Alright. Sine and Cosine.
Sine and cosine and tangent
are all about the ratios
and right triangles.
You have a triangle and,
since, right triangles
right triangles can very easily be similar
because they all have a right angle.
You just need another angle
and then triangles are similar.
Imagine you have a right triangle
with a bigger right triangle.
Both have 90 degree angles,
that's what makes them right triangles.
And, now you need another angle.
So, let's call that angle 30 degrees.
And, the third angle has to automatically be equal,
because the angles always have to add up to 180.
So, once these two are equal
and these two are equal, then
those two have to be equal.
So, in this case they both have to be 60.
Then the idea of trig is
the ratio of any side
and you have a side here
is the same as the ratio from here to here.
on any size right triangle
as long as they have that 30 degree angle.
So, we just give names to the ratios.
The names of the ratios are sine,
and cosine, and tangent
There is actually a reason why we call them sine, cosine and tangent.
But, I will not go into why we call them sine, cosine and tangent.
Okay. Um.
So,
let's remember how we define those.
I am going to erase
ah, I'll leave that there.
So, we have a right triangle.
We call this angle theta
because that is a Greek letter. We can call it x.
We can call it anything we want.
Then the sine of that angle
is the opposite side over the hypotenuse.
Okay?
Because the sine
is the opposite over the hypotenuse.
The cosine
is the adjacent over the hypotenuse.
Tangent
is the opposite over the adjacent.
That is how we have that mneumonic
SOH CAH TOA.
You all should have seen at some point in your life.
You are seeing it again.
Okay. By the way, you could go on in biology
and never see trigonometry.
I don't know how far you are going to go.
You could go to med school and you are not going to find trigonometry.
Um.
Okay, so. This helps us
name the ratio.
There is also some problems
lets step back a minutes.
The three triangles
Can you guys see over here?
When I write on that board? Alright.
So, the 30, 60, 90 triangle.
In order to choose a 30, 60, 90 triangle
so that each side are always the same proportion.
So, the 30, 60, 90 triangle
whatever side is opposite
the 30 degrees is always
half the hypotenuse.
If this is 10 the hypotenuse is 20.
If this is 100 the hypotenuse
is 200. If this is 4
the hypotenuse is 8. It is always half of
what the hypotenuse is.
The third side just comes out of the Pythagorean Theorem.
It is the square root of 3.
Okay?
And that is true, so here, the proportion
is the same.
So, let's say this is 10.
And, that would be 20, and that would be
10 times the square root of 3.
Now, that means that the sine is
always going to have the same ratio.
So, sine is 30 degrees is always going to come out a half.
The cosine of 30 degrees
Is always going to come square root of 3 over 2.
It doesn't matter what the sides are,
because the ratio will come out the same.
So, whatever you put in,
you simplify and reduce.
It is always going to come out the same number.
And, the tangent
will always be 1 over the square root of 3.
(Laughter)
Alright. The sine of 60 degrees
I was sitting with some professors yesterday
and they were complaining about students.
So, I will treat you as adults.
Sine of 60 degrees
square root of 3 over 2.
Cosine of 60 degrees
is a half.
Like I said, these two are the
opposite of each other. Okay? The sine of one
is the cosine of the other
Cosine of one is the sine of the other.
Then, the tangent of 60 degrees
Well, you go tangent of 60 degrees.
And, you say, its square root of 3 over 1.
Here if you do 10 square root of 3
over 10, notice what happens...
The 10s just cancel and you get the square root of 3 again.
Okay? So, that ratio doesn't change.
Now we will go to another type of triangle.
These are called the special angles.
These are the ones you are expected to memorize.
Because, you don't know the sine
of 73 degrees. I don't know the sine of
73 degrees. You need a calculator for that.
You don't need a calculator for these.
These are the ones that we put on tests.
Okay? If we had a
73 degree angle on a test
your answer would be something like sine of 73.
And, I expect you to know that's
you know 8 something.
