So, if I told you that sine of v is
3/11 and I gave you either a or c
this would not be very hard.
Because then what you would do is set
the ratio. You would say
3 is to 11 as this is to this.
In fact we know that the sine of v is
is a over c. So,
three elevenths...3 is to 11
as a is to c.
The problem is that you do not know a and you do not know c.
Right, so that is a bit of an issue.
So, how long, what could we do?
We have 2 variables.
We could use the Pythagoren Theorem
to get another equation.
You also know
That a squared plus 4 squared
equals c squared.
So, now we have to do a kind of nightmarish
algebra that's
the reason we will figure it out.
Um, this is the
same as a squared plus 16
equals c squared.
And if you cross multiply,
3 c is 11 a.
So, we can do a little substituting,
Okay?
and solve it. We are trying to find what a is
so we want to get rid of c.
C is 11 a over 3.
Now, we plug it into here.
So, a squared
plus 16
is (11 over 3 a) squared.
Alright, lets solve that.
A squared plus 16
equals 11 squared, which is 121,
a squared over 9.
Another board...
So, we have...
a squared plus 16
is 121 a squared over 9
We all love fractions.
The best way to deal with a fraction
is to multiply everything by 9.
Now you get...
9 a squared plus 144 is
121 a squared.
You are sitting here saying that I would never do this on an exam.
I have faith.
Subtract the 9 a squared..
You get 144 is
112 a squared.
Divide it by 112.
Then you take the square root of it.
a is
the square root
of 144 over 112.
So,
did I make a mistake?
Sure. Let's review.
We tell you that sine v is
3/11. And, we are trying to find a.
So, you go to your triangle
and you say that you know that the
sine of v is the opposite
over the hypotenuse.
That means that 3/11 is
the same as a over c.
And, as I said before, the problem is
there is 2 variables and we only have one equation.
We need another equation.
You say, "I don't know another equation."
"Except, I love the Pythagorean Theorem. It is the only one I ever memorized."
a squared plus 4 squared equals c squared.
If a is 3
right?! and c is 11,
3 squared plus 4 squared is not going to give you 11 squared.
That is the problem.
Okay?
Um, so, now
we have a second equation.
Now you just have to do messy algebra.
So, the way that you solve this is that you could
isolate a or c in this one to get something squared.
That is what you want to do.
So in this equation, you cross multiply
and isolate c
and you call it 11 a over 3.
And, you place it here for c.
Then you square that.
As I said, you multiply through by 9
and you get this nice, not too hard equation.
Subtract the 9.
Confused?
Another
thing that we seem to have this
TA versus Professor conflict, is
when we give you something like,
theta in between
Pi over 2
and Pi
sin
theta is
4/7
find cosine theta
We did some of these one Monday, but let us do one more.
Okay? There is more than one
way to get this right. What you want
to do is to find the most straight forward way
to get this right.
The first thing is where is Pi over 2?
Well, Pi is
180 degrees so that is 90 degrees.
That is 180 degrees
So it is somewhere in the second quadrant.
Remember what I said, "First you need to figure out where to
place the angle.
Now, the sine is 4 over 7.
do Pythagorean Theorem of 4 and 7.
You could do some other squaring
and Pythagorean stuff but there is no need to.
We need to find,
That side because we need to
find the cosine of theta.
Okay? So, 4 squared
plus b squared is 7 squared.
Okay, 16 plus b squared,
is 49.
b squared is 33.
b is plus or minus the square root of 33.
We are in the second quadrant
so be is in this direction. So, b is going
to be a negative value. So, b is going to be
negative square root of 33.
So, therefore, cosine of theta
is SOH CAH TOA.
Negative square root of 33
over 7.
They are going to capture it on video.
I want to keep that on the DL.
Okay. So, some other stuff
you should know.
Sine of
a negative angle
is negative sine of a positive angle.
What does that mean?
What does the negative angle mean?
Well, this is the angle, theta. Right?
You go this way.
Negative theta just means go the other way.
So, that is the corresponding
negative angle. So, if this is
40 degrees
this is negative 40 degrees. If this is
Pi over 5, this is negative Pi over 5.
So, the sine of
positive theta is
that y.So, the sine of negative theta
is going to be negative y.
If you know that the sine of
17 degrees
is point 2.
Then the sine of negative 17 degrees
is negative point 2.
Okay. Not very hard.
The cosine of a
negative angle is the
same
as the cosine of a positive angle.
Because they have the same x.
