WEBVTT
Kind: captions
Language: en-US
00:00:00.000 --> 00:00:00.640
Alright.
00:00:12.620 --> 00:00:15.100
Lets just warm up and have you do that.
00:00:17.740 --> 00:00:22.060
Alright lets do this one together and make sure we understand it!
00:00:23.520 --> 00:00:26.740
So first, what sine inverse of 2/3 be?
00:00:26.740 --> 00:00:28.400
means we have triangles somewhere.
00:00:30.400 --> 00:00:31.380
This is in the first quadrant
00:00:31.380 --> 00:00:33.100
How do we know its in the first quadrant?
00:00:33.500 --> 00:00:35.760
2/3 is positive. Ok?
00:00:37.460 --> 00:00:42.960
So that means that some angle x, the sine of that angle is 2/3.
00:00:45.380 --> 00:00:45.880
Ok?
00:00:46.820 --> 00:00:48.660
Sine of that angle is 2/3.
00:00:48.660 --> 00:00:52.300
We can now use pythagorean theorem to find this other side
00:00:54.080 --> 00:00:57.980
2 squared plus b squared equal 3 squared.
00:00:59.020 --> 00:01:01.220
b is square root of 5. I will let you do that
00:01:03.060 --> 00:01:04.200
Now you are on your own.
00:01:06.200 --> 00:01:10.240
Ok? Now, we just need to find the tangent of x
00:01:18.420 --> 00:01:21.460
So we had a problem like this in the first exam.
00:01:23.180 --> 00:01:26.180
Except, in the first exam we gave you the actual number.
00:01:26.540 --> 00:01:29.380
ok? We did not do this as sine inverse.
00:01:29.380 --> 00:01:33.420
We said the sine of some angle was 2/3 and then
00:01:33.420 --> 00:01:34.740
we said what is the tangent of that angle.
00:01:34.740 --> 00:01:37.320
Now, we just write, sine inverse of 2/3.
00:01:37.760 --> 00:01:40.020
Same thing. ok?
00:01:46.620 --> 00:01:48.060
Sorry that was cosine
00:01:49.940 --> 00:01:50.660
Here we go!
00:01:51.060 --> 00:01:52.660
Ok lets try this again!.
00:02:12.060 --> 00:02:14.060
Lets see if you can do that one!
00:02:15.580 --> 00:02:18.120
Alright remember the rules.
00:02:19.280 --> 00:02:23.680
If you have an inverse trig function and its about a positive angle
00:02:23.700 --> 00:02:28.020
you always, a positive number you always go to the first quadrant
00:02:28.480 --> 00:02:32.700
But if you do the inverse trig function of a negative number,
00:02:32.700 --> 00:02:35.720
you are in either the second or fourth quadrant.
00:02:35.720 --> 00:02:36.520
And you know where it is.
00:02:39.400 --> 00:02:41.320
I am just going to remind you.
00:02:57.320 --> 00:03:01.220
Ok, if you do the inverse sine, the inverse cosine
00:03:01.220 --> 00:03:06.720
and the inverse tangent of a positive number always will be in quadrant 1
00:03:11.740 --> 00:03:16.460
If you are going do the inverse sin or tangent of a negative value,
00:03:16.540 --> 00:03:18.780
you will go in the fourth quadrant
00:03:19.960 --> 00:03:23.000
And the cosine will be in the second quadrant.
00:03:25.220 --> 00:03:30.880
Ok? So the cosine, sine inverse of -4/11.
00:03:31.220 --> 00:03:33.620
So we will be in the fourth quadrant.
00:03:35.360 --> 00:03:39.660
If you draw an angle the sine is -4/11
00:03:39.660 --> 00:03:42.180
We don't really care about the minus sine.
00:03:42.180 --> 00:03:44.540
If you put the minus is not that important
00:03:45.140 --> 00:03:47.780
you need to know what cosine is.
00:03:49.500 --> 00:03:50.140
This is a.
00:03:50.140 --> 00:03:55.520
So, a squared plus (-4) squared equal 11 squared.
00:03:55.520 --> 00:03:58.940
When you square the -4 you get positive 16
00:04:00.220 --> 00:04:02.420
a squared is 105.