Okay? But if we say,
sine of 60 degrees, you are supposed to know that it
is the square root of 3 over 2.
If you have
A right triangle where these 2 sides are the same,
the two legs,
then the two angles are also the same.
They are both going to be 45 degrees.
And, the ratio of these two sides
are only 1 because they are the same
and that is the square root of 2.
You can find sine and cosine and tangent of 45 degrees.
The sine of 45 degrees and cosine of 45 degrees
they are going to be the same.
Alright. Because if you look at
45 degree angle the opposite side
and the adjacent side are the same.
So, the sine and the cosine are going to come out the same.
They are both going to be 1 over the square root of 2.
For no special reason, we simplify this
the square root of 2 over 2.
I don't think that is more simple.
In fact, if we had to square it. This is
messier, and this is more simple. But,
this fits nicely into a number line.
Okay?
And, the tangent of 45 degrees,
On the tangent of 45
is the opposite over the adjacent.
They are both 1, so, that is just 1.
So, then I came up with a nice handy way
for everyone to memorize this.
Okay. Sine and cosine are always over 2.
So, this is 1,
2
3, and that goes
3, 2, 1
Did any of your teachers teach you this in school?
One. That's it?
That small a group?
Okay. Because this used to be the way to learn it.
And, then they came up with this unit circle stuff.
And, that doesn't work as well.
Because then you guys never memorize this.
And, it isn't as hard
if you get this stuff ingrained.
Then you don't have to think about it. You know where it
comes from and you get this stuff ingrained
and it stays. Just like when
you learn a language or a sport
or music. Okay?
So, the sine
okay, so,
tangent turns out
to be sine over cosine
I will show you why in a minute.
Okay, so, that is your chart.
Um.
So, memorize this. As I said
to you the other day, what you do when
you walk into the exam
pick out a corner of a back page
or wherever you want to put it.
So, that you dont have to think
about the sine and cosine and tangent.
Okay?
Now let's just have a little more fun with this.
Alright.
Ah. Remember that the sine
of that angle theta, A over C,
opposite over hypotenuse.
The cosine
theta, B over C,
and the tangent of theta, we wrote this already.
The tangent of theta
A over B.
If you notice, if you take these two
and divide them. You take A over C
and you divide by B over C,
you get
A over B
The sine of the angle
divided by the cosine of the angle.
And, that is going to be very useful. It is
going to show up a lot. Okay.
What happens is I will give you the sine, I will give you the cosine
and I will say, find the tangent. Just divide them
You know that A squared plus B squared
in this triangle,
equals C squared, right?
Good. Well let's take everything and divide it by C squared.
What is C squared divided by C squared?
One.
Well, you are in for another.
So, neat, because we write this as
A over C squared
and B over C squared
equals 1.
Okay.
So, A over C
is the sine
and B over C is the cosine.
So, we can replace this
with sine of theta squared
plus cosine of theta
squared
equals 1.
That's the second thing to memorize.
Over on this board.
Sine squared of any angle
plus cosine squared of the same angle
will be 1.
That is the Pythagorean Theorem
in the trigonometry world.
Okay? We write the squared here
because although it is sine theta squared,
like this,
then it is not clear if I am squaring theta or squaring sine.
By putting it here,
that tells you I am
squaring sine, not the
angle. If you do it
on your calculator, you do sine of
the angle closed parentheses
then you square it.
If you square it before you
close the parentheses it will square the angle.
Okay? When you use the
calculator and you want
to do sine of an angle, close
parentheses, then square it.
Also, you using your calculator
make sure you are in degree mode.
The calculators have more than one mode.
And often if you are having
problems you are in the wrong mode.
Okay?
So far, so good?
Okay.
I was telling you guys the other day.
You are ancient Babylonians
and you want to knock down a wall
you have to figure out how high it is.
So, a person stands 40 feet from the base of a wall.
That wasn't far enough back then, though, right?
It would be a lot farther.
