WEBVTT
Kind: captions
Language: en-US
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So lets practice a bit of variation of yesterday.
00:00:05.000 --> 00:00:13.700
make sure everybody can do something like that.
00:00:15.140 --> 00:00:15.840
ok?
00:00:17.540 --> 00:00:20.280
Very straight forward, you are in the first quadrant.
00:00:21.000 --> 00:00:23.660
It is a very straight forward problem.
00:00:23.940 --> 00:00:26.660
This is a kind of thing that you could see on the final.
00:00:27.580 --> 00:00:32.720
We had stuff very similar to this [untelligible] on the first exam.
00:00:34.340 --> 00:00:38.100
The only difference is that there we told you cos(x)
00:00:38.100 --> 00:00:41.240
was 1/5 and we asked you the inverse sine.
00:00:41.240 --> 00:00:47.580
This time [untelligible] Cosine inverse of 1/5 means
00:00:47.580 --> 00:00:51.860
we have a triangle in the first quadrant because it is positive.
00:00:51.860 --> 00:00:53.860
The cosine of a positive number [untelligible]
00:00:54.540 --> 00:00:58.260
We know the cosine of that is 1/5.
00:01:00.340 --> 00:01:10.760
[unintelligible]
00:01:10.760 --> 00:01:14.100
Now sine of x, sine of x is opposite over hypotenuse.
00:01:14.960 --> 00:01:17.520
So to find the opposite by Pythagorean theorem
00:01:22.360 --> 00:01:27.420
Sin (x) is radical 24 over 5.
00:01:29.740 --> 00:01:31.740
That is it! That is the all problem!
00:01:31.740 --> 00:01:39.300
All you have to do is to get the two points and to do something like this.
00:01:42.700 --> 00:01:45.240
This is part "oneish" question ok?
00:01:45.680 --> 00:01:48.960
The final, ok so I will review this again.
00:01:48.960 --> 00:01:49.740
I probably said this many times.
00:01:49.740 --> 00:01:51.740
The final has part I and part II.
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If you already passed both part ones so far you do not have to take that on the final,
00:02:00.140 --> 00:02:02.140
proceed directly to part II.
00:02:02.140 --> 00:02:03.380
Leave that blank.
00:02:03.960 --> 00:02:07.640
ok? If haven't passed both, then you to do it.
00:02:08.040 --> 00:02:11.120
This time is on paper with partial credit and all that.
00:02:11.120 --> 00:02:17.180
We found from last year that almost all of you were able to pull it off at the end ok?
00:02:17.800 --> 00:02:20.700
I really try, I look very hard for that partial credit.
00:02:20.700 --> 00:02:24.960
With several of you who I said sure I will give you half credit for this
00:02:24.960 --> 00:02:27.700
and I had no idea who really deserved that half credit.
00:02:27.700 --> 00:02:29.380
You can give it a shot and that is ok.
00:02:29.380 --> 00:02:38.940
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.
00:02:38.940 --> 00:02:45.340
Yes.
So for those you did not pass part I, need to pass part I.
00:02:47.540 --> 00:02:49.540
And then they get a full final grade.
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If they don't, Then they are not going to get a grade for the course.
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[unintelligible].
00:02:57.000 --> 00:03:02.020
If you did pass both part ones you just take the final and then you stop.
00:03:02.920 --> 00:03:14.060
If I say tangent sin inverse of 4/9 equals question mark.
00:03:15.000 --> 00:03:18.960
Go to first quadrant, draw an angle,
00:03:19.920 --> 00:03:22.560
and you need to find the tangent of that angle.
00:03:22.560 --> 00:03:27.460
If you know sine of that angle is 4/9
00:03:31.700 --> 00:03:35.440
Then you find the missing side using Pythagorean theorem.
00:03:41.080 --> 00:03:43.740
try not to mess that up. Use your scrap paper.
00:03:47.240 --> 00:03:49.940
Now you can find tangent of x.
00:03:49.940 --> 00:03:53.480
Tangent of x is square root 65 over 9.
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Because it is positive, it is in the first quadrant.
00:04:04.000 --> 00:04:06.000
Not too bad?
00:04:06.620 --> 00:04:08.620
[unintelligible]
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Oh! I am sorry I wrote the wrong thing, I wrote the cosine, my bad!
