Stony Brook MAT 123 Fall 2015
Lecture 03: Radian measure and trig on the unit circle
August 31, 2015

Start   Basic right triangle stuff. We should do a bit on the unit circle.
I am looking to make sure I have everything.
The next thing we are going to do is play with the unit circle some more.
Lets practice this one more time to make sure everyone can do it.
Lets find sine of 240 degrees.
0:32They say that is an unusual occurence.
Find the angle from the x axis.
Figure out if it is positive or negative.
1:06Find the value.
Alright, so first of all we want to find where is 240 degrees.
1:39So, 240 degrees. 240 degrees is somewhere down here.
That is 180 and that is 270.
So, 240 is somewhere between the 2.
And, 60 degrees past the 180 mark.
So, that is the first thing.
2:01Find the angle with the x axis.
Never with the y axis.
So, the first thing is, this is going to be the same as sine of 60 degrees. That is what is called the reference angle.
The 60 degree angle.
The reference angle is the angle which meets with the x axis.
Now, positive or negative? So, we have some very clever things here.
So, Albany Students Take Crack. (laughter)
2:33We are in this T quadrant where the sign of only Tangent will be positive.
So, sine is negative.
Now, find the negative sine of 60 degrees.
Now you go to your chart.
And, you find the sine of 60 is square root of 3 over 2.
Everyone get that one?
Who wants to teach the next one? So now, let's learn something new.
3:02Remember degrees? How many of you remember the proportion of radians when you were in high school.
Now we will learn what radians are.
So, you know how there is Fahrenheit and Celsius?
There is more than one way to find the measurements of an angle.
The fun one is degrees. Why do we have 360 degrees around circle?
Why is it 360?
There could be one day in a year, not 365 days, right?
3:32It is easy to remember. Why do we do 360 degrees?
The answer is, well, there is lots of numbers that go into 360.
So, 1 goes in, 2 goes in, 3 goes in, 4, 5, 6, so, it just has lots of divisors.
Its why a dozen eggs is better than 10 eggs.
Because a third of 10 is messy, but a third of a dozen is 4.
So, 10 actually doesn't do very well.
You can't do thirds, you can't do quarters of it.
4:0060 minutes in an hour, 360 degrees in a circle.
Um, you can divide the circle into a 100 sections.
Um, or 300.
But, we use 360.
But, the problems is that those aren't, when we are measuring, we are dealing with the length.
Rather than the angle.
And, that is just called radians.
Radians uses the same basic idea.
Not bad.
Some find it like a game show.
4:32Where if you draw a circle you get a prize.
So, how far is it around a circle with a radius of 1?
We all know the formula of a circumference is just 2 Pi.
So, why do we say that this distance of 2 Pi is the same as 360 degrees?
Easy? You sure?
5:00So, these are the radians.
So 2 Pi radians.
Is the equivalent of 360 degrees.
Or, to get more technical, I will use the center board for a minute.
Why do we choose radians?
Well, Calculus doesn't work with degrees.
In the Calculus that we learn you have all these rules. And things.
And it only works if angles are measured in radians.
5:30360 degrees, once around the circle, is 2 Pi.
Means... for that distance.
An angle of 1 degree corresponds to that distance in radians. Okay?
But, this is the theoretical part.
So, what does that mean to you guys?
What that means is this.
That is how you are going to convert. If you divide by 2, 180 degrees is Pi radians.
6:00Not very hard, right?
So, that is how you do it.
If I keep an angle in degrees, you can convert to radians If I give an angle in radians, you can convert into degrees.
1 degree is Pi over 180 radians.
And, 1 radian is 180 over Pi.
180 degrees over Pi.
Then you don't use any units for radians.
If you write a number it just means radians.
Have to have that little circle for degrees.
6:30So, if you want to convert.
An angle from degrees To radians.
What do you do? You multiply by Pi over 180.
So, you take the angle and you multiply it by Pi, and divide it by 180.
7:01Or, first you might want to divide it by 180 and then multiply by Pi.