40 feet from the base of the wall
and you measure the angle of elevation,
this angle, and it is 60 degrees.
How tall is the wall?
So, let's set it up.
Okay, this,
we are standing to face the wall
we are 40 feet away.
We measure that as 60 degrees.
Right. So, how tall is this?
So, let's use a trig function.
So, which trig function would we use?
Tangent.
Good. So, we have the adjacent
side and we want to find
the opposite side. Tangent
is opposite over adjacent.
You look at this and say, "Well, tangent of 60 degrees
is the opposite over the adjacent.
Cross multiply.
And you get 40
times the tangent of 60
is x.
Now, what is the tangent of 60?
Square root of 3.
So, 40
times the square root of 3
is x. Okay? Yes.
Why do I use tangent? Alright.
If I am looking at the wall, and I have this distance,
that is the 40 feet from the bottom
of the wall. And I want to know how high the wall is.
So, I have my adjacent side
that is the one that is touching my feet.
Student: How about the other side?
I don't know what my hypotenuse is and I don't care what my
hypotenuse is. So, which
trig function uses opposite and
adjacent? Tangent.
Okay?
Any other questions?
Feel free to ask.
Yes.
Okay. So let us do that again. (laughter)
That is alright. This is the hardest part
of trigonometry. How do you
know which trig function to use.
First question to ask is:
do you want to know the hypotenuse?
Do you want to know the angle distance?
Or a side distance?
If I don't knowthe hypotenuse,
I have to use tangent.
Because there are only 3 things you know.
SOH CAH TOA
Sine and cosine both use the
hypotenuse. If I don't know the hypotenuse,
I have to use tangent. Okay?
So, opposite side is
I am looking at the triangle, standing at the
corner and it is
the one that is directly opposite me.
The adjacent side is the one under my feet.
The tangent is
the one shooting the bow and arrow
the tangent would be the diagonal.
Where the arrow goes. If I have,
a rope I am
holding up to a flag pole
or something. That would be the diagonal.
Okay. That is the hypotenuse.
So, if you don't know the hypotenuse, I
have to use tangent. Okay?
We will practice this again.
So, one side of a triangle, I know that I am looking at
the side opposite me. And, I have the side
adjacent to me. That is why
I am using tangent. Okay,
so, I say, "Well, what is the tangent?" The tangent is
that opposite side divided by
this adjacent side. That x
divided by the 40.
And, I know that is equal to the square root of 3,
because the tangent of 60
is the square root of 3.
From the chart.
Which you are going to, the chart,
which you are going to write down on your test
so you don't even have to think about it.
Okay. Sure. You want me to do it one more time?
Yes. One more time?
Okay.
Um.
So, once more. When you are
given the word problem you have to
think about what we give you and what we don't give you.
Okay? So, I will give you
another word problem. In a second
and you will see the difference.
So, I give you how
far I am away from a wall and where I am standing.
See I am not standing at the top of the wall and looking down.
I am below the wall and looking up.
The adjacent side is the distance from below my feet
to the bottom of the wall.
Everyone understand that? Adjacent means next to.
Yes.
Not quite the angle of the direction.
Okay, so, if you
hold your hand out straight and you are looking at something.
And, you elevate your head as the angle of elevation.
You drop your head as the angle of depression.
You are depressing your head. So,
that would be the angle of depression. So, the angle of depression
is almost the same thing as the angle of elevation.
Okay?
So, there is 2 ways to measure things.
So, if you are in an airplane and you want to land.
Right. So, you are flying straight and now you are going to
dip down, so, now you are depressing the angle.
That is the angle of depression.
Okay?
Um,
so, again.
So, you look at where you are standing and you imagine
a triangle and you say, "I am standing here."
I will draw a really good representation of me.
My own work.
And I say, "My distance from
the bottom of the wall. Well, that is
adjacent from where I am. That is the
adjacent side. This side is
opposite me. Because,
it is opposite me. This is
the hypotenuse. I have no idea what the
hypotenuse is." Okay?