00:04:21.380 --> 00:04:23.380
here we go.
00:04:25.320 --> 00:04:27.320
You know I am old.
00:04:28.740 --> 00:04:30.740
What is that was negative 1/5
00:04:31.500 --> 00:04:36.460
The only difference now is that you need to go on the second quadrant.
00:04:50.400 --> 00:04:53.980
ok? The sine is positive on the second quadrant.
00:04:54.580 --> 00:04:59.740
This is equal to square root of 24 over 5.
00:05:00.740 --> 00:05:04.560
If I ask you for example for the tangent than you get a negative value.
00:05:04.560 --> 00:05:07.080
You need to pay attention at the quadrant you go to.
00:05:07.080 --> 00:05:08.300
ok?
00:05:08.300 --> 00:05:11.820
Remember if you do the inverse trig function of positive number you are in the first quadrant
00:05:12.580 --> 00:05:15.740
If you do the inverse trig function of a negative number then
00:05:15.740 --> 00:05:20.200
you are in the second or forth quadrant depending on which angle.
00:05:24.900 --> 00:05:31.440
So if I asked for tan cosine inverse of -1/5
00:05:32.080 --> 00:05:34.080
we already have the picture
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it would be negative square root of 24 over 1
00:05:37.380 --> 00:05:39.380
which is negative square root of 24. Ok?
00:05:44.320 --> 00:05:46.320
You got that? One more!
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Suppose this was tangent sine inverse of -4/5.
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Now you go down here to make the triangle.
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It would be the same as negative answer and a positive answer
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How do you feel about these?
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Questions?
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(student) Why did I use the fourth quadrant?
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Why did I use the fourth quadrant?
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When I do the inverse sine,
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where the inverse of a negative number I use the forth quadrant
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When I do the inverse cosine, I use the second quadrant.
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I never use the third quadrant.
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You guys are ok with this? thats great!
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Ok. Lets do something completely different.
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Alright! Let me go erase and show you something new.
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Ok! So we draw the unit circle and we label two different angles.
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First angle A and second angle B.
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The coordinates are (cosA, sinA) on top and (cosB,-sinB) on the bottom.
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ok! So I want to find the distance between those two points.
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The distance is right here.
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Well, imagine the distance formula.
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the distance formula was, is
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Pythagorean theorem.
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You take the difference of x's squared and the difference of y's squared.
00:09:09.960 --> 00:09:11.960
Its sort of Pythagorean theorem
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Distances squared plus distances squared is D squared.
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Ok? So lets plug in!
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This is cosA-cosB squared plus sinA minus, minus sinB squared.
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The square roots are annoying so lets do D squared instead.
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It does not matter. [unintelligible].
00:10:01.120 --> 00:10:03.700
Now, we love FOIL. So let's multiply these out.
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This is cosine squared A - 2cos A cosB plus cosine squared B
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plus sine squared A plus 2 sinA sinB plus sine B squared.
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Remember we are trying to find the distance using the distance formula.
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Cannot leave that!
00:10:34.780 --> 00:10:45.280
That says: cosine squared A - 2cos A cosB plus cosine squared B
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plus sine squared A plus 2 sinA sinB plus sine square B.
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So far so good?
00:10:57.280 --> 00:10:58.660
Where did I make the mistake?
00:11:00.580 --> 00:11:01.740
You still cannot see.
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Already, I will re-write it nice and big
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How is that?
00:11:32.600 --> 00:11:34.020
Can you see it now?
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Alright, now we need to simplify it.
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What is sine squared plus cosine squared equal?
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1 (one). Same angle
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So you can take these two terms and replace it with one.
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And cosine squared B plus sine squared B you replace them with another 1.
00:12:06.580 --> 00:12:16.620
And then you got 2 cosine A cosine B plus 2 sine A sine B.
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One plus one is 2.
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You can do that in your heads.
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How are you ok?
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So far so good? Alright!
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We will leave that alone one second.
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Remember I am trying to find D.
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I also could find D with law of cosine.
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But I use the law of cosine instead.
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D squared equals 1 squared plus 1 squared minus 2 times 1 time 1 times cosine (A+B)
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I need both angles.