So the cosine of this
angle is the same
as the cosine of the negative angle.
Okay? So this is what is called an odd function,
and this is an even function.
Later we will do odd and even functions
So, that is one thing you should know.
Will that show up in a problem? Well,
the problem would be using the sine of a negative
angle in it. And we would convert it
negative sine of a positive angle.
That is all that we would be testing.
Another thing is this
angle is theta
and since the sum of these two angles is 90 degrees
or Pi over 2.This angle is
Pi over 2 minus theta.
So, I pointed this out
last week.
The sine of the one angle is the cosine of the other angle.
Right, this is A, B, and C.
The sine of theta
is A over C.
Cosine of
Pi over 2 minus theta.
Is also A over C.
That tells you that the sine of
theta
is cosine
of Pi over 2 minus theta.
Also, known as 90 degrees minus theta.
That
means that they are complementary angles.
2 angles add up to 90 degrees.
or Pi over 2 ok?
So, if the sine of the one angle
is the cosine of the other angle.
And, of course
cosine of theta
is sine of pi over 2 minus theta.
Suppose that I tell you...
sine of 2x-5 equals
equals 0.
Find the smallest x where that is true.
Smallest positive x.
Well,
for sine of 2x-5 equals
0 sine 0 is 0.
We learned that one. So this just means
2x-5=0.
2x=5
x=5/2
Cosine of 2x-5=0
Where is cosine equal to 0?
90
also known as Pi over 2.
So, when you do these problems, you always do them in radians.
So, the cosine of 90 degrees,
the cosine of Pi over 2 is 0.
So, 2x-5 has to equal Pi over 2.
We can just solve that for x.
So you add 5.
and you divide it by 2.
Pi over 2 plus 5
divided by 2. And, you could leave it like that.
Okay? You do not need to simplify unless we ask.
So, 2x-5 = Pi over 2.
So, to solve that you just add 5 to
both sides
and divide by 2.
So, these are all in terms of radians.
Ah, I said the smallest x.
Smallest positive x where it is true.
Now, yes, you could use 3 Pi over 2.
You could use 5 Pi over 2. There is
an infinite number of answers. So, we want the smallest one.
How about the smallest positive x where
this is true?
Lets see if you could figure it out.
Where is sine equal to 1/2?
Lots of places plus
not degrees
Pi over 6.
Do not do it in degrees, you have to do it in radians.
So this means
the sine of Pi over 6 is 1/2.
2x-1
will give you Pi over 6.
Now you need to figure out what x is
so, add 1
and divide by 2.
How about that one?
The smallest positive x where
that is true. Tangent of 4x
minus 3
is 1.
So, you have to say to yourself,
where is tangent equal to 1?
Lots of places, but the
smallest positive value that is equal to 1
is Pi over 4.
Okay?
So, now...
you just solve for x, and you add 3.
and divide by 4.
You can leave it like that.
Do not
do not write that.
The problem is that it looks like you are writing
3/4.
So, if you want to use that slash
make sure you put parentheses around the
Pi over 4 plus 3. I want to recommend
not doing it that way.
Okay?
So what if I asked you that?
What if I was in a really bad mood? So, I
said, "How about that one?"
It is not as bad as it looks.
Because remember just a couple minutes ago,
I showed you an identity.
What do you know about the relationship
between the sine of an angle
and cosine? We know that
sine of the one angle is the cosine of
the complementary angle.
So, cosine of x is the same as
the sine of
Pi over 2 minus x.
Okay?
So, we could say that the sine of
2x-1
is the same as the sine of
Pi over 2 minus x.
So, remember, the cosine of
the angle is the same as the sine of Pi over 2
minus the same angle.
Now, I just say 2x-1
is Pi over 2 minus x.
Yes?
This is x
that angle is
Pi over 2 minus x.
Sine of this angle
is the cosine of that angle.
The cosine of this angle is the sine of that angle.
So, sine of x is
B/C
Cosine is Pi over 2
minus x
is also B/C
Therefore the sine of x
is the cosine of Pi over 2 minus x.
That is where that came from before.
And, the cosine of x is the sine of Pi over 2 minus x.
But, if we give you something like this
This is your typical Webassign homework
meaning that this caliber of problem.
If you just replace the cosine
with sine of Pi over 2 minus x.
The sine of this is the sine of that
and you can pull away your sines.
So, 2x-1 is just equal to Pi over 2 minus x.
Add x to both sides.