So, now that I know
the adjacent, and I am looking for the opposite,
I am going to use tangent.
Okay?
Then, I set it up. I say, "Well, the
tangent of that angle is the side
opposite divided by the distance
of the adjacent side.
I cross multiply. Because, we love cross multiplying.
By the way, you can cross multiply
in one direction without going in the other direction.
Then you don't have to do both.
But, if you wanted to you can imagine there is a 1 down there.
Now, that I cross multiplied, I know that
x is 40 times the tangent of 60.
The tangent of 60 is just the square root of 3.
Okay. Let us think about another one.
Okay,
a pole
you know that thing that sticks up straight.
Is supported by a rope
that runs from the top of the pole
to the ground. The rope
is 50 feet long, and
I'm sorry, "fitty" feet long, (laughter)
makes an angle with the ground of 30 degrees.
How tall is the pole?
Pole supported by a rope.
well, from the top.
Assuming the pole is standing straight up.
i could leave that out.
The pole could be tilted.
But, assuming that the pole was standing straight up.
And, rope runs from the top of the pole
down to the ground. You are
getting the hypotenuse. Okay?
And, that rope is
"Fitty" feet long. (laughter)
Okay?
And, the angle that it makes with the ground
is 30 degrees.
So, I want to know.
How tall is the pole?
Okay?
Are we actually getting answers?
Alrighty.
Um. Well, you are correct,
but let's see.
What trig function do I use?
Sine!
How do I know I use sine? Because
opposite over hypotenuse.
This is great!
Okay. I have the hypotenuse
that is the rope. That is the diagonal length.
I am looking for the opposite length.
It will be sine
of 30 degrees
is x over 50.
Then, I cross multiply,
and I get 50
times sine of 30
equals x.
Sine of 30 is a half.
So, 50
times a half.
And, x is 25.
Can everyone read that down there?
That is not hidden, but it is low.
Good. Alright.
So, that's basic premise of this course.
How do we make this harder?
We don't use 30 degrees.
We use something like 12 degrees.
Then, you need a calculator. The principal doesn't change.
So, in fact, because this is a special
triangle we do not really need to use trigonometry.
A lot of you just know about a 30 60 90 triangle.
But, you should get the principal.
Because, as I say, what would you do if I gave you 12 degrees?
Well, then the answer would be
50 times the sin of 12.
These would be alone.
If I were to tell you it was an x equation.
It would be 50 sine of x.
okay?
Got that? Should I repeat all of this?
Another round. Okay.
Once again. I am at the top of the pole.
I have a rope that runs to the bottom of the pole.
At the top of the pole, the rope runs down at an angle.
So, that is going to be the hypotenuse of the triangle.
Then, I tell you that the angle that the rope
makes with the ground is 30 degrees.
Okay?
So, now that I know this distance.
I am at this angle.
I am looking for tall the pole is.
That is x. The height of the pole.
I can use sine.
The sine uses opposite over hypotenuse.
I wrote that somewhere. Ah, opposite over hypotenuse.
Okay?
So, I say that the sine of 30 degrees
is the height of the pole
divided by 50.
I cross multiply so that
50 times the sine of 30 degrees equals x.
Then I just solve. Sine of 30 is a half.
I know that there.
50 times a half is 25.
Okay.
Alright. Another word problem or we will move on to more exciting stuff.
Who wants another word problem?
So, we are sitting here and we are saying, "Well, I know
how to do the sine of 30 and 45
and 60 and 12
You can do anything in your right mind.
Then, you say to yourself,
what if I want to find the sine, the cosine, and tangent.
Of something that is not in a right triangle.
How do I...okay...so the angles
in a right triangle are limited to being
between 0 and 90 degrees.
Why is that true?
Think about a right triangle.
This angle is 90 degrees.
The other two angles have to add up to 90 degrees.
Because the total for triangles is 180.
