Start  So remember the double angle formulas we had from last time.
Just a reminder everybody. ok! so we did this last time. You should all have this written down in your notes. 
0:32  Or you should have this in your scrap or both.
I know, we know but I had an easy way. So the sin(A+B) is equal to sin(A)cos(B)+cos(A)sin(B). Right? So, what's the sine of 90 degrees? No, what's the SINE of 90 degrees? ok, so sine of 90 degrees also known as sin(pi/2) equals 1. Right? 
1:10  How can we find it in another way?
So lets convince ourselves. Ok? Well, lets see that's going to be Sin(60)Cos(30)+cos(60)Sin(30). 
1:39  Do we have left overs?
Alright what's sine of 60 degrees? square root of three over 2. 
2:00  What is cos(30 degrees)?
same thing as, radical 3 over 2. What's the cosine of 60? 1/2. Good! What's the sin of 30? So radical three times radical three is 3, 2 times 2 is 4. 1 time 1 over 2 time 2 which is 1/4. You get 1. 
2:32  You should get 1. If you don't get 1 we have a problem.
One way to test the formula let's say you need this on your exam Ok? and you are not sure if it's plus or minus, check by using the formula. ok? Be sure! I don't think you need to check any of this formulas. You know them. Right? Another thing you can do is, suppose you wanted to do sin 15 degrees. 
3:14  So I can I find the sine of 15 degrees?
Jennifer? I can do sine of 45 minus 30. 
3:35  So lets use the formula!
Thats going to be Sin 45 Cos30  Cos45 Sin30 So sin 45 is radical 2 over 2. 
4:01  cosine of 30 is radical 3 over 2.
cosine of 45 is radical 2 over 2. Sine 30 is 1/2. ok! So radical 2 times radical 3 is radical 6 And 1 times 1 is 1. I am sorry radical 2 over 4. Thats our answer. 
4:31  We also know another way of finding the sine of 15 degrees.
We use the half angle formula. So, remember what the half angle formula is? you guys remember that formula? 
5:03  Nick what is that?
Perfect! ok, these will be very useful for example for web assign Ok? So if you want to find the sine of 30/2, just remember sine of theta over 2 
5:34  sine of theta over 2 is plus or mines the square root of 1 minus cosine theta all over 2.
So this will be the square root of 1 minus cosine of 30 all over 2. That is equals to the square root of 1 minus radical 3 over 2, all over 2. These two are the same thing. ok? 
6:19  Of course you can play around with this a bit.
That! 
6:32  If I multiply here top and bottom by 2, you get 4.
Still does not come out the same thing. play with that for a while [unintelligible] which you can convince yourself that is the same answer. 
7:00  But anyway, you can do the half angle formula or you can do it with angle addition formula.
let's us practice a couple of things, make sure that you guys are good at this. Suppose I tell you. 
7:38  See if you can do that for a minute.
So sin(v) is 2/5. that means this tells us that "v" is in the second quadrant. ok? The second quadrant. Then we know that the sin(v) is 2/5. 
8:02  We use Pythagorean theorem to find the missing side.
So 2 square plus this square is 5 squared. We get the square root of 21. Alright, what the formula for sin 2v? 2 sin(v) cos v. 
8:30  The sine of v, we already know it; 2/5.
and the cosine, the cosine has to be negative because we are down here in the second quadrant. So we should really say its negative square root of 21. Multiply by negative radical 21 over 5; geti negative 4 radical 21 over 25. Ok? Now, I want to find the cos(2v). 
9:01  So whats the cosine formula?
cosine squared v minus sine squared v. There is multiple formulas. You can use that one. Cosine squared which is square root of 21 over 5, squared. Technically negative square root, but with the square is not going to matter. 
9:30  And the sine squared is 2/5 squared.
I am running out of board. I am going to move this problem up here. 
10:12  ok. sin(2v) = 2 sin(v)cos(v).
2 times 2/5 times negative radical 21 over 5. is  4 radical 21 over 25. 
10:31  Pretty good.
cos(2) is [cos(v)]^2  [sin(v)]^2. So the cosine v is negative the square root of 21 over 5. 
11:02  and the sine of v is 2/5.
so you are going to get: 21 over 25 minus 4/25 = 17/25. How are you doing on this? Good? Are you doing good? Alright, maybe you can do tan(2v) which is just sine over cosine. 
11:33  So there are tangent formulas.
I don't bother learning them because you have sine and cosine formulas. Just find sine, find the cosine, and divide. Not really valuable to memorize Also very easy to get messed up. So its just going to be... 
12:03  ok? the 25s cancel.
negative 4 radical 21 over 17 That's what you thought, right? 
12:47  ok! Let's find out!