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Do you agree?
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So one squared plus one squared,
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2 times one time one and the cosine of the angle between them.
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So this simplifies to D squared equal to 2 -2 cosine( A+B).
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Now we say that equals to that because they are both equal to D squared
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2 minus 2 cosine (A+B) equals 2 minus 2 cosine A cosine B plus sine A sine B.
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This is fun, this is called derivation
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We derive
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We are not proving, we deriving.
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And we are getting there, we getting there. I promise!
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cancel the 2
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ok?
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and now you get minus 2.
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I lost a 2 here sorry!
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I have to write that down!
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ok?
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So,
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And now I divide by minus 2 and I get cos(A+B)=cosA cosB - sinA sin B.
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Some of you learned this mysterious formula back, last time you took trigonometry
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And of course if you took the regents it was on the formula sheet.
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I divided by -2 here. I divided by -2.
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So there we go!
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So now we derive the formula and we are going to use that formula.
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ok?
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Because what really matters is this.
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That tells you by the way, this says cos(A+B) is not cos A + cos B.
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[unintelligible]
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ok? Some people think that they are multiplying by cosine ok?
00:16:18.020 --> 00:16:23.580
They are not multiplying by cosine. cos(A+B) is NOT cosA +cos B.
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It is this
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Cosine of the first angle cosine of the second angle
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and sine of first angle minus sine of second angle
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We can use it right now.
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So write it down and save that formula.
00:17:08.300 --> 00:17:10.080
So why it that nice to know.
00:17:10.080 --> 00:17:14.860
Well, [unintelligible]
00:17:14.860 --> 00:17:16.360
for example.
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What if I asked to find cosine of 75 degree.
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You say, I heard cosine of 75 degrees but I don't think is in that chart
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It is not on the chart.
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I don't understand it.
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I am going to fail.
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This is unfair.
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This was not on the chart, he told me I did not need to know this.
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How am I going to find cosine of 75.
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Now I am not going to [unintelligible] My mom is supporting me.
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I say, wait a second.
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We can figure out cos 75 degrees.
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because 75 degrees is 45 and 30. You promise!
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It is also 50 and 25 but that is not [unintelligible].
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So now we have a formula for that.
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Is cosine of the fist angle cosine of the second angle
00:18:21.400 --> 00:18:27.320
minus sine of the first angle sine of the second angle.
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you just plug in the formula ok?
00:18:32.960 --> 00:18:34.360
You know the formula
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cos45 cos30-sin45 sin30.
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Now we just plug in the values
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We memorized these.
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We all know these.
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Ok square root of 2 times square root of 3 is square root of 6 so square root of 6 over 4.
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Minus the square root of 2 over 4.
00:19:11.500 --> 00:19:13.500
If you really want to show off:
00:19:15.920 --> 00:19:17.920
square root of 6 minus square root of 2 over 4.
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Square root of 6 minus square root of 2 is NOT square root of 4.
00:19:23.740 --> 00:19:25.740
ok? That would be too simple if they were.
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ok? Any of those are acceptable.
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for those who have asked me do we need to go to this step on thee exam
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No we do NOT.
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Thats fine! But you should be able to do that.
00:19:40.840 --> 00:19:43.460
So that what we will use for cos(A+B).
00:19:43.460 --> 00:19:45.460
Very convenient formula
00:19:45.460 --> 00:19:50.900
[...] right over here.
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ok?
00:19:53.740 --> 00:19:55.420
[unintelligible]
00:19:55.420 --> 00:19:58.840
I like to see you be able to go one more step but,
00:19:58.840 --> 00:20:02.060
you know, I am nervous that you will do the square root of 2
00:20:02.060 --> 00:20:04.060
times the square root of 3 is equal to the square root of 32
00:20:04.660 --> 00:20:06.340
Or something like that.
00:20:06.340 --> 00:20:09.520
Or square root of 5 [unintelligible]
00:20:09.520 --> 00:20:10.920
And then I will feel bad.
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[unintelligible]
00:20:15.300 --> 00:20:16.300
Yes!
00:20:16.300 --> 00:20:19.060
Student: [ where the sine come from?]
00:20:23.860 --> 00:20:25.860
Remember you use that formula.
