BAM!!! Mr. Tarrou. Welcome to one of my favorite subjects ever!! Trig proofs. We are going
to use the Pythagorean Identities, Quotient Identities, Reciprocal Identities, our values
or properties about even and odd functions, and that is it at this point to start working
and learning how to do trigonometry proofs. Trigonometry proofs are going to be not your
nasty two column proofs from Geometry, or maybe loved them like I did:) But, we are
not going to be doing a line of Algebra and then writing on the side the name of what
we are doing. We are just going to do some Algebraic manipulation, we are going to some
substitution with these identities to try and prove identities. So it is going to look
like an equation and that means that you are basically going to have the start of the problem
and the answer given to you at the same time. All you are going to have to do is make those
two sides match. Now, as we use these identities we are going to be allowed to manipulate them
and adjust them, especially these three pythagorean identities, well adjust them for use in our
particular problem or proof that we are doing. Now when you are doing a proof, you are going
to want to again make the two sides match and can do some substitution with these identities.
What you are not allowed to do though, is you are not allowed to move, or do mathematical
functions to both sides of these proofs. Like you are not allowed to add two to both sides,
subtract a sine from both sides, not allowed to square root both sides of the equation...or
the identity that you are proving. You are only going to be able to work with each side
independently and you can work with both sides of these identities, but you cannot move things
from side to side like you are actually solving an equation. You should already have these.
They should be in the back of your PreCalculus book or trig book. So let's erase this and
start doing some problems. I am going to verbally give you a lot of tips and suggestions along
the way. If I say something that you think is very helpful that you have read or heard
before, then write that note down. Ok, let's get this out of here. If need be, I might
erase my bottom set of lights. Yes, it is the winter season, the holiday season and
I am going to be wearing funny shirts for the next seven days. I hope you appreciate
them. I hope that it is not too hot when you are watching these videos thinking...uh.."Why
does he have those lights up and holly?" We have the cosine squared theta times the tangent
squared theta plus one equals one. And in my class, I jump around a bunch when I do
these because I have a lot of chalk boards. I am not going to be able to run over and
point to an identity that we are using, so hopefully you have that formula sheet copied
down from the first few seconds of the video. So we have...What this saying by the way,
by the way.... it is an identity. It does work. But our job is to verify that it works.
To say that no matter what theta is, when you plug it in both sides of this identity
are going to be equal. I can't call it an equation because I am not going to be moving
things around. Like I am not going to divide both sides by cosine squared. So what kind
of substitution are we going to be able to do to make the two sides of this identity
equal? When you go to solve identities..or prove them..verify them, you want to make
sure all the angle measurements are the same, and in all of my examples today they are.
This is the very first video of what is probably going to be long series of lessons for doing
proofs. You want to make sure all these angle measures are the same, and the next thing
you want to do is you want look for the three pythagorean identities. Like any of the big
heavy duty identities and right not we only have three, the three pythagorean identities.
sine squared plus cosine squared equals one. One plus tangent squared equals secant squared.
And one plus cotangent squared equals cosecant squared. See if any of those big identities
help to simply and make these two sides equal. Then if there is, if none of those identities
help which they will with this one, then move on to ok... All the angle measurements are
the same, non of the three pythagorean identities which is all that we know right now to help
us out are going to be of any great help, and there is no algebraic steps to do. Now
we do have parenthesis here so I might be tempted to just distribute the cosine through
and maybe we will. But, you want to look to see if any of the 3 pythagorean identities
will help simply that you can follow that are being asked in the identity, and then
if that is not the case you last step that you want to do is rewrite everything into
sine and cosine. Sine and cosine are the two basic trig functions. They make up the other
four trig functions. So if there is nothing else left to do, make everything in terms
of sine and cosine and again let the algebra to sort of lead your way and attempt to make
the two sides match. And sometimes you will try stuff and nothing happens. You might just
go down a long path and end up...well let's just erase this and start all over again.
Tangent squared theta plus one is equal to secant squared. So we can use one of our Pythagorean
identities to take this part of the expression out and replace it with secant squared theta.
Then we have got cosine squared theta equals one. Ok, well we are really close to final
answer. This is just the beginning introductory type proof so it is nice and short. Secant,
the reciprocal functions that I just erased, or reciprocal identities, is the reciprocal
of cosine. Instead of an x over r, it is r over x if you are dealing with the unit circle.
Well, here we can write this as the cosine squared theta times one over cosine squared
theta equals one. And of course, whenever you divide anything by itself you get one.