Add 1 to both sides.
x equals Pi over 2 plus 1 all over 3.
This should take you a minute.
So, if you want to turn this into cosine.
I could turn this into sine or
I could turn this into cosine.
It would be cosine of Pi over 2
minus whatever
this angle is. So, this angle
is x minus 2.
Be very careful when
you do this it should be the minus theta here.
This is the angle, this is the theta.
So this is
the Pi over 2 minus
theta. Okay?
You sure? I see
a lot of people confused on this.
You have to put it in parentheses because it
would otherwise mess up your minus sign.
Maybe you would not mess up the minus sign,
but most people will.
This means 4x-5
equals
Pi over 2
minus (x-2)
So, 4x-5
is Pi over 2 minus
whatever that angle is.
Okay?
This becomes 4x-5
equals Pi over 2
minus x plus 2.
So, 5x
equals
Pi over 2
plus 7.
Because you divide by 5.
Okay?
x equals
Pi over 2
plus 7.
All over Pi.
Yes.
It is the same thing. Yes.
Not 90 degrees though.
It must be Pi over 2.
Oh, you also
could have made this into sine.
Okay? You could have said sine of x
minus 2 is the sine of Pi
of 2 minus 4x-5.
In the end you get the same angle.
You can convert this to
sine or you can convert this
to cosine. It doesn't matter.
Okay? So, this
is going to be the same as sine
of Pi over 2
minus
2-x.
Distribute the minus sign.
Well, you can throw away the sine.
And you can say, 3x+
4 is Pi over 2
minus 2
plus x.
You must get that minus sign in there.
Pi over 2 minus
whatever that angle is.
Subtract x from both sides.
Subtract 4 from both
sides.
Divide by 2.
Okay?
Feeling happier?
Good.
So, let us see, now, what these would be.
Sine of theta plus Pi.
Cosine of theta plus Pi.
What does that mean to be plus Pi?
Well, let us think about the unit circle for a second.
If that is theta.
If you are going Pi
it is going 180 degrees.
So that is coming around to the other side.
This would be theta plus Pi.
I went Pi. I went half way around the circle.
So, sine
here is the negative sine
here. The cosine
here is the negative cosine
here.
So, by doing theta plus Pi
I am switching the signs
from positive to negative or from
negative to positive. So, again this angle
theta. When I went
this far.
Then, adding another Pi means I am going
half way around the circle. I am doing 180 degrees.
That means now I am down here.
So, all that happens is values
that were positive here, become negative here.
and values that become positive here were negative here.
And, the other way around.
And, of course
if I add 2 Pi,
I get back where I started.
So, what does 2 Pi mean? It means that I went all the way around the circle.
In fact when we have 2 Pi,
we are back where we started and everything starts
repeating. So that is called a
period.
A circle is one full period and it starts over.
So, where is theta plus 2 Pi?
Well, that is theta,
that is theta plus 2 Pi. I went one
full cycle around.
In fact, anytime
I go Pi.
I change the signs
from positive to negative and negative to positive.
Anytime I go 2 Pi,
I get back where I started,
so,
If I take the
sine of theta and I
add k time Pi.
Where k is an integer.
So, I have sine of theta
plus k Pi
k is an integer.
Then from that you pull,
Sine is theta
if
k is even
or negative sine
of theta
if k is odd.
Similarly,
If I have cosine of theta
plus k pi
that will be
cosine of theta
when theta is even.
negative cosine of theta
if k is odd.
So, this helps us figure out something like.
sine of
13 Pi over 6.
One thing you need to do of course
is to turn that into degrees.
Another thing you can do of course is say
that is Pi over 6
plus 12 Pi over 6.
Why do I need the 12 Pi over 6?
Because that is 2 Pi.
That is the same thing as sine
of Pi over 6
Plus, 2 Pi.
And we know that if i add 2 Pi,
and because 2 is an even number
that is just the same
as sine of Pi over 6.
Did we all memorize our charts yet?
NO?!
You should get them tattooed.
Tattoo it right about here.
So, you can cross your leg in the test.
You can look down and "Oh!"
It is equal to a half.
If I give you this, you should be able to convert it to degrees.
We should try to figure out
how many 2 Pis am I adding to this?
That is another way to figure it out.
So, one thing you can do to make your life easier,
just sort of start subtracting
2 Pis
Why 2 Pis?
Anytime I add 2 Pi I go once around the circle.
Okay? So if I say,
how many times around the circle do I go?
Well, let's subtract 2 Pi.