It is actually a little easier.
And, if you want to go from radians to degrees, You are going to multiply by 180 over Pi.
Let's see. Let's do 0 degrees.
If you take 0 degrees and you take 0 and you multiply by Pi over 180,
7:30you get 0.
That wasn't too tough. So, 0 degrees, 0 radians.
We haven't gone anywhere.
We haven't made any angles, we haven't gone any distance around the circle.
How about 90 degrees?
Well, you take 90 and you multiply by Pi over 180 So, 90 goes into 180 2 times.
8:01So, that is Pi over 2.
This isn't that hard, right?
It is not as bad as the whole five ninths.
You subtract, you add.
Nine fifths and you do the whole degree Celsius and Fahrenheit thing.
It is messier.
Where this is fractions.
The key to remember is when you write Pi over 2. You are taking one half of Pi. Okay?
A fraction times Pi.
8:32If we have 180 degrees Well, we already know that is Pi radians.
How about 270 degrees?
Well, if 270 times Pi over 180 cross out the 0s.
27 over 18 is 3 over 2.
So, this is 3 Pi over 2.
9:082 Pi, oops, backwards.
360 degrees is 2 Pi radians.
Let us do some more.
9:30Let us take a minute to see if you could figure those 3 out.
These are very important.
Some of you already know this. So, we might just have a common denominator.
That wasn't so hard. I hope.
So, if you take 30 and you multiply it by Pi and divide it by 180. We should get Pi over 6.
10:00Did any of you get that?
It is one sixth times Pi.
At 45 degrees, 45 divided by 180 is 1/4th. So, this is Pi over 4.
And, 180 over 60 is, ah, 60 over 180 is 1/3.
This is Pi over 3.
These are the ones to memorize.
You should really try to memorize this set of 8.
If you can't memorize them, you need to be able to figure it out quickly.
10:30Because we are going to use these a lot.
Remember you can't have calculators on the exams. We can't give you 17 degrees and convert that.
We can ask you to convert it to radians. What if I said, convert 17 degrees to radians?
We do 17 times Pi over 180.
It is not a very interesting problem.
All you do is literally multiply by Pi and divide by 180.
Okay? So, we have to give you something that is fairly annoying to put it as a test question.
For example, what if I said,
11:07Convert 7 Pi over 6 into degrees.
What do you do?
We look at the person next to us.
We say, what do we do?
Well, we are going to take this and multiply by 180 over Pi.
But, now let us start in radians.
We will go backwards and find, well in the other direction, and find degrees.
11:34Notice, the Pis cancel.
Now, you've got 7 times 180 divided by 6.
How do you do 7 times 180 divided by 6?
What you don't do is first do 7 times 180 and then divide it by 6.
First you do the 180 and you divide it by 6.
We write 30.
12:00Okay? Make your life easy.
First, do division, then do the So, let us practice converting a couple more of these.
Um, let us convert 310 degrees equals x radians and, so
12:34So, if I take 310 degrees, I am literally going to just multiply by Pi over 180 cancel the 0s, and you get 31 Pi over 18.
That is not much fun. Is it?
So, you can do 31 eighteenths and try to reduce that but it doesn't reduce.
You can come up with a fun decimal, but why would I do that to you?
11 Pi over 3
13:03you multiply by 180 over Pi.
The Pis will cancel.
11 times 180 is 1980, but why would you do that to yourself.
When 180 divided by 3 is 60.
So, this comes out 660 degrees.
How did you do on those 2? Okay?
I have one more pair.
Just to make sure that we can do these.
13:31Then, we will do something else that is fun.
This is fun.
14:224 Pi over 15, multiply that by 180 over Pi.
14:32The Pis cancel.
180 divided by 15.
So, this is 4 times 12.
which is 48.
The trouble doing something like 180 divided by 15. Divide it by 3 first.
Get it down to 60 over 5.
Sometimes that is easier than trying to do the high number of digits in divisions.
15:00Okay. 570 degrees.
Multiply that by Pi over 180.
