Lecture 05: Other trig functions & graphs of sine and cosine

September 9, 2015

Start | So, negative 1 and zero is here.
So, then you go counterclockwise, that is this way No, I am wrong. yeah, this way. So the point of the radius corresponding to some number of radians. In this case it is 6 radians. 6 radians is a little less than 2 Pi. How far has the ant walked? |

0:30 | So, I just read this question
3 times.
Now, I finally understand what it is looking for. By the way, Professor Sutherland and I said, I think Webassign is wrong and then we read it and read it and read it, and we said, "Oh..ok." So, he is walking to here. He is not walking 6 radians. He walks to the point that equals 6 radians. You do not even need to know where 6 radians is, but 2 Pi is 6.28. That helps. So, 6 radians is pretty close to there. |

1:01 | Michelle did you start yet? Yes.
Somehow I have kept my job, but I am not sure how. Anyway. So, 6 radians is about here. So, this is where the ant gets stepped on. So, how far did the ant go before it died? Ok, this distance is Pi. Because it is half way around the unit circle. Right, and the unit circle is 2 Pi. So, the distance that the ant walked is, um, |

1:31 |
lets see, how we do this...
okay. So, this
distance Pi that is half way around the circle.
The whole thing is 6 so, the left over distance, this distance is 6 minus Pi. Makes sense? One more time... Okay... All the way around the circle is 2 Pi. So, from here to here is Pi. Then they said that this was 6 radians. So, we want to know the left over piece. |

2:00 | We want to take that Pi away.
Because the ant starts here and goes to here. Okay? So, if the total distance is 6 minus this part which is Pi. 6 radians is 6. How do I know where 6 is? Okay. Okay, so, Pi is a little more than 3. So, 3 radians is pretty close to half way around the circle. 6 radians is pretty close to all the way around the unit circle. |

2:31 | Okay?
One radian is about 60 degrees. A little less, I think it is 57 degrees or something like that. Okay? You take 360 and divide it by 2 Pi. Which is about 6.28 but not exactly and that will tell you how many degrees. Okay? Thoroughly confused, or sorta confused? Alright. So, lets learn how you do that question... |

3:02 | So, you have a circle, and
a circle could be a pizza or not a pizza
it doesn't matter. In a
sector of the circle,
And, the sector has an angle.
Associated with it. You guys remember this from geometry it is called a central angle. And, it has an arc which we will call S, I have no idea why. The sector is called A |

3:30 | and the arc is called S.
You would think that the sector would be S and the arc would be A, but it is not. And then, you have a radius. So, how can we find the length of this arc. Well, the imagine it is an opening angle. As you open the angle the arc gets bigger. As you close the arc and the angle gets smaller. Thats gators? I think that is gators. |

4:01 | So, as you go around the circle,
as you make the angle bigger the arc gets bigger.
In fact the ratio of the angle to 360 degrees would be the same as the ratio of the arc going all the way around. Which is 2 Pi r. Does that make sense? Okay? So, the angle out of the whole thing equals the arc out of the whole thing. Got it? One more time. So, the angle |

4:31 |
Okay, so the S stands for the
length of this arc.
If it is a slice of pizza, it is how long the crust is. If you are going around the earth, it is how far you have traveled around the earth. So that longitude question. If you had gone from 150 degrees to 180 degrees longitude, you have gone 30 degrees of the way around the earth. Okay? I'll get there. Okay, so... One more time. So, the angle is |

5:01 | part to whole. So the angle compared to
all the way around, is the same as the arc
length compared to all the way around.
Okay. Now, you could also do this in radians. So if you do theta over 2 Pi radians equals S over 2 Pi r, Then you cancel the 2 Pis. You multiply everything by 2 Pi. And, you get theta is S over r or S equals r times theta. |

5:31 | Okay?
So, if the theta is in radians you can use this formula. If the theta is in degrees you can use this formula. How are we doing? Alright, now let us do this again, but this time let us do the sector. Which I am going to call A. As I said it makes no sense, |

6:00 | but that is just the way they do these things.
So, A is the amount of pizza. Okay? Yes, and no and the crust, and the crust isn't thin... whatever. Okay? So, you use the same principal. The angle compared to all the way around. equals the sector area, so, A. |

