Stony Brook MAT 123 Fall 2015
Lecture 23: Angle sum, double angle, and half-angle formulae
November 11, 2015

Start   So lets practice a bit of variation of yesterday.
make sure everybody can do something like that.
Very straight forward, you are in the first quadrant.
It is a very straight forward problem.
This is a kind of thing that you could see on the final.
We had stuff very similar to this [untelligible] on the first exam.
0:34The only difference is that there we told you cos(x) was 1/5 and we asked you the inverse sine.
This time [untelligible] Cosine inverse of 1/5 means we have a triangle in the first quadrant because it is positive.
The cosine of a positive number [untelligible] We know the cosine of that is 1/5.
1:00[unintelligible] Now sine of x, sine of x is opposite over hypotenuse.
So to find the opposite by Pythagorean theorem Sin (x) is radical 24 over 5.
That is it! That is the all problem!
1:31All you have to do is to get the two points and to do something like this.
This is part "oneish" question ok?
The final, ok so I will review this again.
I probably said this many times.
The final has part I and part II.
If you already passed both part ones so far you do not have to take that on the final,
2:00proceed directly to part II.
Leave that blank.
ok? If haven't passed both, then you to do it.
This time is on paper with partial credit and all that.
We found from last year that almost all of you were able to pull it off at the end ok?
I really try, I look very hard for that partial credit.
With several of you who I said sure I will give you half credit for this and I had no idea who really deserved that half credit.
You can give it a shot and that is ok.
Yes. If you don't need to do part I on the final than you don't lose any point you just don't have to do it.
So for those you did not pass part I, need to pass part I.
And then they get a full final grade.
If they don't, Then they are not going to get a grade for the course.
If you did pass both part ones you just take the final and then you stop.
3:02If I say tangent sin inverse of 4/9 equals question mark.
Go to first quadrant, draw an angle, and you need to find the tangent of that angle.
If you know sine of that angle is 4/9
3:31Then you find the missing side using Pythagorean theorem.
try not to mess that up. Use your scrap paper.
Now you can find tangent of x.
Tangent of x is square root 65 over 9.
Because it is positive, it is in the first quadrant.
4:04Not too bad?
[unintelligible] Oh! I am sorry I wrote the wrong thing, I wrote the cosine, my bad!
here we go.
You know I am old.
What is that was negative 1/5
4:31The only difference now is that you need to go on the second quadrant.
ok? The sine is positive on the second quadrant.
This is equal to square root of 24 over 5.
5:00If I ask you for example for the tangent than you get a negative value.
You need to pay attention at the quadrant you go to.
Remember if you do the inverse trig function of positive number you are in the first quadrant If you do the inverse trig function of a negative number then you are in the second or forth quadrant depending on which angle.
So if I asked for tan cosine inverse of -1/5
5:32we already have the picture it would be negative square root of 24 over 1 which is negative square root of 24. Ok?
You got that? One more!
Suppose this was tangent sine inverse of -4/5.
Now you go down here to make the triangle.
6:11It would be the same as negative answer and a positive answer How do you feel about these?
(student) Why did I use the fourth quadrant?
Why did I use the fourth quadrant?
6:30When I do the inverse sine, where the inverse of a negative number I use the forth quadrant When I do the inverse cosine, I use the second quadrant.
I never use the third quadrant.
You guys are ok with this? thats great!
Ok. Lets do something completely different.
Alright! Let me go erase and show you something new.
8:15Ok! So we draw the unit circle and we label two different angles.
First angle A and second angle B.
The coordinates are (cosA, sinA) on top and (cosB,-sinB) on the bottom.
8:40ok! So I want to find the distance between those two points.
The distance is right here.
Well, imagine the distance formula.
the distance formula was, is Pythagorean theorem.
9:05You take the difference of x's squared and the difference of y's squared.
Its sort of Pythagorean theorem Distances squared plus distances squared is D squared.
Ok? So lets plug in!
This is cosA-cosB squared plus sinA minus, minus sinB squared.
9:47The square roots are annoying so lets do D squared instead.
It does not matter. [unintelligible].
10:01Now, we love FOIL. So let's multiply these out.
This is cosine squared A - 2cos A cosB plus cosine squared B plus sine squared A plus 2 sinA sinB plus sine B squared.
Remember we are trying to find the distance using the distance formula.
10:33Cannot leave that!
That says: cosine squared A - 2cos A cosB plus cosine squared B plus sine squared A plus 2 sinA sinB plus sine square B.
So far so good?
Where did I make the mistake?