Tangent of x is negative 5/8, tan(x)= 5/8. And we already in the second quadrant. 
13:06  So for tangent we need to do SOH, CAH, TOA. So we do opposite over adjacent.
How do I know that this is negative? Because is the negative xaxis. ok? If you do pythagorean theorem, you get square root of 89. ok? five squared and 8 squared. So far so good? 
13:34  Why is the 8 negative? Look at the coordinates.
Going this way on the xaxis is negative. This way is positive. ok? The sin(x/2). Am I getting the positive or the negative square root? 
14:01  Why am I getting this positive?
Because we are in the second quadrant. Half of that will put us either in the second or in the first. Either way its still going to come out positive. In fact, it's gotta be the first, because half of 180 is 90. Since we are less than half way to 180, ok so less than 180 degree. Half of that is going to be less than 90. So we have to end up in the first quadrant. 
14:30  So we plug in. cosine of x is negative 8 over the square root of 89 over 2.
And the cos(x/2) is square root of 1 plus cos(x) all over 2, plus or minus. So that I am going to end up in the first quadrant. Its going to be positive square root again. So this is the square root of 1 minus 8 over the square root of 89 over 2. 
15:06  How are you doing on this?
Are you able to do those? No, Yes, Maybe. Remember you don't have to simplify, you just have to do it. ok? Simplifying is actually kind of messy. Radical of a radical it's a mess. Ok? Are you able to do it? 
15:30  Why this is plus?
Because the cosine is negative there. The cosine of x is negative 8 over the square root of 89. So here when I do 1 minus is minus minus. Did I answer your question? {yes}. So this is plus and this is negative ok? 
16:06  How would you do something like that?
It kind looks like one of those cosine A plus B thing. right? Its cosine of something plus something. So remember when you want to use that formula? you want to use cos (A+B)= cos(A)cos(B)sin(A)sin(B). 
16:44  cosine cosine, sine sine ok?
So cosine inverse of one quarter must be A and tan inverse of 3 must be B. so, cosine inverse of one quarter is A and tan inverse of 3 is B. 
17:10  cosine inverse of one quarter is A and that is if we have some triangle whose cosine is 1/4.
That's what it means that the cosine is 1/4: 
17:30  If we have some angle A, cosine of that angle is 1/4.
So you are going to look at Pythagorean theorem and get square root of 15. Tan inverse of 3 is B. We have some triangle, with angle B so you should make that 3 over 1. So this is square root of 10. 
18:04  ok?
Now cosine of A is 1/4, cosine of B is 1 over radical 10, sine of A is radical 15 over 4, and sine B is 3 over radical 10. 
18:32  This comes out as 1 minus 3 radical 15 over 4 radical 10.
Ok? Let's do it again. If I tell you that the cosine of, the cosine inverse of something, tangent inverse of something. These are both inverse trig functions. So an inverse trig function always stands for an angle. Because remember you take the trig function of an angle you get a number. 
19:02  If you do the inverse trig of a number you get an angle.
ok? So the inverse cosine of 1/4 is going to be A and the inverse tan of 3 is going to be B. You can call them x or y it does not really matter. So we know our formula for cosine of one angle plus another angle is equal to cos(A)cos(B)sin(A)sin(B). Now we need to figure out what A and B are. Really I don't care where they actually are. We know that the cosine of A is 1/4. 
19:32  That's what this tells us.
You draw some triangle, you put an A and make the cosine base over hypotenuse: 1/4. You look confused... oh, OK. Than Tangent inverse of 3 is B. So we have some triangle where tangent is 3 over 1. So we find the hypotenuse. And now we have a triangle, now we just plug things in our formula. Make sense? Kind of hard! You get it? 
20:03  Lets try one!
Maybe we will do something new. It does not involve trig... well, it sort of involves trig. 
20:45  Alright! So this is just like that.
You have to be able to imitate but you also have to understand, because if you just imitate you will have a problem on the final. ok? You need to understand So this is the sine of the inverse trig plus a inverse trig. 
21:05  This this must be some A plus some B.
ok? So this is the sine (A+B). We know that sin (A+B) = sin(A)cos(B) + cos(A)sin(B) And you know if you play with these enough, you'll easily memorize them. 
21:32  [unintelligible]
So this says we have an angle A and the cosine of that angle is 2/5.
If you are not sure that its in the first quadrant, this is positive so its exactly in the first quadrant. Cosine 2/5. So then you do the pythagorean theorem and this comes out square of 21. 
22:03  Alright?
I know that tangent of B is 3/8. That's angle B and the tangent of that angle is 3/8. So you do the pythagorean theorem and get square root of 73. So far so good? Now you just plug in! Sine A is square root of 21 over 5. 