Cancel the 0 and now you have 57 Pi over 18.
So, we divide both of those by 3.
You get 19 Pi over 6.
This is as far as we can go.
Alright, so radians are going to be useful when we are going to ask you lots of questions. We are going to say things like:
15:30Find the tangent, well, lets stick with sine and cosine.
Find the sine of 7 Pi over 6.
So, let us do the sine of 7 Pi over 6.
So, the first thing that you do.
Is that you convert, you don't have to, you can convert that into degrees.
You need to figure out, where is 7 Pi over 6.
Well, you are going 7 sixths of the way around the unit circle.
That is very hard for most people to process.
16:04multiply by 180 over Pi.
You will end up with sine of 210 degrees.
210 degrees is about there. You don't have to convert it. Remember this is Pi.
Pi is half way around.
2 Pi is all the way around.
You just have to go another sixth past Pi.
Okay, so if you convert it, you get that this
16:30is 30 degrees.
So, that is the first step, you find out where you are.
Now, is it positive or negative?
I get negative.
Because it is in the third quadrant.
So, this is the same as the negative sine of 30 degrees.
What is sine of 30 degrees?
It is a half.
So, this is negative 1/2.
Okay, start memorizing that chart.
17:07A couple other things.
So, when we are in the unit circle, we start at 0 degrees.
At 90, 180, 270 and all the way around is 360 degrees.
You get the circle.
So, unit means 1.
17:30So, the unit circle, so according to that point this is (1,0).
According to that point, (0,1).
(-1,0) (0,-1) So, remember, the coordinates correspond to cosine and sine of the angle that you are going around.
So, we could use these coordinates to find the sine and cosine of 0 degrees, 90 degrees and so on.
So, if I want to find.
18:00The cosine of 0 degrees, it is just the x coordinate right here at zero degrees, which is just 1.
And, the sine of 0 degrees is 0.
Does that make sense?
It intercepts the x axis at point (1,0) The x coordinate is 1 and the y coordinate is 0.
The cosine of 0 is 1.
Sine of 0 is 0.
Remember this is cosine
18:30and sine.
Questions? No.
Alright, let us do 90 degrees.
So, now 90 degrees is straight up.
So, this coordinates well, the cosine is the x coordinate so, the cosine of 90 degree is 0
19:00and the sine of 90 degrees is 1.
So, let us do 180 degrees.
180 degrees is over here. So, the x coordinate is cosine and the y coordinate is sine.
So, cosine of 180 is Is negative 1.
And the sine of 180 is 0.
19:30And, finally, down here is 270 degrees over here.
Cosine is going to be 0 and sine is going to be -1.
So, I don't think we will need our charts.
These are nice and easy to memorize.
20:23Okay, so first of all.
0 degrees is 0 radians.
90 degrees is Pi over 2 radians.
20:30180 degrees is Pi.
And, 270 is 3 Pi over 2.
Okay. 360 just brings you back to 0.
So, 2 Pi is back at 0.
So, the sine is 0 and cosine is 1.
So, remember on the other chart, sine and cosine and tangent sine would go in one direction and cosine would go in the other direction.
1 and 0.
21:08Sine of 180 degrees or Pi radians is just going to be 0.
and sine of 3 Pi over 2 or 270 degrees will be -1.
Notice sine goes 0, -1.
Cosine goes -1, 0.
21:34Okay. Remember, I told you last time that the tangent of an angle is the sine divided by the cosine.
So, 0 divided by 1 is just 0.
1 divided by 0. You can't divide by 0.
So, this is undefined.
There is no tangent of 90 degrees. Well, that is not really true.
The tangent of 90 degrees is undefined.
It gets infinitely large.
And then, it also gets infinitely negative.
22:01So, it is a little messy.
So, it is undefined.
The sine and the cosine here is 0/-1 which brings you back to 0.
And again, you get 0 in the denominator, it is undefined.
Okay. So this is a handy chart to memorize.
I will take a picture of this later and post it up on Instagram because it is a lot of fun.
