okay we want to solve this trig equation we have cosine 2 theta plus 2x sine squared theta this is
equal to 4 what should we do first in what to notice that this right here is 2 data but then
here we have a data the angles are different that's no good we must have all the angles to
be the same in a trick question okay keep that in mind but then this is 2 data in cosine we do have
a double angle formula for cosine right which one should you stop it has two versions right so the
version that I'm going to use right here is that this is the same as 1 minus 2 times sine squared
theta and this is another single to keep in mind because you see we have sine squared theta right
here already I want to use the one that has the light term so I want to use 1 minus 2 sine squared
theta and then once I bring this down you will see we can combine like terms and everything will go
pretty smoothly from there so keep in mind all the angles has to be the same and then ideally
speaking we want to have either all sides or all cosines since attack okay what can we do right
here to solve a trig equation we want to isolate the sine cosine part right here we only have the
sine square theta so let's try to isolate that and then we see that we have a negative 2 sine
squared theta plus 6 sine square theta we can compile a terms right negative 2 plus 6 is 4
and then we have the sine squared theta and what should we do this one that's just minus 1 on both
sides so that is now cancel and then we'll get 4 sine square theta equals 2 4 minus 1 which is 3
and now what we do now want to have this port and this is 4 times that so that divide both sides by
4 so this cancels and then we have this right here and then we are talking about sine squared theta
is equal to 3 over 4 I want to isolate the sign part ok sine-theta part this is sine squared theta
that's not good well we can just take the square root on both sides and don't forget to attach the
plus minus on the numbers side right here so square square root cancels we have two things
to consider this right here let me just read it down I'm going to first worry about sine theta
ok and I want to look at the past aversion first positive square root Street on the top over will
also square root the four on the bottom which is just a tool on the bottom right and then we also
have the negative version right here and then just read it down but that will focus on that
okay negative square root 3/2 okay so sine theta sine alpha angle is equal to square root of 3 over
2 you have two ways to do it one just memorize all the numbers all the angles on the unit circle and
then just tell me the answers and the answers but I would not recommend you to do that this is how
can figure out like how to get the angle right here the only two things that you remember is
the special right triangles and the ratio of the sides that's all and here we go so we are going
to start off with x and y coordinate plane right here and then you have to remember for sine it's
what sine by definition is the Y over R and we're talking about the XY plane right so by looking at
this equation here we know that Y value is square root of 3 and positive so that means I can just
kind of draw like a positive screws through somewhere and that means I have to go above
the x-axis I can just draw right here this is my square root of 3 how is that what does our mean
R is the radius but then in this situation when were talking about triangles it's the hypotenuse
right so R is this and this R is always passed by the way ok just the quick remark the Y power you
can sometimes be negative apart about that the bottom is all is positive for our anyways our
is to that means I can label this right here to be to and now what you see we have square root 3 and
2 what should this I be this should be 1 isn't it and now when you have this picture right here all
we have to do is figure out what this angle is and all we have to do is figure out this angle
PI remember the special right triangle this is going to be a 30-60-90 special right triangle
well this is one that means this has to be 30 degree the smallest angle so that means this
has to be 60 degrees right but then we are going to write the answers in terms of radians so the
60 degrees here is what PI over 3 radians right which is the same as 60 degrees but then do we
have another choice for the past of the Y value yes because right here this is positive square
root of 3 right however on the other side this Y value is also positive as long as we go above the
x axis we can also draw this to be square of 3 and then as I said the R is always positive and that's
the hypotenuse so you have it like this right and once again you can draw a right triangle this way
by this time this die here is technically negative 1 however this is how we are going to figure out
the other answer by the way M should just write on the answer right here the first answer is PI
over 3 we got it right here okay and then this is the second answer from this picture we know
already this is the right triangle that we have so this is the angle which is the same as PI over
3 right just like that one this is not how we count the in google what we're going to do is
you start off with the pass the x-axis and then you have to go from here all the way to here and
then figure out how much to return well you have a few ways to do it up to you by the end the way
I will show you this you know the whole thing here to here is 180 degrees which is PI and you
know this much right here is PI of us already so the blue one draw right here it has to be pi
minus PI over three right pi is what is like one over one PI right so this is the same as 3 over 3
PI 3 over 3 minus 1 over 3 is 5 2 over 3 and of course we have the PI and let's put a PI on the
top that's this angle here that I have in blue for you ok so I'll put that down right here this is 2
PI over 3 and then we just have to do the same thing right here ok that's sleep what can we do
right here draw the x and y axis and then you see that we have the negative version right but then
I want to put a negative on the top because by definition of the sign in the XY plane is y over
R R is always positive so that you always look at the negative on the top in our situation here we
have negative square roots 3 for the Y and let me just first draw it like this right here we just
want to do it counterclockwise this is negative square root of 3 isn't it the 2 on the bottom is
what is R that means it's the hypotenuse right here and you see will be using pretty much the
same angle you know this right here it's going to be PI over 3 however that's not how we count the
angle how do we do it you go from here ok you go from here and you come all the way to this side
right here and this side the hypotenuse you have to figure how much do you return ok on the top
we turn 180 degrees namely PI radians but then we go PI over 3 more so for the red one we are going
to write down pi right and then PI over 3 more so we add it with PI over 3 this is 3 over 3 PI
plus 1 over 2 pi which is going to be 4 PI over 3 isn't it so that's another answer for PI over 2
do we have one more yes we do because negative square root 3 it can also be done here as well
this is negative square root 3 and then we have that 2 right here and then this is still PI over
3 but this time we'll go from here come all the way to this side in blue for the blue part
you see the whole thing is 2 pi right the whole circle the whole revolution is 2 pi 2 pi minus
PI over 3 okay so this right here 2 pi 2 s ms 6 PI over 3 minus 1 PI over 3 right so if you do
the calculations right here 6 minus 1 that's 5 PI over the same denominator 3 and then we are done
PI over 3 2 PI over 3 4 PI over 3 and then the last one that we have is 5 PI over 3 that's it