11:00You still cannot see.
Already, I will re-write it nice and big
11:31How is that?
Can you see it now?
Alright, now we need to simplify it.
What is sine squared plus cosine squared equal?
1 (one). Same angle So you can take these two terms and replace it with one.
12:01And cosine squared B plus sine squared B you replace them with another 1.
And then you got 2 cosine A cosine B plus 2 sine A sine B.
One plus one is 2.
You can do that in your heads.
12:34How are you ok?
So far so good? Alright!
We will leave that alone one second.
Remember I am trying to find D.
I also could find D with law of cosine.
But I use the law of cosine instead.
13:06D squared equals 1 squared plus 1 squared minus 2 times 1 time 1 times cosine (A+B) I need both angles.
Do you agree?
So one squared plus one squared,
13:302 times one time one and the cosine of the angle between them.
So this simplifies to D squared equal to 2 -2 cosine( A+B).
Now we say that equals to that because they are both equal to D squared 2 minus 2 cosine (A+B) equals 2 minus 2 cosine A cosine B plus sine A sine B.
14:07This is fun, this is called derivation We derive We are not proving, we deriving.
And we are getting there, we getting there. I promise!
cancel the 2 ok?
14:31and now you get minus 2.
I lost a 2 here sorry!
I have to write that down!
So, And now I divide by minus 2 and I get cos(A+B)=cosA cosB - sinA sin B.
15:14Some of you learned this mysterious formula back, last time you took trigonometry And of course if you took the regents it was on the formula sheet.
I divided by -2 here. I divided by -2.
15:31So there we go!
So now we derive the formula and we are going to use that formula.
Because what really matters is this.
That tells you by the way, this says cos(A+B) is not cos A + cos B.
16:04[unintelligible] ok? Some people think that they are multiplying by cosine ok?
They are not multiplying by cosine. cos(A+B) is NOT cosA +cos B.
It is this Cosine of the first angle cosine of the second angle and sine of first angle minus sine of second angle We can use it right now.
16:30So write it down and save that formula.
17:08So why it that nice to know.
Well, [unintelligible] for example.
What if I asked to find cosine of 75 degree.
17:34You say, I heard cosine of 75 degrees but I don't think is in that chart It is not on the chart.
I don't understand it.
I am going to fail.
This is unfair.
This was not on the chart, he told me I did not need to know this.
How am I going to find cosine of 75.
Now I am not going to [unintelligible] My mom is supporting me.
I say, wait a second.
We can figure out cos 75 degrees.
because 75 degrees is 45 and 30. You promise!
18:07It is also 50 and 25 but that is not [unintelligible].
So now we have a formula for that.
Is cosine of the fist angle cosine of the second angle minus sine of the first angle sine of the second angle.
you just plug in the formula ok?
18:32You know the formula cos45 cos30-sin45 sin30.
Now we just plug in the values We memorized these.
We all know these.
19:03Ok square root of 2 times square root of 3 is square root of 6 so square root of 6 over 4.
Minus the square root of 2 over 4.
If you really want to show off: square root of 6 minus square root of 2 over 4.
Square root of 6 minus square root of 2 is NOT square root of 4.
ok? That would be too simple if they were.
ok? Any of those are acceptable.
for those who have asked me do we need to go to this step on thee exam
19:34No we do NOT.
Thats fine! But you should be able to do that.
So that what we will use for cos(A+B).
Very convenient formula [...] right over here.
[unintelligible] I like to see you be able to go one more step but, you know, I am nervous that you will do the square root of 2
20:02times the square root of 3 is equal to the square root of 32 Or something like that.
Or square root of 5 [unintelligible] And then I will feel bad.
[unintelligible] Yes!
Student: [ where the sine come from?] Remember you use that formula.
Alright! Now that we know how to do cos(A+B) what is Cos(A-B)?
20:35Where do I use this?
Suppose I want to find the cosine of 25 degrees.
I will erase!
21:00So if I want to find the cosine of 15 degrees.
You say well, how do I find 15 degrees?
I only know 30, 45, and 60.
You do 45 minus 30.
You can also do 60 minus 45 by the way.
But thats good enough.
with the formula: cos45 cos30 +sin45 sin30.
21:41And now we plug in the values.
cosine of 45 is radical 2 over 2.
cosine of 30 is radical 3 over 2, sin 45 is radical 2 over 2, sin 30 is 1/2.
that simplifies radical 6 over 4, minus radical 2 over 4
22:05Radical 6 minus radical 2 over 4.
22:37Ok so now we did cosine formula.