So...
these have to add up to 90.
If 1 is 1 then the other is 89.
If this is .1 than the other is 89.9.
You can't get bigger than 90 degrees.
So this is limited to only being able to find
sines, cosines, and tangents
of angles between 0 and 90.
So, some very clever people said,
"Well, let's see if we can find them of an angle
more than 0."
More than 90, sorry. or less than 0.
What they did was they went off
the coordinate axes and they said
ok, let's pretend the angle
I showed you this last time. But, we will do it again.
The angle is say here.
And, I want to find
the sine, cosine, and tangent of that angle.
But, that angle is more than 90 degrees.
It is a hundred and something.
We are looking for
sine and cosine of this angle
we will call theta.
People looked around and said, "Well,
I could just
use this angle instead.
180 minus theta.
The opposite and adjacent
do not make any sense.
Why don't they make any sense?
Well, because I do not have a 90 degree angle to work with.
If you did this instead,
let's just define it,
as the coordinates of the circle.
Oh, could you do that again.
(laughter)
So, why don't we just call sine and cosine the coordinates of that point.
Why can we do that?
I will do a little erasing.
Well, we got the circle.
Let us just look at the first quadrant for a minute.
I got a triangle that is just touching.
If I make the radius of this circle 1,
the coordinates of this point are (x,y).
And, now I have found some angle, theta.
And, sine
theta
is what? Is the opposite over the hypotenuse.
It is y divided by 1.
I don't need to write the divided by 1. It is just 1.
And, similarly
if I want to find the
x value, the cosine of theta
is x divided by 1.
So, the coordinates of that point
are really (cos (theta), sin (theta)).
So, any point on the unit circle
the coordinates of the point are cosine of theta, sine of theta.
This is very easy because now you can say
okay now I can find sine and cosine of angles
other than
between 0 and 90 degrees.
Now, let us go back to our example.
I will draw another circle.
Now, I find the sine
and the cosine
of this angle.
But remember it is just the coordinates of this point.
So, the coordinates of that point
are this distance and this distance.
It is the x distance and the y distance.
If I wanted to find
some angle down here.
Again, they are just the cosine and sine at that point.
For that angle.
So, in general, if I wanted to find
sine and cosine of angles
other than between
0 and 90 degrees
I would put them on the unit circle.
So, let us do an example.
Let's find the sine
of 120 degrees.
It is better to have examples.
Yeah, that is terrible.
So where is 120 degrees.
Well, I can start wherever I want.
But it is tradition to say,
0 degrees is on the positive x axis.
That makes this 90,
that makes 180
and, that makes this.
270 degrees, and then
360 degrees
is going all the way around the circle.
So, where would 120 degrees be?
Well, if this is 0
and this is 90
you just keep going until you get to about
there.
Okay? Imagine you are 0.
And you rotate to 90.
and you keep rotating until you get to 120.
So, if I want to find the sine of 120 degrees.
I need to the the sine and cosine
This angle, well,
that is going to be our theta.
How big is theta?
60 degrees. Why is it 60?
Because that is 180 and that is 120.
There is 60 degrees left over.
So, what is the sine of 60 degrees?
Well, it is radical 3 over 2.
You have to be careful now.
Somewhere over here in the second quadrant.
So, this distance
sine is still going to be positive.
because it is still
pointing up. The y direction
is pointing up is positive and negative is down.
If I wanted to find the
x value the cosine
is going to be negative because I am
pointing this way on the x-axis.
If I said, the cosine of 120 degrees,
that would be the same as the
cosine of 60 degrees.
so it is negative.
Cosine of 60 degrees is a half.
So, this would be negative one half.
Should I repeat that?
Okay. You think about the coordinates
on the x axis and the y axis.
As long as I am pointing up,
I have a positive value.
Positive y value.
So, the sine
is the opposite of this angle.
Sine is that length and that is a
positive distance which is just up the y-axis.
However, when I am going
out the x axis
and going in the negative x direction.