6:30 | divided by
the total amount of area.
So, Pi r squared. So, a bigger angle a bigger piece of the area. Right? To get a bigger slice of pizza for yourself. You are going to take the nice big fat slice. If you are slicing for your friend, oh, you know, you get a little bit. If you are in the mood. Or, you give them the crust if you are not a crust person. Okay? Um, and similarly, if you want to put this in degrees, if you want to do it in radians, |

7:00 |
You do theta is to 2Pi,
since 2 Pi is the same thing as 360 degrees,
equals A over Pi r squared.
Cancel the Pis, do a little rearranging. And, you get A is 1/2 r squared theta. That is in radians. 0.3 radians, remember 0.3 is 0.3. |

7:31 | Okay? So,
1 radian is about
60 degrees. It is a little bit
less than 60 degrees. So, 0.3 radians
is about 0.3 time 60,
so about 18 degrees.
So, if you are supposed to guess, yeah, you can kind of guess. So, just to give you a feel. A, B, C, needs improvement, I got a lot of needs improvement. Its okay, I am I am working a home. I can draw a good circle. |

8:01 |
So, if this is 0 radians,
it is about 1 radian
2 radians
3 radians
4 radians
5 radians
6 radians. About like that. Okay?
So, you see it is about 60 degrees each one. Because 6.28 blah, blah, blah is all the way. Yes, so miles and longitude is a sector |

8:32 | okay,
So, if you are on the earth.
and you know the diameter of the earth is 7,962 miles okay, so first you have to use radius not diameter. so half of 7,962 is 3,981 to you all that sounds really impressive okay? So, thats going to go in your radius. Okay? But, you are doing it in degrees so if you are going from say 120 to 150 |

9:00 | longitude you are probably 30 degrees.
So, you would have 30 is to 360 is the distance you go through Pi 3,981. I can't explain it any better than that. So, here this is the problem, sine of v is 3/13 the sine of v is also a over c. We know that a over c |

9:30 | is 3 over 13.
You know that is nice, but, how does that help me find a that is 2 variables, I want only one variable. So, we learned back when we started with algebra. That to get 2 variables, you need 2 equations. They have to be different equations. We have another equation though. We have a squared plus b squared equals c squared. Or, a squared plus 4 squared equals c squared. 4 squared is 16. Right? |

10:00 |
Okay, so now you have 2 equations and 2 variables.
And, we are trying to find a, so let us get rid of c. Um... We could do 13 a equals 3 c so, 13 a over 3 equals c. So, you plug in there and you get, a squared plus 16 equals 13 a over 3 |

10:30 | squared.
So, the Pythagorean Theorem. love is Pythagoreas, right? One theorem you all know. You don't know that theorem? You do know the theorem. You know more than one. You know the isoceles theorem. You know that one. The sides are equal so the base angles are equal. Sum of the angles of a triangle. You know that one that is 3. That is a lot now. You are theorem happy at this point. |

11:01 | Okay...a squared
plus 16
now equals 169 a squared over 9.
Multiply across, you get 9 a squared plus 144 equals 169 a squared. Solve. Solve for a. Solve for a! Okay? I leave the rest as an exercise for you guys to figure out. a equals 3 square root of 10 over 10. |

11:30 | There is something called reciprocal functions.
Secant, Cosecant and Cotangent. We all know sine, cosine, and tangent by now. So we have 3 more functions. We have cotangent of an angle. We have cotangent of an angle is 1 divided by the tangent of the angle. Also, since tangent is sine over cosine, cotangent is cosine over sine. |

12:01 |
These are just definitions. You could say
this is what we are now going to call cotangent.
The secant of an angle. is 1 over the cosine of the angle. And, of course, cosine is 1 over the secant. So, they are reciprocals of each other. And... Cosecant of the angle is 1 over the sine of the angle. Why would you ever want those? |

12:32 | Well, especially when you get into Calculus
there will have problems where we will have
blah on top and sine
on the bottom. Okay?
So, sine will be in the denominator. And, you say, "I do not want sine in the denominator." "Its annoying down there. So if I change it to cosecant, now its in the numerator. Okay? So, the reason that these are reciprocal functions is you have |

13:03 | You could change that to tangent theta
times
cosecant theta. Why would you want
that? You are learning Calculus.
Sometimes it is handy to switch from one to the other. By the way, you might want to move the numerator to the denominator. So, it depends on what is going on. Okay. Um. It just depends on the problem. So, how do you use the reciprocal function? You really only need sine. You do not need any of the other 5 functions. You could just write everything in terms of sine if you wanted to. |

