BAM!
Mr. Tarrou.
Welcome to my third video about dealing with half angle and double angle identities.
My goal in this video is to start working on some and show how half angle and double
angle identities they work inside some proofs or verify some identities.
I am going to do four examples and hopefully get done in the time span of this video.
So let's get started.
The sine of two theta is equal to two cotangent of theta over one plus cotangent squared of
theta.
Ok.
Well, the sine of two theta you might know, you should know if you have your memorized
and if not start working on it or at least look at your formula sheet, is two times sine
theta cosine theta.
So I need to have all of this written in terms of sine and cosine and do some algebra and
see what happens to get the two sides to match.
Well, what to do...what to do.
The first thing to do is go BAM!
this is cosine over sine and this is cosine over sine, find common denominators and do
all that good stuff, and it is going to look something like this.
Two cosine theta over sine theta over one plus cosine squared theta over sine squared
theta.
We need a common denominator so we are going to take one and make it sine squared theta
over sine squared theta.
That is going to become two cosine theta over sine theta all over sine squared theta plus
cosine squared theta over sine squared theta.
Do you know your identities?
Sine squared plus cosine square is equal to one and any time you have one fraction over
one fraction you can flip that bottom up.
We are going to flip the bottom up and write two cosine theta over sine theta times sine
squared theta when we flipped the bottom up, and don't forget that sine squared plus cosine
squared equals one.
The sines will cancel out.
This one sine in the denominator will cancel out with one of the two sines in the numerator.
Remember you cancel out factors but not terms.
We get two times sine theta cosine theta.
I am just turning the multiplication around to perfectly match the identity.
That is ok because three times five and five times three is fifteen.
This is the sine of two theta.
Now, if you are really at proofs and you just stumbled upon this video or you are just getting
used to them, or you are using this to teach yourself how to do these.
There is a reason the back of your book says that proofs may vary.
Because, well they may vary:)
That is why like teaching trig proofs so much, because there is no just use these five steps
and if you follow them in a certain order you get the right answer.
You have to think a bit more.
Play with the algebra, do some substitution, see what helps, and for goodness sakes if
you have a pythagorean identity that is going to help you right off the bat...maybe you
should use it.
So, I have shown you a proof where I immediately went into sine and cosine and I did an ok
amount of work.
That is fine and I get the right answer.
You know what, this is full credit...BAM!
Well, here is another way.
Somebody might be going, why didn't you do it like this?
And you are going to go, because did it MY WAY!!!
One plus...
I know I am bad at singing:)
One plus cotangent squared, let's write this, two cotangent theta over one plus cotangent
squared is cosecant squared.
Now when I turn everything in into terms of sine and cosine, I will not have to find common
denominators and I will be done much quicker.
This is still going to be two times cosine over sine, so it is two cosine theta over
sine theta.
Cosecant squared is one over, it is one of the reciprocal identities, that is one over
sine squared theta.
Look, see I originally had sine squared plus cosine squared there after finding common
denominators and after you applied that first Pythagorean Identity, so I kind of needed
to know those Pythagorean Identities any way regardless of how I do this proof.
Any way, I am rambling.
The denominator is going to get flipped up becoming sine squared theta over one.
Once again, the sines will cancel out and once again we have the sine of two theta.
So, I did one proof two ways.
Is that two examples or is that one?
I don't know:)
But, when you are doing proofs, when you are verifying identities you want to...
Well, look at the two sides and see what is going on, are the angle measures the same,
are the degrees the same.
The only way you can change the degree is through factoring and canceling (or Power
Reducing Identities).
The way you can change the size of the angle measures is the substitution of the half angle
or double angle identities (or sum and difference identities) or at least that is all I am covering.
See if there is any Pythagorean Identities or any of those heavy duty identities whether
it is sum or difference, half angle or double angle, power reducing, whatever that is going
to help you right off the bat, then go to turning everything into sine and cosine...not
as a last resort because that is a really common skill or process.
But sometimes you can shorten your proof up by applying some identities early on if there
is any there to help you.
Ok, I have a feeling this is going to be a two part lesson for proofs.
How about we take a look at the cosine squared of theta over two, and let's see if we can
prove that this equals the secant of theta plus one all over two secant theta.
Well, cosine of theta over two is the square root of one plus cosine theta over two, not
anything to do with secant.
So there are no identities that are going to help me immediately, no heavy duty identities,
there is no algebra to do, so we are going to have to go into sine and cosine and hope
that the algebra allows us to get the two sides closer and help this click so that we
can see what is going to happen.
So secant is equal to one over cosine theta.
One, I am going to make common denominators now.
One is going to be rewritten as cosine theta over cosine theta, common denominator, over
two times one over cosine theta or just two over cosine theta.
I am going to put the top together and get one plus cosine theta all over cosine theta
over two over cosine theta.
Any time again you have one fraction over one fraction, you want to flip that denominator
fraction up next to the numerator with multiplication.
Any time your denominators are an exact perfect match, they are going to cancel out.
So, we have...wrong color...one plus the cosine of theta over two.
Hmm.
Ok, so you could stop there or you can keep manipulating.
If you are not sure how to make this equal to that and you are totally stuck, then maybe
just playing with the other side.
You know, what is the cosine of theta over two?
Well, it is the square root of one plus cosine theta over two, and then there is a power
of two up here.