So, the x value would be negative.
So, the cosine would be negative, not positive.
It would be exactly the same as if
it was pointing this way
but it is negative instead of positive.
How are we doing back there?
Are you getting it?
Watching a video on your phone?
So, how do I get the 60 degrees?
This is 180 degrees.
And, I want 120.
So, 60 is the left over angle.
Well, I was looking for sine.
Another question.
I am going to do a few more examples.
If I do a few more you guys will get it.
So, what is the answer of sine of 120? Square root of 3 over 2.
And the reason is it is the same thing
as the sine of 60 degrees.
Cosine of 120 is negative a half.
As demonstrated,
cosine is negative there and sine is positive.
Okay, so let us go over why these things are positive or negative.
Maybe we will just use 210.
Maybe that will work.
How do you find the sine of 210 degrees?
First, I have to find
where am I going to put 210 degrees.
Okay, 210 degrees
is where? Well, this is 0.
and this is 90.
This is 180.
So, 210 is somewhere
down in the 3rd quadrant.
Okay. If I pick a point in the 3rd quadrant.
What do I know about the sine
and the x and the y coordinates?
Well...
Any point down here is going to have a negative
x-coordinate and a negative
y-coordinate. So, negative x
and a negative y.
So, since cosine of theta is the
x coordinate it is going to be
negative. The sine of theta
is the y coordinate. Is also
going to be negative. So, in the previous problem,
I was in the second quadrant.
X coordinates are negative
and y coordinates are positive.
So, since the x coordinates are negative,
cosines come out negative.
Y coordinates are positive
so, sines come out positive.
So, I go here and I say, sine of 210
well, this is 180,
and this is 210,
so this is 30 degrees.
These all have to come out 30 and 45 and
60 because we don't know any other angles.
So, the sine of 210,
is the same
as the sine of 30 degrees.
I am down here
in the third quadrant.
In the third quadrant, I know sine is negative.
Because y coordinates are negative.
So, what is the sine of 30 degrees?
It is a half.
This is negative a half.
Do I need to go through that again?
Once again, to find the sine of 210 degrees,
I draw a unit circle
and I go around until I have gotten
to 210 degrees. This is
180 that is 210
so I have gone 30 degrees past the x axis.
We are always going to measure angles around the x axis.
Never from the y axis.
Never.
Okay.
In the third quadrant,
sines and cosines
both have negative values. So,
I will get exactly what I would have
gotten for the sine of 30 degrees.
Except it will be negative
because I am pointing down instead of pointing up.
Y coordinate when it is pointing up it is positive.
When it is pointing down it is negative.
The sine of 30 degrees
this will be the same as the sine of 30 degrees.
except it will be negative. It is negative a half.
If I want to find the cosine of 210 degrees.
It is the same angle as the sine has
it will also be
of 30 degrees.
And, it will also be negative
because it is found in the third quadrant.
Cosine of 30 is the square root of 3 over 2.
So, this turns out negative square root of 3 over 2.
First I want to find
cosine of 315 degrees.
315 degrees, now where is 315 degrees?
Well...
Well, let us see. 270 is
straight down. So,
315, I am going to keep going
Until I pass 270.
And I get to 315 degrees.
So, now what is the left over angle
between this line, this radius and the x axis?
45 degrees.
So, the cosine of 315 will be
the same as
cosine of 45. Now in the 4th
quadrant. Is x positive or negative?
X is positive.
So, you are going this way. Is Y positive or negative?
Y is negative.
So, cosine will be positive
and sine will be negative.
Because x will be positive
and y will be negative.
That means cosine
positive
Sine is negative. Okay?
And what is the cosine of 45 degrees?
1 over square root of 2. So this will be
rad of 2 over 2.
What if I wanted to find the
sine of 315 degrees?
Well, that is going to be the same as the sine of 45 degrees.
Plus, the y coordinate is negative.
So, the sine would be negative.
