine theta now this is not silical this is calcilly so this is 1 over tan so root three equals one over tan theta so that means tan theta equals one over three okay or you could rearrange it the other way but either way you should get this so if you do shift tan one over root three we're going to have theta equals 30 degrees now again i'm going to draw my cast diagram i know this is 30 degrees we're going to reflect it here because it's positive 30 is my first one 180 plus 30 is 210. so there's my answers and you can see how quick i'm doing this and when i'm doing my cast diagram i'm not writing 0 90 180 or anything like this but as you can see with me being so fast and rushing it i haven't read the question properly because the question says between minus 180 and 80. so let's start that again 210 is not quite the answer but that's okay that's me going positive let's go negative shall we so if i go negative this is 0 minus 19 minus 180 so minus 150. i'm now going to look at my values for theta and the correct answers are between -180 and 80 is minus 150 degrees and 30 degrees those are your two solutions so even though we can do this fast again make sure you read the question just in case now we're going to look at these types of hard questions and this is going to come really handy into why the cast diagram is even more beneficial for us okay because even though there are these tricks it is a lot easier doing it this way now the difference between this question and the other types of questions are the fact that this time you can see here we've normally just seen questions where we have cos of x or cos of theta here we have cos of three x or cos of three theta now what does that actually mean though in terms of the graph okay this is a graph transformation so if you think about the cos graph it's been transformed all of our x coordinates have been divided by three so normally our cos graph would look like well it definitely wouldn't look like that sorry my pen is jolted it would look like that between 0 and 360. but if all of our values are divided by 3 we're actually going to get three lots of that between zero and three sixties it's gonna look like this okay between zero and 360. so if we were using the rules and the symmetry etc and we were thinking okay so where does it cross at minus a half minus a half is roughly here so we're going to go across we're going to have six solutions now trying to figure out oh that was you know where does this cross at this normally crosses at 90. now this crosses at 30. where does this normally cross at this normally crosses at 270. now it crosses out 90. okay where does this normally cross out so instead of doing all of this you can see it's going to take a while isn't it to do all of that we're not going to do that what we're going to do is use our cast diagram and this is why our cast diagram comes in handy okay because no matter for what transformation now sometimes we might have 3x plus 10. okay so it doesn't matter what transformation and the cast diagram can help us do it so this is how we do these types of questions step one because of three x equals negative a half so what are we going to do we are going to do inverse cos of your hair looks lovely by the way i really like it okay so that means three x equals inverse cos of negative a half just like we normally do just the normal step and we get 120 degrees okay now a lot of people right now automatically want to divide by three don't they we're not going to do that okay we're not going to do that yet so we have 3x equals 120 degrees okay our interval is between 0 is less than 3x is less than 360. but we no longer have x what do we have 3x so we're going to change it how do i change this x into a 3x multiplied by 3. whatever i do to this i have to do to the whole expression so x has been multiplied by 3 so 0 times 3 is 0. x times 3 is 3x 360 times 3 1000 and is it 1080. okay so that is the crucial step before we divide because what happens some people divide here okay and in a minute i will show you what all happens if we divide i'm just going to show you the right way and then i'm going to show you what would happen the wrong way okay so now we know that 0 is less than 3x is less than 1080. so when i draw my cast diagram so i start my cast diagram okay 120 we know the angle is roughly this big isn't it now remember we're cast it's c-a-s-t so then my other i always reflect in essentially the horizontal line so this is 120 as well okay so this means that this is 60 and this is 60. so here we go i've reflected it and everyone should be happy with these reflections so now i have my starting point okay so now think about this i start at zero because this is zero isn't it i'm gonna do some play and i get to my first angle which is 120 which i've already written down 3x equals 120. okay i carry on i get to this angle this is 180 plus 60 which is 240. i carry on when i get to here i'm now at 360 aren't i i've gone round one whole circle and we know how many degrees are in one circle 360. so now i say that this is 360 and then i know i go up in increments of 90. so this is 450 540 630 and then i'm going to get to 720. so now i carry on going around if i'm at 360 and i get this point again i've done 360 plus 120 which is 480 so that's my next one when i carry on i have 540 plus 60 which is 600 and then i carry on and i reach 720 but remember the end goal is 1080 isn't it so don't stop there i keep going okay and if i think about it after 720 i have 810 900 990 1080 okay so i fill in my next lot so i know i have to go around but really you know we've multiplied x by three so you know you have to go around the circle three times yeah so let's go around again i'm now at 720 i go round to here 720 plus 120 840. i keep going 900 plus 60 960. keep going i stop once i get to 1080 because that is our maximum so done i found all the possible volumes but remember this is what 3x gave me i want to find what x gives me so now i have all the values now i divide by 3. so i divide everything by 3. 