00:20:26.060 --> 00:20:32.700
Alright! Now that we know how to do cos(A+B) what is Cos(A-B)?
00:20:35.040 --> 00:20:36.180
Where do I use this?
00:20:36.180 --> 00:20:41.920
Suppose I want to find the cosine of 25 degrees.
00:20:45.160 --> 00:20:46.600
I will erase!
00:21:00.960 --> 00:21:03.500
So if I want to find the cosine of 15 degrees.
00:21:05.180 --> 00:21:08.580
You say well, how do I find 15 degrees?
00:21:10.040 --> 00:21:12.040
I only know 30, 45, and 60.
00:21:13.400 --> 00:21:14.460
You do 45 minus 30.
00:21:14.460 --> 00:21:16.460
You can also do 60 minus 45 by the way.
00:21:17.600 --> 00:21:19.600
But thats good enough.
00:21:25.020 --> 00:21:36.540
with the formula: cos45 cos30 +sin45 sin30.
00:21:41.960 --> 00:21:43.340
And now we plug in the values.
00:21:43.340 --> 00:21:46.360
cosine of 45 is radical 2 over 2.
00:21:48.520 --> 00:21:58.820
cosine of 30 is radical 3 over 2, sin 45 is radical 2 over 2, sin 30 is 1/2.
00:21:59.760 --> 00:22:04.720
that simplifies radical 6 over 4, minus radical 2 over 4
00:22:05.520 --> 00:22:10.260
Radical 6 minus radical 2 over 4.
00:22:14.380 --> 00:22:16.380
Plus!
00:22:37.840 --> 00:22:40.220
Ok so now we did cosine formula.
00:22:41.440 --> 00:22:46.420
What if the sine affected the cosine ? wellI, I will just spoil it.
00:22:48.820 --> 00:22:52.300
I just tell you sin (A+B).
00:22:55.300 --> 00:23:03.540
Sin (A+B)= sin A cos B + cos A sin B.
00:23:04.800 --> 00:23:06.800
And if that is a negative sign.
00:23:06.800 --> 00:23:19.240
Sin (A-B)= sin A cos B - cos A sin B.
00:23:31.640 --> 00:23:33.200
Are you ready?
00:23:33.200 --> 00:23:37.660
Why don't you find sine of 75 degrees?
00:23:39.120 --> 00:23:40.020
Sine of 15 degrees?
00:23:41.500 --> 00:23:43.500
See how you do it!
00:23:44.960 --> 00:23:56.080
So if you want to find the sine of 75, that would be sin(45+30).
00:23:56.080 --> 00:23:59.340
But you can certainly think of it as 30+45.
00:23:59.340 --> 00:24:01.840
That does not matter. Right?
00:24:02.440 --> 00:24:11.540
If you do sin 15, you have to do sin (45-30) or (60-45).
00:24:11.540 --> 00:24:15.500
Cannot have it backward, you will get -15.
00:24:20.820 --> 00:24:23.880
So which one is A? A is the first one and B is the second one.
00:24:26.880 --> 00:24:39.040
Sin (45+30) = sin 45 cos 30 +cos 45sin 30
00:24:39.960 --> 00:24:43.940
You can switch 30 and 45 because we are adding.
00:24:45.960 --> 00:24:52.780
What is the sin 45 is radical 2 over 2, cos 30 is radical 3 over 2.
00:24:52.780 --> 00:24:54.780
You just follow the other one we did.
00:24:55.600 --> 00:24:58.640
You should because sin 75 is cos 15
00:25:01.520 --> 00:25:05.180
Radical 2 over 2 and that's 1/2.
00:25:06.400 --> 00:25:12.200
You get radical 6 over 4 (plus) radical 2 over 4.
00:25:15.580 --> 00:25:17.200
[unintelligible]
00:25:17.200 --> 00:25:19.200
You just follow the formula
00:25:19.200 --> 00:25:23.740
The key will be when we give you and angle and figure out which 2 angles is equal to.
00:25:24.220 --> 00:25:30.620
Yes! There is a plus. A plus.
00:25:32.160 --> 00:25:35.220
Same question? Yea!
00:25:35.700 --> 00:25:46.680
Now we know, [unintelligible]
00:25:47.380 --> 00:26:02.800
Alright! The sine of (45-30) is sin 45 cos 30 minus cos 45 sin 30.
