Mastering Trigonometric Identities, Equations, and the CAST Diagram

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