What if the sine affected the cosine ? wellI, I will just spoil it.
I just tell you sin (A+B).
Sin (A+B)= sin A cos B + cos A sin B.
23:04And if that is a negative sign.
Sin (A-B)= sin A cos B - cos A sin B.
23:31Are you ready?
Why don't you find sine of 75 degrees?
Sine of 15 degrees?
See how you do it!
So if you want to find the sine of 75, that would be sin(45+30).
But you can certainly think of it as 30+45.
That does not matter. Right?
24:02If you do sin 15, you have to do sin (45-30) or (60-45).
Cannot have it backward, you will get -15.
So which one is A? A is the first one and B is the second one.
Sin (45+30) = sin 45 cos 30 +cos 45sin 30
24:39You can switch 30 and 45 because we are adding.
What is the sin 45 is radical 2 over 2, cos 30 is radical 3 over 2.
You just follow the other one we did.
You should because sin 75 is cos 15
25:01Radical 2 over 2 and that's 1/2.
You get radical 6 over 4 (plus) radical 2 over 4.
[unintelligible] You just follow the formula The key will be when we give you and angle and figure out which 2 angles is equal to.
Yes! There is a plus. A plus.
25:32Same question? Yea!
Now we know, [unintelligible] Alright! The sine of (45-30) is sin 45 cos 30 minus cos 45 sin 30.
26:02Ok! So radical 2 over 2, radical 3 over 2 minus radical 2 over 2 times 1/2.
Which is radical 2 over 4 minus radical 2 over 4.
How are you doing on these?
We always get a special angle.
We kind of have to because if we gave you sin 78,
26:32there is nothing you can do.
There is no pair of "special angles" that is equal to 78 Alright? So far so good?
Lets practice more!
27:16Sure why not!
so if i want to do sin of 105 is pair of angles pair angles that we know the trig values of that add up to 105.
27:30(60+45) is a good one.
Write it like this sin (60+45).
You can also write (45+60).
This is sin 60 cos 45 + cos 60 sin 45 Plug in values. radical 3 over 2 radical 2 over 2 plus 1/2, radical 2 over 2.
28:14This is equals to radical 6 over 4 plus radical 2 over 4.
Does it look familiar?
It should.
Is the same thing as sine of 75.
Its in the second quadrant.
Sine in the second quadrant is positive.
28:31Just like in the first one Alright! If you do the cos 105.
That is cos (60+45) And that is cos 60 cos 45 minus sin 60 sin 45.
29:02And that is radical 3 over 2.
Sorry 1/2, radical 2 over 2 minus radical 3 over 2, radical 2 over 2.
This is radical 2 minus radical 6 over 4.
Which is the negative of what we had before.
ok? Of course we might ask you to do these in radiants,
29:33sometimes instead of degrees we ask you to do these in radiants.
I hope I gave you guys some clues just to me your life a little easier.
Everybody copied these stuff down?
I see you are still writing. I'll wait!
We keep working with the same angles.
So 30 degrees is pi/6, 45 degrees is pi/4.
Take pi/6 and pi/4 this is 5pi/12.
30:33That is 2 over 12 and 3 over 12.
So what we will do is we will ask you for sine and cosine of 5pi/12 We hope you know that this is pi/6 plus pi/4.
If you are not sure, convert into degrees.
What about pi/4 plus pi/3.
Well that is 7pi/12
31:14And lets's see.
What about pi/ 4 minus pi/6 That's equal to 15 degrees.
That is pi/12.
See now we can ask you questions like What is the sine of pi/12?
31:32Or do you know what I can do if I was really nasty?
I can ask you for tan 105 What would the tan105 degree?
Find the sine and find the cosine and then you divide.
Tan 105 would be square root of 6 plus square root of 2 over 4 divide by square root of 6 minus square root of 2 over 4
32:03Which is square root of 6 plus square root of 2 over square root of 6 minus square root of 2.
Not so bad! Ok?
So I could ask you for the tangent.
It is a lot of work to ask you for that but it could be a paper homework problem maybe a test problem.
Alright! Some other stuff.
Suppose I want to find what is called the double angle formula?
Suppose I want to find the sine of 2A.
32:42Why do I want to find the double angle?
33:03So how would I find the sine of 2A?
Well, I will make my sin (A+B) formula and write it as sin (A+A).
Sin(A+A) is a very complicated formula.
Sin A cos A plus cos A sin A.
33:32These two are the same thing the sin A and cos A.
So this is just 2 sin A cosA.
And now we have the double angle.