00:26:02.840 --> 00:26:16.020
Ok! So radical 2 over 2, radical 3 over 2 minus radical 2 over 2 times 1/2.
00:26:17.080 --> 00:26:21.680
Which is radical 2 over 4 minus radical 2 over 4.
00:26:21.680 --> 00:26:23.680
How are you doing on these?
00:26:26.500 --> 00:26:28.100
We always get a special angle.
00:26:28.100 --> 00:26:32.380
We kind of have to because if we gave you sin 78,
00:26:32.380 --> 00:26:33.860
there is nothing you can do.
00:26:33.860 --> 00:26:37.720
There is no pair of "special angles" that is equal to 78
00:26:38.900 --> 00:26:41.900
Alright? So far so good?
00:26:43.480 --> 00:26:45.480
Lets practice more!
00:27:16.440 --> 00:27:18.440
Sure why not!
00:27:19.160 --> 00:27:20.220
Ok!
00:27:20.900 --> 00:27:23.120
so if i want to do sin of 105 is pair of angles
00:27:23.120 --> 00:27:30.120
pair angles that we know the trig values of that add up to 105.
00:27:30.120 --> 00:27:32.120
(60+45) is a good one.
00:27:35.380 --> 00:27:41.180
Write it like this sin (60+45).
00:27:41.180 --> 00:27:43.640
You can also write (45+60).
00:27:44.780 --> 00:27:58.220
This is sin 60 cos 45 + cos 60 sin 45
00:27:58.840 --> 00:28:10.180
Plug in values. radical 3 over 2 radical 2 over 2 plus 1/2, radical 2 over 2.
00:28:14.720 --> 00:28:20.660
This is equals to radical 6 over 4 plus radical 2 over 4.
00:28:21.560 --> 00:28:22.680
Does it look familiar?
00:28:22.680 --> 00:28:23.580
It should.
00:28:24.380 --> 00:28:26.680
Is the same thing as sine of 75.
00:28:27.920 --> 00:28:29.920
Its in the second quadrant.
00:28:29.920 --> 00:28:31.340
Sine in the second quadrant is positive.
00:28:31.340 --> 00:28:32.440
Just like in the first one
00:28:33.420 --> 00:28:35.420
Alright! If you do the cos 105.
00:28:35.420 --> 00:28:59.920
That is cos (60+45) And that is cos 60 cos 45 minus sin 60 sin 45.
00:29:02.500 --> 00:29:06.000
And that is radical 3 over 2.
00:29:06.000 --> 00:29:21.320
Sorry 1/2, radical 2 over 2 minus radical 3 over 2, radical 2 over 2.
00:29:22.600 --> 00:29:26.060
This is radical 2 minus radical 6 over 4.
00:29:26.060 --> 00:29:29.360
Which is the negative of what we had before.
00:29:29.360 --> 00:29:33.900
ok? Of course we might ask you to do these in radiants,
00:29:33.900 --> 00:29:38.760
sometimes instead of degrees we ask you to do these in radiants.
00:29:40.280 --> 00:29:45.700
I hope I gave you guys some clues just to me your life a little easier.
00:29:45.700 --> 00:29:47.700
Everybody copied these stuff down?
00:29:49.200 --> 00:29:51.200
I see you are still writing. I'll wait!
00:29:52.880 --> 00:30:04.100
[unintelligible]
00:30:04.100 --> 00:30:06.100
alright!
00:30:10.200 --> 00:30:12.200
We keep working with the same angles.
00:30:19.460 --> 00:30:24.160
So 30 degrees is pi/6, 45 degrees is pi/4.
00:30:24.160 --> 00:30:30.660
Take pi/6 and pi/4 this is 5pi/12.
00:30:33.040 --> 00:30:36.820
That is 2 over 12 and 3 over 12.
00:30:36.820 --> 00:30:41.760
So what we will do is we will ask you for sine and cosine of 5pi/12
00:30:41.760 --> 00:30:45.920
We hope you know that this is pi/6 plus pi/4.
00:30:45.920 --> 00:30:48.820
If you are not sure, convert into degrees.