So, this would be negative sine of 45 degrees.
This is negative radical 2 over 2.
How about if I did the
sine of 420 degrees?
420 is everyone's favorite number. (laughter)
This is 360 degrees right?
That is all the way around.
So, I keep going another 60 degrees
until I get to there.
Okay. So, why is that 60?
Well, I went all the way around
360 and I kept going
until I got to 420 degrees.
I am in the first quadrant.
Is sine going to be positive or negative?
It is going to be positive. Because it is
the y value and the y value
is positive in the first quadrant.
So is the x value. This will be the same thing
as the sine of 60 degrees.
Which is?
Radical 3 over 2.
Let's find the sine of 300 degrees.
So, how do you find the sine of 300 degrees?
Well, we have to figure out where 300 degrees is.
300 degrees is
not quite 360, but
past 270.
How much farther than 270 do we go?
We go another 30 degrees.
So, this angle is 60 degrees.
Remember you always use the angle
with the x axis. You never
use the angle with the y axis.
Always the x. Never the y.
So, in the fourth quadrant, here,
the x coordinates are positive.
Y coordinates are negative.
Sine is the y coordinate.
So, this would be a negative value.
Negative sine of 60 degrees.
Sine of 60 degrees is radical 3 over 2.
There is a mnemonic for this.
Some of you may have learned.
All students take crack.
Or, All students take calculus.
Or, All Seawolves,
take
Calculus.
I learned a lot of these.
Feel free to pick up whichever one you want.
The goal of the mnemonic is that you
remember it. That is why lots of them
are obscene, because then you remember it.
If they are a little off color, then you have a better chance of remembering it.
What does this stand for?
A is for All
trig functions are positive if found in the first quadrant.
So, you get an angle.
You end up in the first quadrant. You get a positive answer.
S stands for sine.
If you are in the second quadrant
sine will be positive.
That means the tangent and
cosine will not be positive.
In the third quadrant,
tangent is positive. That means
sine and cosine will be negative.
The tangent will be positive.
The fourth quadrant, cosines are
positive. The sines and
tangents will be negative.
So it is a very handy way to remember things.
So if I asked you for,
tangent of 135 degrees.
Tangent is positive in the third quadrant.
But, not in the fourth quadrant.
Everything is positive in the first.
Sine is positive in the second.
Tan is positive in the third.
Cosine is positive in the fourth.
So, in the fourth cosine is positive, so sine
and tan are negative.
So, take tan of 135 degrees.
So, where is 135 degrees.
You don't really need to draw a circle
all the time. You can.
135 degrees is somewhere there.
How do I know that?
That's 90.
That is 180.
So, 135
That comes out 45 degrees.
Notice, practice the subtraction,
but you can't get like 55 degrees.
You don't know what to do with 55 degrees.
It has got to be 45 degrees.
So, the tangent of 135 degrees
would be the same
as the tangent of 45 degrees.
Except, we are in the second quadrant.
And, in the second quadrant,
we are up there with Seawolves, okay?
Only sine is positive.
Tangent would be negative.
So, this is negative.
What is the tan of 45?
The tan of 45 is 1.
Well, this is negative 1.
Well, these are very important to get down.
Sine, cosine and
tangent of
315 degrees.
Where is the angle at 315 degrees?
315 degrees is down
here.
It gets kind of boring drawing that circle all the time.
So I am not going to draw it.
So here the angle is down here.
Because that is 360
and that is 270.
So what is my left over angle?
45 degrees.
Okay, so this will be the same
so sine of 315
will be the same
as sine of 45, except
we are in quadrant, the fourth quadrant.
only cosine is positive.
All of these will be at the 45 degree angle.
But, only cosine will be positive.
These two will be negative.
What is the sine of 45?
radical 2 over 2.
So this is negative radical 2 over 2.
Cosine of 45 is radical 2 over 2.
This is positive.
Tangent would be negative 1.
You got that one?
Alright, it is close enough to 6:50.