120 divided by 3 forty two forty divided by three eighty four eighty divided by three one sixty six hundred divided by three two hundred eight forty divided by 3 280 and 960 divided by 3 320. now you can see all of these values are in between 0 and 360 which is what our interval is it makes sense because we draw we drew the cos graph earlier and if we think the transformation okay um we knew it should cross six times so this is my recommendation of how to do it now the reason why i say we don't divide first okay and i'm going to do it here classic mistake are we ready for this classic mistake okay this here is a classic mistake so they've done because of 3x equals negative a half they've got 3x equals 120 so that means x equals 40 degrees perfect now it says the interval between 0 and 360. great so they've gone this is 40. i reflect here okay let's go round first one 40 check done it all the way around this is 360 minus 40 320 perfect then stop at 360 because that's what the interval says great i've got two solutions perfect right off i go now as you can see that is completely wrong isn't it because remember the cast diagram is like a little trick it's there to help you but it only helps you if you understand what's going on you need to realize that cos of 3x is a transformation how does that transform the graph that transforms the graph by dividing all of our x coordinates by three so therefore um we're going to have three graphs in our normal zero to 360. so if we put a minus a half here we know our our minimum point here is minus one minus one so we know between zero and minus one lies a half so we can easily see there would be six solutions so this is where we need to make sure we're using our common sense and this is where the cast diagram can let you down if you don't fully understand the question the cast diagram is here to help us okay the cast diagram is here to help us but you still need to understand what's going on and you really need to understand trig okay so i've got two more examples for you so i'm going to show you these two examples step one solve sine of 2x plus 30 degrees equals one over two so step one what we're going to do is we're going to once we know it's in the correct format we are now going to inverse sine we have sine of something equals a number so now we can inverse sine so i'm going to have 2x plus 30 equals the inverse of 1 over root 2 which is 45 degrees okay stop once we've inverse sine we stop why because now we need to change our interval what have i done to x here i've multiplied it by 2 and then i've added 30. so if i need to do that here i need to do that to every single side 0 multiplied by 2 is 0 plus 30 is 30. x multiplied by 2 is 2x plus 30 perfect 360 multiplied by 2 720 plus 30 750 this is our new interval and when we're doing our cast we're going to keep going round so that it fits in this interval okay now with our new interval we do cast so the angle we're working with is 45 degrees okay we're using sign so we know it's here now again if you're not sure please write the words cast c a s t okay but the more you do it the more you'll be able to understand so right we have an imaginary line which is 30 here we know it's below this is our starting point here isn't it 30 yeah because our least value is 30. so i go round 45 is my first one okay the next go round remember this angle is 45 degrees this is 180 minus 45 which is 135 degrees okay i keep going round i hit 360. do i stop no i'm just going to fill the rest in for now so this is 4 50. this is 5 40. this is 6 30. this is 7 20 and then if this is my 30 this becomes 750 okay because remember we're stopping at 750 aren't we so i carry on going round i'm now at 360 i go round i hit my angle line 360 plus 45 405 degrees keep going i hit it again 540 minus 45 495 degrees i keep going round i've hit 7 20 but i do not stop at 7 20. i keep going so i get 7 15. lucky for us i stop at 7 15 we haven't intersected again okay okay so here we have we have 2x plus 30. now step one take away 30 from everything so i'm going to have 2x equals fifteen hundred and five three hundred and seventy-five four hundred and sixty five so i've taken away thirty degrees from everything now what am i gonna do divide by 2 absolutely so i'm going to divide by 2. so my x equals 7.5 degrees 52.5 degrees 187.5 degrees 232.5 degrees and there is the final answer now if you think about this this sine graph here all of our exponents have been divided by 2 and then we've shifted it to the left 30 degrees okay so we are roughly expecting around four we could maybe a bit more because normally our sine graph looks like this but what's happened to us is uh we've halved it so now it looks like that not only have we done that we've now shifted it to the left 30 degrees so it's gonna look like that can you see and one over root two it could be you know we could have four solutions so roughly we know we're on the right track okay perfect okay and then the next question okay next up so sine x equals two cos x so i've got sine of cos what am i thinking yeah divide by cos so sine x equals two cos x sine x over cos x equals two sine over cos tan x equals two okay