00:30:48.820 --> 00:30:52.140
What about pi/4 plus pi/3.
00:30:57.800 --> 00:30:59.800
Well that is 7pi/12
00:31:14.600 --> 00:31:15.280
And lets's see.
00:31:15.280 --> 00:31:19.320
What about pi/ 4 minus pi/6
00:31:20.620 --> 00:31:22.620
That's equal to 15 degrees.
00:31:22.620 --> 00:31:24.620
That is pi/12.
00:31:26.380 --> 00:31:28.420
See now we can ask you questions like
00:31:28.420 --> 00:31:30.420
What is the sine of pi/12?
00:31:32.280 --> 00:31:34.780
Or do you know what I can do if I was really nasty?
00:31:37.940 --> 00:31:39.940
I can ask you for tan 105
00:31:41.740 --> 00:31:43.740
What would the tan105 degree?
00:31:45.040 --> 00:31:49.180
Find the sine and find the cosine and then you divide.
00:31:49.960 --> 00:31:58.340
Tan 105 would be square root of 6 plus square root of 2 over 4
00:31:58.340 --> 00:32:02.560
divide by square root of 6 minus square root of 2 over 4
00:32:03.680 --> 00:32:09.120
Which is square root of 6 plus square root of 2 over square root of 6 minus square root of 2.
00:32:09.120 --> 00:32:12.760
Not so bad! Ok?
00:32:13.540 --> 00:32:15.020
So I could ask you for the tangent.
00:32:15.020 --> 00:32:17.360
It is a lot of work to ask you for that
00:32:17.360 --> 00:32:21.600
but it could be a paper homework problem maybe a test problem.
00:32:21.600 --> 00:32:23.600
Alright! Some other stuff.
00:32:25.000 --> 00:32:27.640
Suppose I want to find what is called the double angle formula?
00:32:29.720 --> 00:32:31.720
Suppose I want to find the sine of 2A.
00:32:42.600 --> 00:32:44.240
Why do I want to find the double angle?
00:32:44.240 --> 00:33:02.360
[unintelligible]
00:33:03.840 --> 00:33:06.880
So how would I find the sine of 2A?
00:33:07.520 --> 00:33:14.300
Well, I will make my sin (A+B) formula and write it as sin (A+A).
00:33:19.680 --> 00:33:22.280
Sin(A+A) is a very complicated formula.
00:33:22.860 --> 00:33:31.660
Sin A cos A plus cos A sin A.
00:33:32.940 --> 00:33:36.280
These two are the same thing the sin A and cos A.
00:33:36.280 --> 00:33:44.040
So this is just 2 sin A cosA.
00:33:47.800 --> 00:33:50.660
And now we have the double angle.
00:34:03.160 --> 00:34:07.320
See what happens is, when you doing calculus you will have some problems.
00:34:08.180 --> 00:34:11.860
If it would be in this form 2sin A cos A.
00:34:12.800 --> 00:34:16.220
And you can write it instead in this form sin 2A.
00:34:16.220 --> 00:34:17.800
Or the other way around.
00:34:17.800 --> 00:34:21.100
And one will make the problem easier than the other one.
00:34:21.100 --> 00:34:24.880
So you just want to be able to shift from on form to the other form.
00:34:28.120 --> 00:34:31.020
What if I want to find the cosine of 2A?
00:34:35.780 --> 00:34:37.780
That's is the cos (A+A)
00:34:41.960 --> 00:34:56.100
That is: cosA cosA minus sin A sin A.
00:34:56.100 --> 00:35:04.000
And that simplifies to cos squared A minus sine squared A.
00:35:04.000 --> 00:35:07.420
Don't confuse that with cosine squared plus sine squared.
00:35:08.580 --> 00:35:10.580
We have cosine squared "minus" sine squared.
00:35:10.580 --> 00:35:14.020
Cosine squared plus sine squared is 1.
00:35:14.020 --> 00:35:19.360
Cosine squared minus 2 sine square is just the cos 2A.
00:35:24.360 --> 00:35:27.820
So far so good? ok, here is a fun fact!
00:35:36.900 --> 00:35:40.380
Lets go back to cos 2A.
00:35:49.300 --> 00:35:54.600
Notice here we have 2 times the angle and here we have a single angle.