13:31 | They invented these other functions to just make them smoother.
Okay? After all sine squared plus cosine squared equals 1. So, once you know the sine, cosine is just square root of 1 minus sine squared. But that is messy, so why not invent a new function that is more simple. So, how are we going to use these? We are not going to ask you to do very much with these. Because, the truth is that you do not really need them. But... Just to make sure you understand. |

14:00 | what is going on.
Alright, how do we do this? Well, Pi over 2 to Pi is in which quadrant? Second quadrant, because Pi over 2 is 90 degrees Pi is 180 degrees. We do that to remind ourselves that we are in the second quadrant. So what is positive? Sine is positive. Which means cosecant will also be positive. |

14:32 | Because it is 1 over sine.
So, it is positive, it doesn't change from positive to negative. We are just putting a one over it. We got our angle. Let me erase that part. And, we know that the sine of the angle is is 5 over 13. So, So, now we can find the missing side here. |

15:01 | Because we know that 5 squared plus
b squared is 13 squared.
So, magically the b is 12. Plus or minus. Okay? You guys should know the Pythagorean Theorem by now. Okay, so we need to find cosine of theta. And that was sine theta, cosine |

15:31 | To do this well, you want to be a little systematic about it.
Okay. You know that the sine is 5 over 13. So, the cosecant is 13 over 5. The cosine is now going to be negative 12 over 13. In fact we know |

16:00 | sine is positive
these other 4 functions
are all going to be negative.
Okay? Because only sine is positive in the second quadrant, and cosecant is positive because sine is positive. So, cosine is 12/13 So, negative 12/13, therefore secant is negative 13/12 And, tangent which is opposite over adjacent is negative |

16:30 | 5 over 12.
And, therefore cotangent is negative 12 over 5. Okay, take a couple minutes to figure that out. Alright. Pi is 180 degrees. |

17:00 | 3 Pi over 2 is 270 degrees.
You should start getting used to 0, Pi over 2, Pi 3 Pi over 2, 2 Pi. You should know where they are. O, Pi over 2 Pi, 3 Pi over 2, 2 Pi You have to get comfortable with that. So, between Pi and 3 Pi over 2, down there in the 3rd quadrant Okay? So, if you put this in the wrong quadrant and do everything else |

17:30 | correctly, you will get partial credit.
You know that the cosine of theta is negative 15 over 17. You can't read that. Negative 15 over 17. Okay So if you want to find the missing side, you know that a squared plus 15 |

18:01 | squared equals
17 squared. You do not need the minus sign
when you are squaring it.
So , a comes out 8. It actually comes out plus or minus 8. So, this 8 down there Yes? If this is tangent of theta than that number would have to be positive to be in the third quadrant. |

18:31 | Because you would be in the 3rd quadrant.
But, it's cosine, negative. Ah, the radius is always positive. These legs can be positive or negative. The simplest thing you can do, if you get confused about all that is don't worry about the minus sign until you are done. And then, just remember that you are in the 3rd quadrant. In the 3rd quadrant, What do we know? In the 3rd quadrant |

19:00 | tangent is positive
and sine and cosine
will both be negative.
And, 1 over sine is cosecant. So, that will be negative. 1 over cosine is secant so that will be negative. And, 1 over tangent is cotangent. So, that will be positive. Okay? You filled in the triangle, so now you can fill in the equations. This will be negative 8 over 17. So cosecant will be negative 17 |

19:31 | over 8.
Cosine you already know is negative 15 over 17. So, secant is negative 17 over 15. And tangent is they actually both will be negative. If you think about the coordinates. This is negative and this is negative. Your tangent will come out 8 over 15 or you should just know you are in the 3rd quadrant, the tangent has to be positive. |

20:01 | 8 over 15
Cotangent is
15 over 8.
How are we doing? Okay? Ask me questions. Go ahead. Remember the legs are negative because you are taking them in the negative x direction. Then going down in the negative y direction. So, it is really not that important to worry about the sign |

20:30 | S-I-G-N. So it goes.
Because, you can adjust the answer knowing what quadrant you are in. Okay? But, if you want to be technically correct, you got 15 to the left, so that is negative 15 You have gone down 8, so that is negative 8. So, when you are doing tangent, the minus signs cancel. So, that is why you get back to a positive answer. That is why we have the a, b, c. Good? Okay? |

21:00 | Do you feel good about this or should we do one more?
One more? Hate ourselves? (laughter) So, one more. What would happen if you started graphing the sine curve? Okay? |