00:04:04.220 --> 00:04:06.140
a square root of 105.
00:04:08.240 --> 00:04:09.540
So the cosine of x
00:04:11.120 --> 00:04:15.560
will then be: square root of 105 over 11
00:04:18.340 --> 00:04:19.300
So far so good?
00:04:19.880 --> 00:04:21.880
everybody understands this?
00:04:21.880 --> 00:04:25.380
I would expect to see a question like this again in less than a month.
00:04:25.880 --> 00:04:27.080
Or exactly a month
00:04:28.120 --> 00:04:29.240
Exactly a month.
00:04:31.520 --> 00:04:34.420
Part 1? No Part II
00:04:34.420 --> 00:04:37.920
everything we do now is in part II problems.
00:04:47.760 --> 00:04:53.100
What if I ask the sine of the tan inverse x?
00:04:56.320 --> 00:04:57.440
So what I will do?
00:04:58.340 --> 00:04:58.840
Well,
00:05:08.760 --> 00:05:10.440
I am in the first quadrant
00:05:10.440 --> 00:05:11.880
I guess I need a different letter.
00:05:12.560 --> 00:05:14.080
So lets call this theta
00:05:14.880 --> 00:05:19.420
And I know that the tangent of theta is x.
00:05:19.700 --> 00:05:21.660
So we can make it x/1.
00:05:27.100 --> 00:05:29.600
ok? I can make it 2x/2,
00:05:29.600 --> 00:05:33.380
I can make it pix/pi or just 1x over 1
00:05:33.880 --> 00:05:36.280
So if I want to find the hypothenuse.
00:05:39.940 --> 00:05:43.480
c is the square root of 1 plus x squared.
00:05:48.180 --> 00:05:54.320
And then the sine will be equal to x/ square root of 1 plus x squared.
00:05:57.340 --> 00:06:00.940
ok? So do you understand where I get the x and the 1 from?
00:06:01.280 --> 00:06:05.820
Ok? tangent of that angle, tan inverse of x has some angle
00:06:06.680 --> 00:06:12.780
And tangent of theta is x over 1.
00:06:14.640 --> 00:06:18.300
Because x over 1 is just x.
00:06:18.760 --> 00:06:20.520
So that means that tangent
00:06:20.520 --> 00:06:22.500
opposite over adjacent is x/1.
00:06:22.760 --> 00:06:27.320
Then we use pythagorean theorem to find the missing hypothenuse ok?
00:06:27.320 --> 00:06:29.240
And now you you get the sine of that.
00:06:32.940 --> 00:06:35.140
You got it? One more time?
00:06:39.580 --> 00:06:40.620
tan inverse of x
00:06:40.840 --> 00:06:44.120
If we have some angle theta, tangent of theta is x.
00:06:44.980 --> 00:06:46.500
that would be x/1.
00:06:46.580 --> 00:06:49.700
I can make it 2x/2 but I still get x.
00:06:49.700 --> 00:06:53.360
I can use pythagorean theorem to find the missing side, the hypotheses.
00:06:53.500 --> 00:06:57.580
Sine of that angle is x over the square root of 1 plus x squared.
00:07:10.040 --> 00:07:12.200
This shows up in calculus.
00:07:19.400 --> 00:07:21.800
What is the sine cosine inverse of x?
00:07:21.800 --> 00:07:24.460
So whats the sin cosine inverse x?
00:07:27.180 --> 00:07:28.980
We can just assume that is in the first quadrant
00:07:28.980 --> 00:07:31.020
there is no reason to think that x is not in the first quadrant
00:07:35.440 --> 00:07:40.860
The cosine inverse of x. cosine of theta equal to x.
00:07:42.060 --> 00:07:43.800
That is an x and that is 1.
00:07:46.060 --> 00:07:47.920
Why is the hypothenuse 1?
00:07:48.600 --> 00:07:52.440
Because opposite, cosine is adjacent over hypothenuse.
00:07:53.020 --> 00:07:55.460
This is x and that will be 1.
00:07:56.160 --> 00:07:57.980
An this is x over 1
00:07:58.640 --> 00:08:04.020
Ok? I can use 2x/2 if I want to, the would be unnecessary complicated.