22:31  Cosine of B is 8 over the square root of 73.
Cosine of A is 2 over 5. Sine of B is 3 over the square root of 73. By the way when you do these The denominator of this piece is the same denominator as this piece. If you don't have the same denominator, you screwed it up! So this is 8 radical 21 over 5 radical 73 plus 6 over 5 radical 73. 
23:12  ok?
So far so good? How did you do on this one? you got it? Or we just copy the other one. you are actually understand? Ok! without using your calculator. 
23:47  This is the graph of y= sin x.
What is the graph of absolute value of sine x look like? I was talking to Professor Sutherland, and the TRUE purpose of this class is to get you ready for calculus because you need to take calculus so you can and become doctors, 
24:03  PA's, nurses and whatever.
Otherwise you wouldn't be here So you are here because you have to do well in calculus, you cannot just take it. We have to give you grades otherwise you have to explain mom and dad next thanksgiving and its really a mess, very awkward. So if you want to do well, you have to have certain set of skills to go to calculus. One of them is you have to know trig ok? Another reason you have to be pretty good with all that factoring and simplifying and rearranging 
24:33  Calculus in and of itself is not that hard, its using all the calculus stuff that is hard.
The third thing that people are weak at is graphing. So I am going to try to make you a little better at graphing. And maybe I will succeed. The problem a lot of you have with graphing is, graph is just something that you draw. You guys don't always think of it as TELLING you something You have to say, "what does the graph tell me?" I want to do the absolute value of sine x. 
25:00  What does the absolute value mean?
what do you do when you are in absolute value? ok? So the absolute value of 5 is? [5] the absolute value of negative 5 is? [5] So if you have a negative value, just make it positive. Well so the real definition of the absolute value is how far am I from the origin. 
25:30  ok? but in one dimension, which is what we care about right now
...in fact, throughout this this course...
ok? as far as you are concerned, absolute value just means throw away the minus sign.
Because if I only care about how far I am from zero, I don't care if you are gong to the left or to the right, I don't care if its positive or negative, I just want to know how far you are. Ok? So if you are 10 miles from Stony Brook, I don't care in which direction you are, I just care that its 10 miles. That's an absolute value. So how to graph sine x. If I just look at the absolute value of sine x, then I don't want any, 
26:02  I just want to take all the negative values and make them positive.
So what does it look like? Well, right here I don't have any negative values. In here, I have negative values and I make them positive. It looks like that! ok? Like the golden arches... I just got hungry. Alright! so that's what the absolute value of sine x looks like. 
26:31  Does that make sense?
Because what I am doing is I am taking the negative values. From where am I getting the negative values? Not from the values of x, negative value of sine x. I am taking, where the function gives you negative values below the xaxis and make them positive. I'm flipping them up above the axis. Above the xaxis. ok? Make sense? uh? That's what the absolute of sin x looks like. 
27:06  What if I ask you ... just to confirm...
the absolute value of cosine?
They look very similar except .. at the start. Basically the cosine graph. It does this!ok! 
27:37  So when you take the places where it's below the xaxis and I need to make them positive.
So basically the same thing as the sine graph except it's shifted. ok? So if I have any function, the absolute value of that function takes whatever is below the xaxis and reflects above the xaxis. 
28:02  Because what is below the xaxis are negative values of the function and I want to make them positive.
So if I had y= x^3. Do you know what y=x^3 look like? You all remember what y=x^3 look like? ok? y=x^3 looks like that. 
28:35  You should develop family of functions,
little bank of functions in your brain because when you get to calculus,
you will see square, cube, log, e, sine, cosine, 1/x. I only know 7
ok? so you should know x^2, x^3, e^x, ln x, sin x, cos x, 1/x.

29:09  You should make sure that you can graph all 7 of these functions.
Those are all obviously f(x). Ok? Those are the once we like to do the most. Of course every once on a while we mess you up with another function. You should make sure you can do this, we will work with those on the next few days. ok? You don't have to worry about now but those are the 7 that you should know. You can give them names: Happy, Sleepy, Dopey, Doc, Bashful, Sneezy, and ... 
29:42  Alright! So if I want do the absolute value of this, I will take what's below the xaxis and I'd flip it above just like that.
Something funny probably goes on at the origin. ok? Well, it kind looks like a parabola, not quite. Ok? 
30:03  Because this is a parabola, this is the end of a parabola.
This is sort looks like a parabola. ok? So that is what the absolute value of x^3 look like. What about the absolute value of x^2? It looks the same as x^2 because x^2 is always positive. That is sort of a trick. 
30:48  What if I do y=sin(x)?
What does that look like? 
31:14  I think you should work on this!