21:31 |
The good news is that we to do
very basic graphs for now.
Don't get all nervous. Okay? Very straight forward. x is going to be our angle. And, y is going to be sine of x. |

22:01 |
You can do this in degrees or in radians, it doesn't matter.
I am going to go once around. Okay? We will go from 0 all the way around from 2 Pi radians. So at 0 radians, and just starting At 0 degrees, 0 radians, what is the sign of this angle? Well, the sine of that angle is 0. You learned this from the chart. You know that the sine of 0 is 0. Now I start moving in this direction. |

22:31 | Before, I have gone
30 degrees,
or Pi over 6.
Okay? 30 degrees or Pi over 6 the sine is just 1/2. So let us say, At 45 degrees, And, now Pi over 4. |

23:01 |
Redrawing triangles.
Sine is the opposite side. Sine is getting bigger. In fact, as I go to here sines are going to keep getting bigger, because sine is the vertical piece, the y coordinate. As I head to 1, Pi over 2 as you get closer and closer to 1 becomes sticking straight |

23:31 | up.
And, at Pi over 2 radians, 90 degrees are sticking straight up and that is 1. So, the sine of that angle is 1. So, sine starts climbing until it gets to 1. And notice, first it goes up very fast, and then these start getting closer and closer to each other. Now what happens? Now it comes down the other side of the circle. So, my triangle is going to look like this. |

24:01 |
Remember sine is the y coordinate. It is the
vertical piece.
The sine starts coming back down. So, okay for this angle. It is just Pi radians. Now I get these vertical pieces. The values are negative. Remember our mnemonic. Then you go back around until you get back. |

24:31 | 2 Pi
Okay?
So as they start going around again. Remember what I am graphing. The x coordinate is the angle. The x coordinate is how far I have gone around the circle. The y coordinate is the sine of that angle. So, as the angle gets bigger and bigger, going toward 90 degrees. The vertical piece the y coordinate starts gets bigger and bigger. When I am exactly straight up, I am at exactly 90 degrees, the y coordinate |

25:00 | the y coordinate is 1.
Then the y coordinate starts coming back down until you get to 0. Then, the y coordinate is a negative number starting to get toward negative 1. And a y coordinate is getting bigger and bigger it gets closer and closer to 0 until it gets back to 0. That is why sine has the shape. And then, if I keep going around the circle, the graph repeats. I will just graph it. But it might work. It might work. |

25:31 | You can try.
So, this is what a sine curve looks like. Okay. You start at the origin. You don't really start anywhere the sine curve kind of goes like that forever. It has no beginning or end. But, you pretend we start at the origin. And you go around the unit circle. Continue to 90 degrees or Pi over 2. |

26:02 | You keep getting bigger and bigger values of sine
until you get to 1. Do this in your calculator.
Start taking values of sine from 0 to 90 degrees and watching as the values get bigger. You go to 1 and then go to Pi. You come back down until you get back to 0. Then it becomes negative numbers you get down to negative 1 and back to 0. Then it repeats. So, this would be 4 Pi, 3 Pi, 5 Pi over 2. 7 Pi over 2 |

26:30 | and so on. These were the problems that we had
the other day where we had like
23 Pi over 6.That would just keep going
on the circle. So, notice
we start with the values
the values keep showing up again and again.
This is the first quadrant value. This is the second quadrant value. Here would be a third quadrant value. This would be a fourth quadrant value. And, then you start over again. |

27:00 | So, you can get a first quadrant value forever.
So, you know sine of theta is 1/2. You got 30 degrees. You got 150 degrees. You got 390 degrees. You have got 510 degrees. 750 degrees. And, so on. So that is why there is an infinite number answers for each problem. Okay. So we usually ask for the smallest positive answer. You see that by asking for the smallest |

27:30 | positive answer, that means
first coordinate that you get
starting from 0.
How are we doing so far? You could make, oh, something that looks like that. Which is what sound looks like. So, your phones or the computer of your phone shoves together a whole bunch of sine and cosine instantly. To make sound waves. By doing a process called Fourier transforms. Okay? By a process you can take |

28:01 | any sound pattern
break it into sines and cosines
Wow. And who said you will
never use trigonometry. If you are going to
be an engineer you will use trigonometry.
Any electroengineers in the room? Physics? ONe. Any way. Okay. So, But anyway. So that is the sine curve. We also need to learn what the cosine curve looks like. |