00:35:56.620 --> 00:36:01.560
So one of the things you are doing is you reducing the angle by a factor of 2.
00:36:01.560 --> 00:36:02.840
You working out the problem.
00:36:02.840 --> 00:36:04.840
Sometimes you have an x.
00:36:06.580 --> 00:36:10.540
I also know that cosine squared plus sine squared is 1.
00:36:10.540 --> 00:36:16.860
So I can replace cos^2 A with (1-sin^2 A) or I can replace sin ^2 A with (1-cos^2 A).
00:36:17.820 --> 00:36:20.380
I can replace this with (1- sin^2 ).
00:36:25.200 --> 00:36:27.200
I will get 1-2sin^2 A.
00:36:33.900 --> 00:36:38.820
All I did is I replaced cos^2 A with (1-sin^2 A) Right here.
00:36:48.740 --> 00:36:54.120
Similarly, I can take the sin^2 A and write it as (1-cos^2 A).
00:37:05.180 --> 00:37:09.440
So, double angle formulas have a separate board.
00:37:13.020 --> 00:37:17.720
[unintelligible]
00:37:56.760 --> 00:37:57.820
So far so good?
00:38:01.200 --> 00:38:07.100
It is convenient to know all of three of those double angle formulas because
00:38:08.480 --> 00:38:10.480
different values depend on the situation
00:38:10.480 --> 00:38:17.440
But you really have to memorize one of them and you can always figure out the other two
00:38:18.380 --> 00:38:20.380
In fact, if you know the addition formulas
00:38:20.380 --> 00:38:22.380
[unintelligible]
00:38:22.380 --> 00:38:24.380
you don't need to memorize them.
00:38:24.380 --> 00:38:26.380
You can always derive them. ok?
00:38:26.380 --> 00:38:28.380
I erase them.
00:38:28.380 --> 00:38:30.380
You can just derive them.
00:38:30.380 --> 00:38:32.380
[unintelligible]
00:38:47.560 --> 00:38:49.560
Alright, lets do some other things with this.
00:38:50.480 --> 00:38:52.480
Just bunch of formulas today!
00:39:01.580 --> 00:39:03.580
What if instead of double angle,
00:39:03.580 --> 00:39:05.580
I want to find half angle?
00:39:08.640 --> 00:39:10.980
Instead of double angle I want to find half an angle.
00:39:11.380 --> 00:39:16.080
Well notice this A is 1/2 of that 2A.
00:39:16.080 --> 00:39:18.080
This A is 1/2 of that 2A.
00:39:21.960 --> 00:39:23.960
So,
00:39:43.520 --> 00:39:45.520
Lets say I start with this formula.
00:39:47.880 --> 00:39:50.220
And I want to isolate the cosine of 2A.
00:39:50.220 --> 00:39:52.220
Because I want it to be half of the angle.
00:39:53.040 --> 00:39:55.040
Well, add 1 on both sides and
00:40:04.060 --> 00:40:05.420
divide by 2.
00:40:15.580 --> 00:40:16.580
next you take the square root.
00:40:18.140 --> 00:40:30.920
cos A plus or minus the square root of
(1+cos 2A / 2) ok?
00:40:30.920 --> 00:40:32.360
Why Plus or minus?
00:40:32.360 --> 00:40:34.360
it depends in what quadrant you end up with.
00:40:34.360 --> 00:40:36.760
Ok? Don't get too worry about that!
00:40:37.540 --> 00:40:42.220
Notice this angle is half of that angle
00:40:43.660 --> 00:40:47.620
This is cos A and this is cos 2A
00:40:50.140 --> 00:40:52.920
So if I made this A, that would be 1/2 of A.
00:40:54.900 --> 00:40:56.900
The half angle formula.
00:41:13.960 --> 00:41:15.100
ok?
00:41:16.260 --> 00:41:18.380
Did all see I made that switch
00:41:18.380 --> 00:41:24.060
All I did, I said A is 1/2 of 2A.
00:41:25.020 --> 00:41:28.440
And if this is A, that is A/2.
00:41:28.440 --> 00:41:30.440
[unintelligible]
00:41:35.740 --> 00:41:37.740
We almost did too much work!