so shift tan of two is 63 0.4 degrees okay i'm going to do my cast 63 63 remember this is my angle okay now we're starting at zero we finish at 300 okay so where is 300 we know this is 270. so 300 stops here doesn't it this is where we stop at 300. so let's go around zero what do i get 63.4 keep going round 180 plus 63.4 243.5 keep going i stop at 300 degrees okay remember we're stopping at 300 why because this is our interval now actually in our case this hasn't actually affected us the 300 but i'm being dramatic about it because it may affect you in different questions okay so you have to just remember you've got all the information you need to have it all down okay this is a seven mark exam question you have all the skills needed to do this seven marks seven minutes off you go let's see how many marks you would have got so first of all it says solve for zeros necessarily equal to x is less than 180 cos of 3x minus 10 equals minus 4.4 okay we know it's a transformation so i've got cos of 3x minus 10 equals negative 0.4 so 3x minus 10 equals and in your calculator you're going to do the inverse of negative 0.4 i get 113.57 i'm just gonna do 113.6 okay you would have got one mark for that okay the second mark now we have figured that out we need to change our interval don't we what have i done to x here multiplied by three then took away 10. so what am i going to do here multiply by 3 then take away 10. 0 multiplied by 3 0 take away 10 minus 10. x multiplied by 3. 3x take away 10. okay i know i'm right because it gives me this which is what i want is less than 180 multiplied by 3 is 540 take away 10 5 30. okay check i now have my new interval i have my answer i now need to set up my diagram so 130 degrees this is if this is zero we're going to be over here somewhere okay this is a hundred and thirteen degrees that angle now this is cos and with cos we reflect it in this line here so that means this is 113.6 so now we're ready this is zero and we're gonna go all the way around now remember where are we starting from minus ten so it's just good habit this is minus ten here now luckily it doesn't intersect anything so we don't have to worry okay but always just double check you don't have to draw it like i did but just double check okay start at zero my first angle is 113.6 keep going second angle what is this 16 minus 113.6 which is 246.4 okay keep going i'm now at 360. do i stop no what's my next one after 360 450 then 5 40. but wait a minute it says stop at 5 30. so my new stop is here 5 30. so i keep going around 360 i get here stop so 360 plus 113.6 is 473.6 okay i haven't stopped there though what do i now now i keep going stop at 5 30 because that is the last part of our interval so now what do i do add 10 to everything one two three point six 256.4 483.6 now what do i do divide everything by three 41.2 two five six point four divided by three eighty-five point five and then four eight three point six divided by three 161.2 now double check it says give your answers to one decimal place which we did we showed our working we did seven marks quadratics with sine cos and tan so as you know we can have hidden quadratics and we can also have quadratics with our trig functions so there's two methods method number one is where we can use substitution so we can let y equal sine of x we can just then substitute this into our equation to get 5 y squared plus 3y minus 2 equals 0. from there i can factorize or solve this using my calculator 5y minus 2 and y plus 1 equals 0 so y equals 2 over 5 and y equals negative 1. therefore using my substitution y equals sine x i'm going to have sine of x equals 2 over 5 and sine of x equals negative 1. so now from there i can use my cast diagram to find my degrees and i'm going to do that in a second the other method is factorize without substitution you should be fairly happy that instead of you know having this as a y we can just write sine x so we're going to have 5 sine squared x plus 3 sine x minus 2 equals 0. we can put this into brackets 5 sine x minus 2 bracket oops sine x plus one equals zero and then from there we know that sine x is going to equal two over five or sine x equals negative one and we're right back um we're back at the completely same answers okay so now what we have to check is that 0 is less than or equal to x is less than or equal to 360 degrees so what we can do now is we can use our cast diagram so i have sine of x equals 2 over 5 so that would mean that x equals inverse sine 2 over 5 which is 23.6 degrees 23.6 degrees now i'm using my cast diagram 23.6 this is sine this is 23.6 so if i go around i'm going to start at zero my first angle is 23.6 i'm going to continue round start at zero this is 180 minus 23.6 which is 156.4 degrees and then continue all the row round and i'm going to stop because i've reached 360 degrees and my interval is between 0 and 360. the next one i have is sine of x equals negative 1. so i'm going to inverse sine negative 1. now we should all know this is negative 90 on our calculator and this is where your calculator gives you the degrees which is closest to zero and if you think about our sine graph here negative one is going to give us our minimum point which we know is at 270. but our calculator is going to give us the answer closest to zero which in this case is negative 19. now what we can do is we get x equals negative 90. we see this is not in our interval but that's okay so on our cast diagram we draw our cast diagram we draw our angle negative 90. we're going to here's our 90 degrees start from zero we're going to go all the way around there's no