34:03See what happens is, when you doing calculus you will have some problems.
If it would be in this form 2sin A cos A.
And you can write it instead in this form sin 2A.
Or the other way around.
And one will make the problem easier than the other one.
So you just want to be able to shift from on form to the other form.
What if I want to find the cosine of 2A?
34:35That's is the cos (A+A) That is: cosA cosA minus sin A sin A.
And that simplifies to cos squared A minus sine squared A.
35:04Don't confuse that with cosine squared plus sine squared.
We have cosine squared "minus" sine squared.
Cosine squared plus sine squared is 1.
Cosine squared minus 2 sine square is just the cos 2A.
So far so good? ok, here is a fun fact!
35:36Lets go back to cos 2A.
Notice here we have 2 times the angle and here we have a single angle.
So one of the things you are doing is you reducing the angle by a factor of 2.
36:01You working out the problem.
Sometimes you have an x.
I also know that cosine squared plus sine squared is 1.
So I can replace cos^2 A with (1-sin^2 A) or I can replace sin ^2 A with (1-cos^2 A).
I can replace this with (1- sin^2 ).
I will get 1-2sin^2 A.
36:33All I did is I replaced cos^2 A with (1-sin^2 A) Right here.
Similarly, I can take the sin^2 A and write it as (1-cos^2 A).
37:05So, double angle formulas have a separate board.
37:56So far so good?
38:01It is convenient to know all of three of those double angle formulas because different values depend on the situation But you really have to memorize one of them and you can always figure out the other two In fact, if you know the addition formulas [unintelligible] you don't need to memorize them.
You can always derive them. ok?
I erase them.
You can just derive them.
38:30[unintelligible] Alright, lets do some other things with this.
Just bunch of formulas today!
39:01What if instead of double angle, I want to find half angle?
Instead of double angle I want to find half an angle.
Well notice this A is 1/2 of that 2A.
This A is 1/2 of that 2A.
39:43Lets say I start with this formula.
And I want to isolate the cosine of 2A.
Because I want it to be half of the angle.
Well, add 1 on both sides and
40:04divide by 2.
next you take the square root.
cos A plus or minus the square root of (1+cos 2A / 2) ok?
40:30Why Plus or minus?
it depends in what quadrant you end up with.
Ok? Don't get too worry about that!
Notice this angle is half of that angle This is cos A and this is cos 2A So if I made this A, that would be 1/2 of A.
The half angle formula.
Did all see I made that switch All I did, I said A is 1/2 of 2A.
And if this is A, that is A/2.
41:35We almost did too much work!
Alright! What if I want to do sine of half and angle?
Well I take the cos 2A formula again.
And now I isolate sin of A.
42:04So 2 sin ^2 A is 1- cos2A.
Divide it by 2.
And take the square root of it.
42:40In other words: sin (A/2) is plus or minus the square root of (1-cos A)/ 2.
So lets do an example!
How are you guys doing
This is A/2 and this is just single A.
43:32What ever this angle is, is just double of that.
Suppose I say to you guys, find the sin (pi/8).
sin (pi/8) the sine of pi/8 is half of pi/4.
44:10So I can use the half angle formula!
How do I know its the positive square root?
Where is pi/8 located? pi/8 is located in the first quadrant. Right?
44:33So half of it can also be located in the first quadrant.
So I use the positive square root.
Now what it cos pi/4?
You can leave it like that.
ok? You don't have to simplify.
45:01It is not so easy to simplify How about cos of 5pi/ 8?
So 5pi/8 is half of pi/4.
45:37ok! 5pi/4 is 225 degrees so 5pi/8 is 112.5 degrees.
That is in the second quadrant. Ok?
So this cosine (5pi/4)/2 is going to be a second quadrant angle
46:04so we will use a negative square root of (1+ cos5pi/4) /2.
That is how I know to make it negative. ok?
[unintelligible] Now, what is the cos of 5pi/4?
46:30ok? 5pi/4 is 225 degrees.
Thats a third quadrant angle.
Half of it is 112.5 degrees its in the second quadrant because its more than 90.
Where is cos 5pi/4?
where is cos 5pi/4?
look at the derivation of it [unintelligible] You can always give it a shot!
47:01What is it? cos, positive or negative?
negative square root of 2 over 2.
And you can leave it like that!
Why is it square root of 2 over 2?
You suppose know how to find cos of 5pi/4 ok?
draw the angle, pick the right quadrant and figure out if its positive or negative
47:32All that stuff.
Alright I will see everybody on Monday.