00:08:07.620 --> 00:08:09.920
Now, If I want to find the missing side?
00:08:11.000 --> 00:08:17.660
x squared plus b squared equal 1 squared. b squared is 1 minus x squared.
00:08:18.680 --> 00:08:21.640
Would be the square root of 1 minus x squared.
00:08:28.060 --> 00:08:32.020
This is the square root of 1 minus x squared.
00:08:32.060 --> 00:08:34.060
plus or minus the square root.
00:08:34.620 --> 00:08:37.200
So now, if I want to find the sine of theta,
00:08:38.640 --> 00:08:43.720
its going to be square root of 1 minus x squared over 1.
00:08:43.860 --> 00:08:46.580
Which is just the square root of 1 minus x squared.
00:08:46.580 --> 00:08:47.340
You don't need 1.
00:08:51.020 --> 00:08:51.980
This shows up.
00:08:51.980 --> 00:08:53.240
This will show up in calculus
00:08:53.300 --> 00:08:56.960
They show up on final. You do not know yet [unintelligible]
00:08:56.960 --> 00:09:00.220
This is something you want to make sure you can do.
00:09:02.520 --> 00:09:03.800
You understand it?
00:09:03.800 --> 00:09:04.920
Should I do one more?
00:09:06.120 --> 00:09:07.400
Who wants one more?
00:09:09.200 --> 00:09:10.320
Oh thats not bad!
00:09:10.320 --> 00:09:11.500
This is very a few of you!
00:09:11.500 --> 00:09:13.180
Because there aren't anymore.
00:09:15.820 --> 00:09:19.340
How many of you think that the Monday before thanksgiving is not good for class?
00:09:21.080 --> 00:09:23.720
I know many of you have already tickets.
00:09:25.640 --> 00:09:27.960
What is the sine, what is the sine sine inverse of 1/2?
00:09:30.100 --> 00:09:30.980
Why is it 1/2?
00:09:38.400 --> 00:09:39.040
Perfect!
00:09:39.040 --> 00:09:44.160
Ok, this says give me and angle whose sine is 1/2 and take the sine of it.
00:09:44.700 --> 00:09:48.700
But if you work it out, you say alright sin x is 1/2.
00:09:50.480 --> 00:09:57.740
You rewrite this as sin x, this will be 1/2 and then sin x is 1/2.
00:09:57.740 --> 00:09:59.360
They sort of cancel them out.
00:09:59.360 --> 00:10:00.720
Because they are inverse functions.
00:10:01.820 --> 00:10:04.640
ok? What if I said.
00:10:22.700 --> 00:10:25.480
What if we do sine inverse of sine of pi/6?
00:10:26.240 --> 00:10:27.820
[unintelligible]
00:10:29.200 --> 00:10:31.180
What is the sine of pi/6?
00:10:33.020 --> 00:10:34.560
its 1/2 ok?
00:10:35.160 --> 00:10:36.920
Sine 30, pi/6 is 30.
00:10:38.080 --> 00:10:41.060
sine of pi/6 sine of 30 is 1/2.
00:10:41.740 --> 00:10:48.320
Sine inverse of 1/2 is pi/6.
00:10:49.220 --> 00:10:49.780
Because
00:10:53.080 --> 00:10:56.760
remember what this would be. sine of pi/6 is 1/2
00:10:57.060 --> 00:11:01.960
Sine inverse of 1/2 is what angle the sine is pi/6.
00:11:03.080 --> 00:11:05.420
ok? Got is? Good.
00:11:05.420 --> 00:11:11.480
What is the sine inverse of sine 5pi/ 6?
00:11:18.440 --> 00:11:19.880
You want to say 5pi/6.
00:11:20.200 --> 00:11:21.880
That would be incorrect.
00:11:27.060 --> 00:11:30.100
ok? Think about that for a couple more seconds
00:11:33.840 --> 00:11:35.640
What is the sine of 5pi/6?
00:11:44.840 --> 00:11:45.880
Where is 5pi/6?
00:11:47.340 --> 00:11:49.180
Its in the second quadrant.