See it?
So that means that it is the square root of one plus cosine theta over two in that square
root, squared.
Well, that power of two is going to cancel out the radical and if your teacher wants
to only see the work on one side of the proof, then you can sort of manipulate both sides
on your scratch work, then anything on this other left hand side you can go in reverse
on the right hand side until you get cosine squared theta over two.
As far as I am concerned, the left and right hand sides match and I am done.
BAM!!!
ALL RIGHTY THEN, let's do another one:)
How about one that is pretty straight forward and then we will do a hard one.
Tangent of x over two equals, you might think that these are all hard, cosecant of x...if
this is your first time learning them...minus the cotangent of x.
Ok, if you look at your formula sheet, you will notice that the tangent of a half angle
identity only has sine and cosine in the three versions of that identity.
So, no cosecants and cotangent and we are going to work on the right hand side.
Remember, and also I have not said this in this video, when you are work on proofs you
always want to simply.
Work down the tree, work your way down to the main trunk.
Don't things more complicated and have a thousand branches you can go up to.
Ok, whatever, we are going to make this in terms of sine and cosine.
Cosecant is one over sine x.
Cotangent is cosine x over sine x.
Well, shoot...we are done.
One minus cosine x all over sine x is one of the half angle identities for tangent.
Also don't stop until the two sides are an exact match please!
Do work your proofs straight down using nice and clear hand writing, no scribble all over
the place.
Your teacher, your test papers are giving you the start and finish of the problem, we
have to grade everything in the middle.
We have to see every single step, make sure that your algebra is right and you logic is
correct.
We are not just looking at the answer.
We gave you the answer.
Make sure that your test is gradable, otherwise you may not like what you get back.
Ok, at least my kids won't.
I am not going to try to decipher their hieroglyphics.
The sine of three x over the sine of x minus the cosine of 3x over the cosine x and
all of this is supposed to equal two.
I hope I don't have to erase anything on my board because this is going to be a lot of
writing though not a very long proof.
Well I guarantee you that some of my students are going to want to say that 3x over x makes
2x, or the x's cancel out, or the sines cancel out.
This is not sine times three times x, it is not cosine times three times x, the cosine
and sine are math functions.
It is the sine of (3x) and the cosine of (3x), and no nothing is going to cancel.
Alright, so get that out of your head.
We want to prove that this equals two.
Man, when I look at my formula sheets I don't have three x's.
I have half angle and double angle identities.
So, this must also include some sum and difference identities.
I am going to rewrite this so that it is the sine of x plus two x, that is equal to the
sine of three x, minus cosine of x plus two x, and our denominators of sine x and cosine
x.
I am not even going to mess with the right hand side.
I think I am going to need a lot more room, so I am going to work down and possibly need
some room over here.
I am going to apply the sum identity for sine.
It is the sine of the first times the cosine of the second plus cosine of the first times
the sine of the second minus, don't forget the parenthesis because I am going to write
all this...I am not going to do that right now...all of that is over the sine of x.
This is minus, applying this sum identity, cosine x times cosine of 2x, don't forget
with your cosine function with sum and difference identities to change that middle sign, so
minus sine of x times sine of 2x all over cosine of x.
Now I have got two humungous fractions that need to come together and clearly a lot of
stuff is going to cancel because the answer is two, right.
We need a common denominator so I am going to multiply this entire fraction by cosine
on the top and bottom and sine x on the top and bottom.
WHOOO!!!
Man, now I have to write an even longer sentence.
Ok, so we have...let me get down here... we have cosine x times sine x times cosine 2x
plus cosine times cosine is cosine squared x times sine 2x.
Now I am making common denominators here, so I am going to put a minus sign and a parenthesis
because there are two terms in the second numerators.
Sine x times cosine x times cosine 2x.
I am getting really quiet here because I don't want to make a sign error here.
All of that is over cosine x sine x.
Ok, now what do we have here.
We have cosine x sine x cosine 2x, cosine x sine x cosine 2x, this one is positive and
after we distribute the negative sign that is going to be positive.
So, cosine x sin x cosine 2x is going to cancel with minus sine x cosine x cosine 2x.
That is going to save some writing!
We got cosine squared x times sine of 2x.
I am going to need to go up here now.
I hate jumping around like this.
I hope that you can follow it.
Cosine squared x times sine 2x, negative times negative is positive
sine squared x times sine of 2x all over the denominator cosine x times sine x.
Good thing...
I might still need to erase part of this.
You might want to copy what I am doing here.
Got it?
Ok, so I probably should erase the holiday lights for this.
We have got the sine of 2x and the sine of 2x in both terms and we are going to factor
that out.
We are going to pull this out and get the sine of 2x times cosine squared x plus sine
squared x.
This is all over cosine x times sine x.
Are you starting to feel it.
I am getting excited!
We are almost done.
Cosine squared x plus sine squared x, that is the first Pythagorean Identity and that
is equal to one.
Now we have the sine of 2x over cosine x times sine x.
That becomes....
We have a double angle and that is a single angle, so nothing is going to happen until
we fix that.
This is sine of 2x is two sine x cosine x all over cosine x sine x.
The cosines cancel out, the sines cancel out, and I just did all that work to get an answer
of two.
Are you kidding me?!
BAM!!!
Mr. Tarrou.
Go Do Your Homework!