Alright! What do we do about the sine of absolute value of x, well first of all sine x is positive it look like it always look. So, this side of the graph will look like that. 
31:30  Now work on the other side of the graph.
Well the left side, its going to look like the right side because if you took any negative number and you plug into x, the absolute value of x is positive. So it looks like that. So, what you do is you essentially take the sine and you reflect it over the other side. Ok? Its not the same as the absolute value of sin x. This is sine of absolute value of x. And what it is, is a reflection of whats to the right of the yaxis onto whats to the left of the yaxis. 
32:04  And what used to be on the left vanishes because you cannot have negative numbers plugged in here,
into the function because you take the absolute value of it.
Every time that you take a negative number and you put Into the bars becomes positive. Ok? So If you have some wired function. Let's move that over there. Let's say I have a function like that. 
32:37  On the other side, its not actually a function, we will mirror it.
That is as good as my art gets. So this will look like that. Ok? Does that make sense? Because here let's say at 4 you get this value, sorry positive 4. And at 4 you get the same picture. I knew that the positive numbers here that we would get the same value over here. ok? 
33:05  So every absolute value of x if harder that f(x).
Then the absolute value of f(x). Alright! You know whats coming. What if I want to do cosine of absolute value of x? 
33:36  The cosine of absolute value of x looks exactly the same as cosine of x. Ok?
That's because what's on the right side. What does the cosine look like? Looks the same on the right/left side. That's because the cosine of a negative angle is the same as the cosine of a positive angle. We learned that two weeks ago. That's is one of those identities. 
34:00  Cosine of negative theta is the same as cosine of positive theta. oK?
So cosine of absolute value of x is exactly the same as the cosine of x. So that is an easy question! Alright! Let's do a couple of others. 
34:48  y is equal to x. f(x)=x.
The line through the origin, slope of 45 degrees. y=x. Of course if y=x just goes like this. Ok? 
35:02  Make sure you know these graphs.
We looove to ask this stuff. ok? it shows up in the first semester of calculus, it shows up in the second semester of calculus, it even shows up in the third. You should know what they look like, because then if I give you 4x, you know thats it is just a variation on x. ok? If I gave you x^3, If you know what x^3 look like then x^3 plus 1, you just have to move up 1. 
35:31  What you want to do, you want to have a little bank in you head with basic graphs so that you can play with them when you need to.
x^2 is a parabola that goes through the origin. Ok? I made it pointy at the bottom.. it shouldn't be pointy at the bottom. x^3, we saw it a minute ago. 
36:05  And it should be sort of flat when it goes through the origin. ok?
Tangent goes through the origin at 45 degree angle, but x^3 goes through the origin. This is sort of flat there. It is not that important, let me try again. more like that! 
36:35  ok?
What does the square root of x look like? How about 1/x? 
37:11  Ok, notice the square root stops at the origin or starts at the origin depending on your point of view. ok?
why? Because we cannot have square root of negative numbers. There are no values over here. And what you get out is always positive. Because the radical stands for the positive square root 
37:31  ok? If we wanted the negative square root, we'd put a negative in front. ok, 1/x looks like this.
So 1/x^2 looks a lot like 1/x except you move the negative piece up on the second quadrant. Because we can't get negative values anymore, or absolute value of 1/x almost the same. 
38:00  e^x ok? first the asymptote: e^x has a horizontal asymptote at the xaxis.
So you cannot get a zero or a negative value. Remember that e^x is never zero. It is very useful in calculus. e^x is never zero. It is always a positive number. ok? I need to move the board so I have to make sure that everyone copied it. 
38:37  Alright!
sin(x) and cos(x), we will draw those a lot. 
39:13  Notice that I am sketching these, I am not really putting any points or anything right?
I guess you should know some points. 
39:34  You should have some basic idea of some important points. Cosine, same kind of thing.

40:14  Natural log of x, natural log of x may look like that has a horizontal asymptote but its not.
This keeps going up, very slowly. It does have one. It has a vertical asymptote in the yaxis. 
40:30  You cannot take the log of a negative number
Remember with e^x you can't get OUT a negative number, so with ln you can't put IN a negative number. With e^x you don't get out zero, with natural log you can't put in zero. That's why there are these asymptotes.
OK? Lets look at e^x again for a second. It passes through 1 on the yaxis ok? We only have positive values and this goes up. Natural log goes through 1, you can get negative values that you get out, but you put in only positive values 
41:07  That is why they are inverse of each other. "e" and natural log.
I would suggest that you make sure you know all those graphs. So other stuff you should be able to do with graphs... Ok? We will practice some more on Monday. You should be able to make sure that you get good at transforming the graphs. 
41:32  Figure out what happens when you change x, multiply by numbers, subtract, divide etc. Ok?