reflection and we're going to hit it at 270. so this is 270 degrees so finally our answer is x equals 23.6 degrees 156.4 degrees and 270 degrees and as you can see here i haven't included the minus 90 because it's not in our interval range and i've also put this in ascending order okay two more examples people get caught up on this one solve ten squared theta equals four so we know that tan squared theta is is actually tan theta all squared this equals four it's just the way that we write it so that means tan theta equals plus or minus the square root of four which we know tan theta equals plus or minus 2. so when tan theta equals 2 theta equals inverse tan of 2 which is 63.4 degrees now remember this is an our only solution so when we have tan we have 63.4 and 63.5 so i'm going to go around start at zero 63.4 and then i'm going to do 180 plus 63.4 which is 243 degrees so i've done my first one now for my second angle i have tan theta equals negative two so shift turn negative two my answer is negative 63.4 it's going to reflect this here and here's my angle again then if i start at zero and go all the way around the first time i hit it is 180 minus 63.4 so my theta equals negative 63.4 180 minus 63.4 is 116.6 and then if i was to carry on going all the way around i would hit it again here 360 minus 63.4 which is 296.6 degrees so now i'm going to double check my interval my interval is between 0 and 360. so now i'm going to write my final answer so therefore theta equals 63.4 degrees because the negative's not going to be included 116.6 degrees 243.4 degrees and 296.6 degrees and then we have we should have four answers second question solve 2 cos squared x plus sine x equals 3 sine squared x now i have a sine x and a sine squared x and a cos squared x so i've got more signs essentially so what i'm going to do is i'm going to substitute in remember our trig identities that sine squared plus cos squared equals one therefore cos squared equals one minus sine squared so i'm going to substitute that in to where cos squared is it's going to have two lots of one minus sine squared x plus nine sine x equals three sine squared x what i'm now going to do is expand and simplify and move everything to have equals zero so we have a quadratic so i'm going to have 2 minus 2 sine squared x plus 9 sine x equals 3 sine squared x and then going to have 5 sine squared x minus nine sine x minus two equals zero i can now factorize this so i'm going to have five sine x plus one and sine x minus two equals zero so this means that sine of x equals negative one over five and sine of x equals two now we know that sine of x can't equal 2 because we know our maximum value of sine x is 1. and if we have a look we can see that on the graph this is 1 negative 1. so we're going to reject this and we're going to continue with sine x equals negative 1 over 5. so when i put this into my calculator now shift sine negative 1 over 5 i get minus 11.5 degrees x equals minus 11.5 degrees so with my cast diagram minus 11.5 degrees is here remember i'm going to reflect when it's sine i reflect in this vertical line i can have here this is my angle minus 11.5 well it's not minus 11.5 this is 11.5 is my angle it's only negative if we go backwards now let's have a look at the interval the interval says minus 180 is less than or equal to x is less than equal to 180. so actually what i'm going to do is i'm going to do between 0 is less than or equal to x is less than or equal to 180 so if i start at zero and go all the way around i haven't hit anything so there's no solutions in that interval and i'm going to do minus 180 is less than equal to zero is less than sorry it's less than or equal to x less than equal to zero so if i go backwards so this is going to be negative 11.5 and then if i carry on i'm going to have negative 180 plus 11.5 which is negative 168. remember i'm figuring out this angle here which is 168.5 and then if i carry on i'm going to stop this is my negative 180. so i have a total of two angles so x equals negative 11.5 or x equals negative 168.5 okay what i'd like you to do is test your understanding this is a typical exam question and it's worth six marks i see they've been quite nice because they've said even if you couldn't show that the equation can be written in this form they've given you so that you can solve this okay so step one we can clearly see that we want a sine squared and we know the identity involving sine squared and cos squared is that sine squared plus cos squared equals 1. so i'm going to substitute in 5 sine x equals 1 plus 2 lots of and we know that cos squared equals 1 minus sine squared x so we're going to have 5 sine x equals 1 plus 2 minus 2 sine squared x if i rearrange this i'm going to have 2 sine squared x plus 5 sine of x minus three equals zero so therefore shown so that's part a nice and simple part b it says solve the interval between zero is less than or equal to x is less than 360. so i'm going to factorize this so i'm going to have 2 sine x minus 1 sine x plus 3 equals 0. so sine of x equals a half sine of x equals negative 3. we're going to reject this because we know our maximum or minimum value of sine x is plus or minus 1. sine of x equals a half if we inverse half we should know it equals 30 degrees but if you didn't you will come to learn this and then again if i was to do my cast diagram 30 30 for sine it's 180 minus now this is quite easy quite an easy question in terms of the angles because if you know the sine diagram of the symmetry you won't need to use cast and we see that is the final answer nice and easy six marks you