00:11:49.180 --> 00:11:50.180
whats the reference angle
00:11:58.880 --> 00:12:00.740
close. 150 oK?
00:12:02.380 --> 00:12:04.500
So 150 gives you a reference angle of how much?
00:12:05.240 --> 00:12:08.000
30. So whats the sine of 30? 1/2.
00:12:08.300 --> 00:12:12.220
So pi/6 is here
00:12:16.740 --> 00:12:18.980
the reference angle is then pi/6.
00:12:19.320 --> 00:12:21.100
The sine of pi/6 is 1/2.
00:12:21.100 --> 00:12:23.500
So this is becomes the sine inverse of 1/2.
00:12:26.180 --> 00:12:29.920
And what it the sine inverse of 1/2? pi/6.
00:12:31.340 --> 00:12:34.980
Ok? So you do not end up back where you started.
00:12:34.980 --> 00:12:38.140
The reason is remember when we you do inverse sine
00:12:39.380 --> 00:12:44.260
If you do the inverse sine of positive quantity we always use the first quadrant angle.
00:12:45.140 --> 00:12:48.680
ok? So even though we get a different angle.
00:12:48.680 --> 00:12:51.260
We took the sine and got a positive number
00:12:51.260 --> 00:12:55.260
The inverse sine of a positive number is in the first quadrant.
00:12:55.880 --> 00:12:58.840
You have to decide what angle you want to use.
00:12:59.020 --> 00:13:00.700
Even inverse sine of 1/2.
00:13:00.700 --> 00:13:04.460
There are infinite number of angle that are sine equal to 1/2
00:13:04.460 --> 00:13:07.660
We always use the first quadrant. ok?
00:13:07.660 --> 00:13:09.880
There is a lot of types of trouble with this!
00:13:10.000 --> 00:13:11.280
I am doing it again.
00:13:12.880 --> 00:13:15.480
The sine of 5pi/6 is 1/2.
00:13:15.880 --> 00:13:17.400
the inverse sine of 1/2
00:13:17.400 --> 00:13:19.060
you alway use the first quadrant angle
00:13:19.060 --> 00:13:20.320
You don't use any other quadrant.
00:13:20.940 --> 00:13:22.860
ok? If this was negative 1/2,
00:13:22.860 --> 00:13:26.360
you were only use the fourth quadrant, negative pi/6.
00:13:28.900 --> 00:13:32.920
ok! Lets do another one of these.
00:13:45.280 --> 00:13:47.920
Ok! The sine of pi/4 is radical 2 over 2 or
00:13:49.540 --> 00:13:55.700
So you get sine inverse of radical 2 over 2. Ok?
00:13:56.080 --> 00:13:59.180
Sine of what angle is radical 2 over 2
00:14:00.420 --> 00:14:06.100
For those who memorized it, 45 degrees or also known as pi/4.
00:14:06.100 --> 00:14:09.160
So notice, you right back where you started!
00:14:10.720 --> 00:14:11.220
ok?
00:14:12.180 --> 00:14:23.240
So, what if I asked for sine inverse, sine of 9pi/4?
00:14:26.820 --> 00:14:31.460
Ok? You want to say it is 9pi/4 but you cannot!
00:14:31.460 --> 00:14:33.920
Because you have to get first quadrant answer.
00:14:35.100 --> 00:14:37.340
So, what is sine 9pi/4?
00:14:38.180 --> 00:14:44.100
Well, thats 9pi/4 is the same thing as pi/4.
00:14:46.540 --> 00:14:49.500
Since you go all the way around to 8pi/4 (2pi)
00:14:49.500 --> 00:14:50.980
and then you go another pi/4.
00:14:50.980 --> 00:14:53.160
Do with degrees if it bothers you!
00:14:53.160 --> 00:14:59.360
Ok? So the sine of 9pi/4 has the same value as sin of pi/4.
00:14:59.360 --> 00:15:03.180
This is still sine inverse radical 2 over 2.
00:15:04.520 --> 00:15:06.040
which is equal to pi/4.
00:15:08.020 --> 00:15:10.220
ok? Yes!