28:30 | Cosine curve looks exactly like
the sine curve, except it
starts in a different place.
Okay. Now we are going to do the cosine graph. It is the same principle as before. We will do the angle and you get to keep your outputs going on with the horizontal piece. you want the x coordinate. As you go around. |

29:01 |
So, let us see.
We start here around 0 degrees. The x coordinate is 1. We would be up here at 1. As you start to move in this direction. Draw some triangles. As you start to move the thetas, The x piece, the x coordinate, is decreasing throughout these triangles. It starts out there and it gets smaller and smaller until |

29:31 | it gets to 0. So, the cosine
of that
until it gets to 0.
See that? Cosine is here and cosine is here and there. I used color chalk because it is easier to see. So, that is the x coordinate the base of the triangle and moving up around the circle. Until I am pointing straight up And, there is no x coordinate, well the x coordinate is 0. There is no lenght to the base of the triangle. |

30:02 | Now, I go down the other side.
So, each angle The cosines are these horizontal pieces. Until I get all the way over to 180 degrees or Pi, then the cosine is negative 1. And, then the angles |

30:30 |
As you go in this direction,
the x pieces, the horizontal
get smaller again going back toward the origin.
So, it kind of goes like this. And, finally we go back Alright, okay? The cosine curve is exactly the same as the sine curve |

31:00 |
Except it is shifted.
It is shifted 90 degrees. The cosine curve is the same as the sine curve shifted Pi over 2 to the right. It looks like that. |

31:30 |
And again, the curve goes on forever.
The co in cosine stands for complement. Remember we learned the |

32:00 | cosine is just the sine of the complementary
angle.They are literally the same curve.
They are just slightly shifted. Again as with sine, with cosine you get multiple answers If I knew that this was 1/2, there is these places where cosine equals 1/2. There is between 0 and Pi which is the first quadrant. Then it is not again until you get to 3 Pi over 2 and 2 Pi. So, that is the fourth quadrant. |

32:31 |
Okay?
That is why we have a graph. That is why we need to draw it this way. If you just think about it in terms of the unit circle, or we can just turn to the sine graph. So far so good? |

33:00 |
2 Pi over 3?
3rd quadrant would not be 2 Pi over 3, it would be 4 Pi over 3, okay? The cosine of 4 Pi over 3 would be negative. Oh, 3 Pi over 2. 3 Pi over 2 is 270 and the cosine of 270 degrees is 0. |

33:32 | Okay? So that is what the sine and cosine
graph look like you can graph them
next to each other. That is
the sine graph and that
is the cosine graph.
That is Pi over 2 That is Pi. That is 3 Pi over 2. That is 2 Pi. That is the sine graph. |

34:01 | Cosine graph.
Okay? So, they are the same graph. Just imagine it is shifted. One over the other. Okay? Correct. So, as the angle gets closer to 90 degrees sine gets closer to 1. Because the sine is going up. And cosine is getting closer to 0 when cosine is going down. |

34:31 | Okay?
So, and that is what happens. That is why we have that chart for that for that case. When one is going up and the other is going down and vice versa all the way all the time. You are asking a lot of questions that is good. Why do we need this graph? We don't. I am just making a deeper and better picture. |

35:02 | So, what is going to happen is you should
know exactly the point where
sine and cosine are equal. Make sure you
know that.
What is going to happen is later we are going to start playing around with graphs. Alright, if you did the paper homework. Okay. Paper homework number 1. You are standing on a beach, |

35:32 | and you look at a house, and you spot
a big chair.
Somewhere over there, right? You make the angle to the chair 88 degrees. You look at the big chair on the beach and you say, "It is 88 degrees that way." It is actually kind of horizontal, but same thing. Okay? Then you back up 50 meters |

36:01 |
And, you measure again.
And, you get 84 degrees. This picture is not drawn to scale. Okay? Because really, It is kind of like this. Okay? That is no fun, so we are going to pretend that it never happened. So this going to be 50 meters to back up. We want to find x. I will give you a clue. |

36:31 | You need to use trigonometry
To easily figure
out these 2 distances.
These are tangents. Once you have figured out a and b you can use the Pythagorean theorem to find x. Okay? So, you can use your trig stuff. You don't actually have to use but you can do it in terms of sines and cosines, so basically that is what it looks like. That is not what you drew. Ah! Okay? But, I only started with like |

37:01 | 6 minutes to go.
So, I know some of you like it when I end a few minutes early. So, I ended a few minutes early. See you next class. |