00:41:40.900 --> 00:41:45.180
Alright! What if I want to do sine of half and angle?
00:41:47.480 --> 00:41:49.480
Well I take the cos 2A formula again.
00:41:59.620 --> 00:42:01.620
And now I isolate sin of A.
00:42:04.680 --> 00:42:10.860
So 2 sin ^2 A is 1- cos2A.
00:42:12.760 --> 00:42:13.940
Divide it by 2.
00:42:27.960 --> 00:42:29.960
And take the square root of it.
00:42:40.500 --> 00:42:55.200
In other words: sin (A/2) is plus or minus the square root of (1-cos A)/ 2.
00:42:55.900 --> 00:42:57.900
So lets do an example!
00:42:59.380 --> 00:43:01.380
How are you guys doing
00:43:22.000 --> 00:43:24.000
[unintelligible]?
00:43:28.480 --> 00:43:32.140
This is A/2 and this is just single A.
00:43:32.800 --> 00:43:35.100
What ever this angle is, is just double of that.
00:43:39.520 --> 00:43:41.520
Suppose I say to you guys,
00:43:51.500 --> 00:43:55.200
find the sin (pi/8).
00:43:57.860 --> 00:44:09.760
sin (pi/8) the sine of pi/8 is half of pi/4.
00:44:10.640 --> 00:44:12.640
So I can use the half angle formula!
00:44:23.520 --> 00:44:26.640
How do I know its the positive square root?
00:44:26.640 --> 00:44:32.280
Where is pi/8 located? pi/8 is located in the first quadrant. Right?
00:44:33.200 --> 00:44:35.200
So half of it can also be located in the first quadrant.
00:44:36.120 --> 00:44:38.120
So I use the positive square root.
00:44:42.360 --> 00:44:45.000
Now what it cos pi/4?
00:44:49.360 --> 00:44:51.360
You can leave it like that.
00:44:59.440 --> 00:45:01.620
ok? You don't have to simplify.
00:45:01.620 --> 00:45:03.620
It is not so easy to simplify
00:45:15.480 --> 00:45:24.920
How about cos of 5pi/ 8?
00:45:27.420 --> 00:45:33.440
So 5pi/8 is half of pi/4.
00:45:37.860 --> 00:45:53.900
ok! 5pi/4 is 225 degrees so 5pi/8 is 112.5 degrees.
00:45:53.900 --> 00:45:57.340
That is in the second quadrant. Ok?
00:45:57.340 --> 00:46:03.200
So this cosine (5pi/4)/2 is going to be a second quadrant angle
00:46:04.360 --> 00:46:13.260
so we will use a negative square root of (1+ cos5pi/4) /2.
00:46:14.180 --> 00:46:17.560
That is how I know to make it negative. ok?
00:46:19.540 --> 00:46:21.540
[unintelligible]
00:46:26.060 --> 00:46:28.060
Now, what is the cos of 5pi/4?
00:46:30.440 --> 00:46:33.780
ok? 5pi/4 is 225 degrees.
00:46:34.380 --> 00:46:36.380
Thats a third quadrant angle.
00:46:36.860 --> 00:46:42.600
Half of it is 112.5 degrees its in the second quadrant because its more than 90.
00:46:48.900 --> 00:46:50.900
Where is cos 5pi/4?
00:46:53.520 --> 00:46:55.520
where is cos 5pi/4?
00:46:55.520 --> 00:46:57.520
look at the derivation of it
00:46:57.520 --> 00:46:59.520
[unintelligible]
00:46:59.520 --> 00:47:01.520
You can always give it a shot!
00:47:01.520 --> 00:47:08.260
What is it? cos, positive or negative?
00:47:09.160 --> 00:47:11.160
negative square root of 2 over 2.
00:47:16.500 --> 00:47:18.500
And you can leave it like that!
00:47:20.720 --> 00:47:22.720
Why is it square root of 2 over 2?
00:47:22.720 --> 00:47:26.980
You suppose know how to find cos of 5pi/4 ok?
00:47:26.980 --> 00:47:32.100
draw the angle, pick the right quadrant and figure out if its positive or negative
00:47:32.820 --> 00:47:34.820
All that stuff.
00:47:42.580 --> 00:47:45.680
Alright I will see everybody on Monday.