That is 17 Pi over 4 minus
8 Pi over 4.
Which is 9 Pi over 4.
Okay?
So,
sure.
17 Pi over 4 is where?
Well, I have to start
somewhere and
it is bigger than 2 Pi, so I must have gone around the circle
more than once.
The question is how far did I go?
I do not know. You just take the 2 Pi away
and see where I am. It is like subtracting
360 degrees.
Absolutely!
Yes. Okay.
Let us subtract another 2 Pi.
Which is 8 Pi over 4.
Which just leaves Pi over 4.
So, that means that
cosine of 17 Pi over 4,
is going to be the same thing
as cosine of Pi over 4.
Because 17 Pi over 4
is really Pi over 4
plus 4 Pi.
What is the cosine of Pi over 4?
How did I get the 8 Pi over 4? 2 Pi is 8 Pi over 4.
2 is 8 over 4.
It could mean 12 Pi over 6, but I want
to have Pi over 4 when I am done.
How about
41 Pi over 6
41 over 6 is bigger than 2 right?
So, it is 41 Pi over 6 is
bigger than 2 Pi.
So, I have to figure out what angle this is.
It is less than 2 Pi.
Everybody agree? It gets down to
some angle less than 2 Pi.
So, 41 Pi over 6
I will subtract 2 Pis.
So, 2 Pi
is 12 Pi over 6.
I am just doing fraction work.
I am going to take 41 6ths and I am going
to subtract
12 Pi over 6.
and I get
29 Pi over 6.
Is 29 over 6 bigger than 2?
Yes!
So I need to take away 12 over 6 again.
Is 17 over 6 bigger than 2?
Yes! I need to take away another 12 over 6.
I could have done it all in one step of course.
Now I am down to 5 Pi over 6.
5 Pi over 6 is
not. Now I know where to stop.
Cause now I am working with a number that is
less than 12 Pi over 6. Less than 2 Pi.
Which is 12 over 6.
So, sine of 41 Pi over 6
is the same as saying,
sine of 5 Pi over 6.
So, then, we need to figure out what
is the sine of 5 Pi over 6.
150 degrees?
You do not have to convert it to degrees.
But, most of you are more comfortable with that.
So, 150 degrees is
about there.
Our reference angle is 30 degrees.
So, this is the same as the sine of 30 degrees
or the sine of Pi over 6.
The sine of 30 degrees is...
1/2.
Now is it positive or negative?
Positive because we are in the second quadrant.
So, therefore,
the sine of 41 Pi over 6
is 1/2.
By the way, we could always turn this into degrees
and subtract 360. It is
exactly the same thing.
You can turn it into degrees
subtract 360 or you can
leave it in radians and subtract 2 Pi.
They are the same thing.
We got to subtract 2 Pis.
Alright, now, you can
do it all at once, by the way. You could
look at this and say,
8 Pi over 4 is 2 Pi.
So, you look at this and say, okay now you know I will need
more than one 8. You may not notice that I can subtract 16 right away,
but you will.
So,
23 Pi over 4
minus 2 Pi
is 23 Pi over 4
minus 8 over 4
is 15 Pi over 4.
Take away an 8 over 4 again.
Now I have 7 Pi over 4.
7 Pi over 4 is
less than 2 Pi. So
this is the same as the cosine
of 7 Pi
over 4.Let's not convert this
into degrees. Let us do it in radians.
All the way around,
is 8 fourths.
So this is one fourth less than that.
So, 1 Pi over 4
less than
M 8 Pi over 4.
So, that is going to be Pi over 4.
Okay?
You understand that?
It is 8 fourths all the way around.
So, 7 fourths we have
1 fourth left over.
So, this is the same as
the cosine of Pi over 4
which is
Why is it not 3 Pi over 4?
Because, once I subtracted
the 2 Pis I got down to 7 Pi over 4
Okay, and that is where I stopped because that number
is less than 2 Pi.
Well, it is okay to say
negative Pi over 4.
Yes.
Is it positive or negative?
Good.
Okay.
So, where did I get 8 fourths from? I am doing
fractions. Okay? That is 23 fourths.
And I need to take away 2.
So, 23 fourths minus 2
is 23 fourths minus
8 fourths is 15 fourths.
Okay?
15 fourths minus 2.
Is 15 fourths minus 8 fourths.
7 fourths.
So, I covered almost everything
that I wanted to cover today.
But, not everything.
So, Wednesday we are going to do graphs.