22:30Remember you have some coordinate.
And this is cosine theta.
and that is sine theta.
because when you find the coordinate it is (x,y).
So, this is (cos theta, sin theta).
So, that is theta.
That is the way that we define this as we move around the circle.
The coordinates, notice one goes in the x direction, one goes in the y direction.
23:01So the tangent.
SOH CAH TOA, so it is the opposite over the adjacent.
So, it is the sine divided by the cosine.
Do you agree?
So, that is why tangent is sine over cosine.
What else do you know about the ratio of this to this?
When you are graphing a line, what do you call this?
That is the slope. Okay.
So, if we want to find the slope of the line
23:31and we know the angle that it makes with the x axis, The tangent of that angle is the slope.
tangent of theta is the slope of the line.
You know that I am going to ask you a question on that.
So... A line goes through (8,19) and makes an angle of 60 degrees with the x-axis.
Find the equation of the line.
So, in order to find the equation of the line
24:00I need slope, y=mx+b.
Let us see if we can figure this out.
That we really don't know.
It goes through (8,19) and this is the angle it makes.
It could go through this part, I am not actually concentrated on what it looks like.
So, you can find this slope all you have to do is take the tangent of that angle.
That angle is 60 degrees.
24:30So, what is the tangent of 60 degrees?
You have to memorize it.
Radical 3.
Okay? So this just means So, the slope is tan of 60 degrees which equals square root of 3.
That's annoying. Yeah.
Its an annoying value, but whatever.
So, y = radical 3 x +b.
Or, better.
Point-slope formula.
25:00The point-slope formula says.
y minus y one equals m times x minus x one.
That is the equation of a line.
Okay? You did that You should have done it in 103.
So, I have the y coordinate that is 19.
I have the x coordinate that is 8.
So, (y-19)= radical 3 (x-8).
25:31You do not have to do any more than that.
Okay. That is your equation of the line.
You will see this, for those of you who continue on, you will see this in Calculus.
Where you need to find the equation of a line and you have a point and a slope and you want to do it this way.
You really do not want to do it the other way.
Okay, so let us say that a line goes through the point (3,23) and makes the angle of Pi over 5.
26:04Then the slope is just tangent of Pi over 5.
Pi over 5 is 36 degrees.
We don't know the tangent of 36 degrees, do we?
So, if this was homework, you would take out your calculator, and you would find the tangent of 36 degrees.
You could use your phone or use a calculator and make sure that your calculator is in radian mode.
And you would use Pi over 5.
Your calculator has a way to switch
26:30modes from degrees to radians and radians to degrees.
Make sure you are in the right one.
Excellent question, "Could I give you something like this on the test?" Well, if I did, you would leave the answer as tangent of Pi over 5.
Okay? So... y -y one=m (x-x one) y minus 23
27:00is tan of Pi over 5 times x minus 3.
So, on a test, you can just leave it like that. Yes?
You did not put tan of 36 degrees you put tan of Pi over 5.
You know, you want to get used to not converting from radians to degrees.
But, you are still early in the course.
Because, in Calculus everything is going to be in radians.
By the way, physics uses everything in degrees.
Chemistry when you do the steps it is in degrees.
27:37One step at a time.
Now, what is the tangent of Pi over 5?
Anybody find it?
77? Well, there you go.
You have to say it louder.
.727 Is that enough sig figs?
28:02Let us do one more brand new thing.
To answer people's questions, this is a perfectly acceptable answer on an exam.
All you will be tested on is 2 things, you will be tested if you know the equation of a line, and do you know that the slope is the tangent of the angle that is really all that we will be looking for on the test.
28:35Here's another type of question that I love to put on tests.
We ask these kinds of things a lot.
The sine of x is 2/3 and x is between Pi over 2 and Pi.
Find the cosine of x.
So, what does that mean. Well,
29:01x is somewhere between Pi over 2 and Pi.
So, Pi over 2 is 90 degrees Pi is 180 degrees.