So one equals one. BAM!!! The two sides match. That says, no matter what the theta measure
is, whatever the angle measure is, cosine squared times tangent squared theta plus one
will always equal one. So we just did our first proof. Pretty cool:) Alright... So,
I am getting excited because I love these problems. Let's take a look at cosine theta
over secant theta plus sine theta over cosecant theta equals one. Now, on the left side I
have two terms and on the right side I only have one. A lot of times when you have a different
number of terms on each side of the identity you are going to want to find common denominators
and add those two terms together and see what happens. And indeed we can do this, but um...
And that is another thing too. When, like when I am just teaching algebra one and I
am teaching my students to solve linear equations I can just give them five steps, and my students
follow those five steps in a particular order they are pretty much always going to get the
answer. Maybe a really difficult question might take an extra step, but I can say here
are the five steps, do them in this order and know that 95% of the time you are going
to get right answer. That is not the case with proofs. You might be at home finding
common denominators, or whatever, and putting these two fractions together to make one term.
And that can very well be a correct answer. Please don't think that what I am writing
is THE solution. Many of these proofs have three, four, five, six, maybe more ways to
prove or make the two sides match. I am just giving you examples. That is why the back
of your textbook says solutions will vary. Well secant is one over cosine, so cosine
theta over one over cosine theta plus sine theta over one over sine theta. You see what
I am doing? I am taking out secant and replacing it's reciprocal identity and doing the same
thing with cosecant. Now you can take these bottom fraction and flip them up. You know
that there is really no true division of fractions, you multiply by the reciprocal of the second
fraction. Well that is always going to be your denominator. The cosine of theta times,
this bottom flips up, cosine of theta times the cosine of theta is going to be cosine
squared... Plus, again this bottom is going to flip up. You are dividing by a fraction,
so you multiply by the reciprocal. So sine times sine is sine squared theta. If you don't
know your Pythagorean identities yet, go look at those notes you just copied, or the back
of the book, what is the first Pythagorean identity? Sine squared theta plus cosine squared
theta equals one. Now I did a different looking proof for this in another video, Introduction
to Fundamental Theorems with Proofs. You can prove the Pythagorean identity is correct
by using the x over r ratios, the y over r ratios off the unit circle, do a little bit
of algebra and incorporate the pythagorean theorem and prove that this does equal one.
However we are not going to proving the basic fundamental theorems or the pythagorean identities.
That was done in a previous lesson, we doing some more in depth proofs. Not proofs that
are going to be using the ratios of x, y, and r, though you could do that as well. Let's
take a look at another one. We have cosecant squared of x over cotangent of x equals cosecant
of x times secant of x. Ok.. Well, let's see what happens. Not much! There is no addition
to work on, there is no common denominators to worry about trying to make, there is no
multiplication, or parenthesis with exponents, there is no algebraic steps that we can do
right now. So, well... I don't know. When you get to a point where there seems to be
nothing that you can work on, that is when you make every in terms of sine and cosine.
Because again those are the building blocks trig functions. And you also when you do proofs,
you want to attempt to work or start with the more complicated side and simplify. Think
about a tree. If you are at the top of the tree where it is all complicated and there
are branches all over the place and you work your way down the tree, you are going to come
down that one trunk. You are going to simplify down to that one answer at the bottom. Where
if you start at the bottom at something simple and you start adding stuff to make it more
complicated, you can go in all kinds of different directions. You can be anywhere up inside
the tree that you lead yourself. So you always want to try and simplify and these kind of
honestly look equally difficult or complicated. So I am just going to pick a side and write
it terms of sine and cosine. This does have a power of two so let's work on that. Cosecant
squared over cotangent. That is going to be, cosecant is one over sine, so that is going
to b one over sine squared x over cosine of x over the sine of x. Well again, fraction,
and a fraction bar means division, so you have a fraction divided by a fraction and
when you divide fractions you multiply by the reciprocal of the second fraction. If
it was written side by side, the denominator would be the second fraction. I make this
little pig-tail notation, not that this a real math notation to remind you that we are
going to take the bottom fraction and flip it up....multiply by the reciprocal. This
is going to be one over sine squared times the sine of x over cosine of x. Ok, well now
you can cancel common factors, you cannot cancel common terms though. Let's take a look
at... We have sine of the first on top, we have sine to the second on the bottom, when
you divide like bases what do you do with those exponents? Or you might just be saying,
"Come man go ahead and cancel those out!" The one sine on top will cancel with one of
the two sines in the denominator. This gives us one over sine x times cosine x. And one
over sine x is cosecant x. And one over cosine x is secant x. We only worked with the left
side and we got it to perfectly match the right hand side. I hear my alarm going off,
so that means I am getting close to being out of time. Sign me up or look me over...LOOK
ME OVER??!!! How about check out part 2 where we do some more examples. Bam!!! Mr. Tarrou
Go Do Your Homework!