So, we are somewhere in the second quadrant.
That is all that second piece of information means.
Ah, How did I think I was in the same quadrant? Pi over 2 is 90 degrees and Pi is 180 degrees.
29:33Those are the angles converted.
Now, I would not expect you guys to be good at it initially.
You need at least 5 more minutes before you are supposed to master this.
Okay. So x is somewhere in the second quadrant.
And the sine of x is 2 over 3.
So we use SOH CAH TOA .
And we have 2 over 3.
Okay. So, we want to find the cosine.
30:00But, the cosine is going to be this distance divided by 3.
SOH CAH TOA So, how can I find that distance?
That is pretty loud. Okay... Pythagorean Theorem Use your game voice.
Okay, so, 2 squared plus
30:30b squared is 3 squared.
So, b squared is 5 So, b is square root of 5. Well, technically Plus, or minus the square root of 5.
Remember when your teachers took off for that in 7th grade?
Right, so it could be positive or it could be negative.
If we are facing this way on the x axis
31:00it is actually negative.
So, the cosine of x is negative square root of 5 over 3.
We also know that it is negative because it is in the second quadrant. In the second quadrant cosines are negative.
Alright, then let us do another one of these.
If the cosine of x is 3/7 and x is between Pi over 2 and 2 Pi
31:32Find the sine of x.
Where is x?
3 Pi over 2 is 270 degrees.
We convert it.
We take 3 Pi over 2 times 180 over Pi.
Do a little magic and we get 270 degrees.
2 Pi is 360 degrees.
When you just start to get the feel, you won't get it right away,
32:00of where the radians are on the unit circle.
So that means that we are somewhere down here.
We have x.
What do we know about the cosine of x?
It is 3/7. By the way, I could make this 3/7 and make this 1.
Technically, that would be the coordinates would be 3/7 and we are looking for the other coordinate.
But, because we hate fractions well, I don't hate fractions, but many of you do.
32:30I can also do this.
And, I get the same result.
Now, I have to find the missing side. What letter should I use for this?
The letter b? How about b?
So, 3 squared plus b squared is 7 squared.
So, 9 plus b squared is 49.
So, b squared is 40.
So, b plus or minus the square root of 40.
You do not need to reduce that to 2 radical 10.
33:01Leave it as the square root of 40.
Okay, we are down here in the fourth quadrant, sine is going to come out negative that would be negative square root of 40.
It is pointing down to the x axis.
So, the sine of x Negative square root of 40 over 7.
33:32Let's do another one of these.
That is a 9 if you can't read my beautiful hand writing.
Okay, so, are looking for x between Pi over 2 and Pi.
Pi over 2 is 90 degrees and Pi is 180 degrees.
34:02There aren't too many combinations of this.
This is 0.
This is Pi over 2 because it is half way to Pi.
This is Pi.
And, this is 3/2 and all the way around is 2 Pi.
We do not really have to convert it.
We just have to remember it. Half way.
Pi, 3/2, and 2 Pi.
34:31So, I am in the second quadrant and I know tangent if you try to draw the angle to the y-axis.
I guarantee you will get the wrong answer.
Tangent of that angle is negative 5/9.
So, the opposite over the adjacent and this is the negative value. Why is it negative 9? Because going this way is negative on the x-axis.
That is a positive number So, I need to find the hypotenuse.
35:01So, what letter should I give that one?
c So, we like c, huh?
So, 5 squared plus 9 squared equals c squared.
Really it is negative 9 squared but negative 9 squared is the same as 9 squared.
So, 45 plus 81 is c squared.
C squared is 106.
So, c is plus or minus the square
35:31root of 106.
It is never negative, the hypotenuse.
The hypotenuse is always positive.
It is going out.
It is always positive.
Okay, so the cosine is the adjacent over the hypotenuse cosine of x is negative 9 over the square root of 106.
If I were to ask you the sine, it is 5 over the square root of 106.
36:01How do we feel about these?
Alright, so, I am going to end this early today.
See you on Wednesday.