00:15:15.680 --> 00:15:21.200
If start off and you are doing the inverse trig function
00:15:21.200 --> 00:15:26.080
of a first quadrant angle you will end up right back where you started.
00:15:26.080 --> 00:15:30.300
If you are not in the first quadrant angle and positive you have to
00:15:30.420 --> 00:15:31.460
[unintelligible]
00:15:31.460 --> 00:15:32.780
in the first quadrant. ok?
00:15:35.860 --> 00:15:39.460
ok? so we practice some more of these until we grasp it.
00:15:52.960 --> 00:15:55.620
cosine inverse of cosine pi/3.
00:15:56.480 --> 00:15:59.400
Cosine of pi/3 is 1/2.
00:15:59.980 --> 00:16:02.140
So this is cosine inverse of 1/2.
00:16:08.080 --> 00:16:12.260
And cosine inverse of 1/2 is in fact pi/3
00:16:12.260 --> 00:16:15.980
Because we just wanted to know the sine of what angle is 1/2?
00:16:27.880 --> 00:16:31.800
Last, the way you do these you work them out.
00:16:33.040 --> 00:16:35.080
What is the cosine of 5pi/3?
00:16:35.080 --> 00:16:36.480
You all got this on the first exam
00:16:37.600 --> 00:16:40.800
If you are not sure whats 5pi/3 is, you turn it into degrees.
00:16:40.800 --> 00:16:41.940
That is 300 degrees.
00:16:47.440 --> 00:16:51.040
ok? 300 degrees will get you 60 as the reference angle.
00:16:51.780 --> 00:16:54.780
5pi/3 is down here.
00:16:54.780 --> 00:16:59.060
Cosine is positive down here. pi/3 left over
00:17:00.560 --> 00:17:02.240
So you will get 1/2.
00:17:04.100 --> 00:17:10.400
Cosine inverse of 1/2 is back to pi/3.
00:17:11.040 --> 00:17:13.600
ok? This looks messy but it isn't.
00:17:13.600 --> 00:17:15.560
What you have to do is to find what the middle,
00:17:15.980 --> 00:17:18.300
what the angle in the middle first.
00:17:22.100 --> 00:17:25.380
Then, you take the angle of that value in the middle
00:17:25.380 --> 00:17:28.600
and then you do the inverse trig function to make sure you get the right quadrant.
00:17:33.500 --> 00:17:34.460
What if I have,
00:17:40.160 --> 00:17:41.380
well lets see!
00:17:42.580 --> 00:17:47.700
cosine inverse cosine 2pi/3.
00:17:50.100 --> 00:17:52.900
Where is 2pi/3 located? Second quadrant.
00:18:01.320 --> 00:18:03.080
Also known as 120 degrees.
00:18:04.960 --> 00:18:09.620
So this is 60 degrees (pi/3) So that is negative 1/2.
00:18:15.640 --> 00:18:19.480
Cosine inverse of negative 1/2 is 2pi/3.
00:18:19.480 --> 00:18:21.040
You are in the second quadrant.
00:18:21.040 --> 00:18:25.220
Back to were you started a few minutes ago. ok?
00:18:26.940 --> 00:18:29.380
You have to use the second quadrant because its a negative value.
00:18:30.940 --> 00:18:32.900
ok? It is not a positive value its a negative value.
00:18:36.840 --> 00:18:38.280
That is negative 1/2.
00:18:39.940 --> 00:18:41.940
So remember inverse trig functions,
00:18:42.120 --> 00:18:46.840
inverse cosine of a negative values you use the second quadrant angle.
00:18:47.540 --> 00:18:49.140
You want to try it again?
00:19:05.800 --> 00:19:16.140
ok! if x is a positive number, for all the inverse trig function you always use first quadrant angle
00:19:18.300 --> 00:19:24.880
ok? if its a negative number, inverse sine and inverse tan
00:19:26.560 --> 00:19:28.240
use the fourth quadrant.
00:19:28.860 --> 00:19:31.340
Inverse cosine use second quadrant.
00:19:34.060 --> 00:19:35.740
Everybody see that ok?
00:19:39.380 --> 00:19:40.820
You guys can see that?
00:19:42.180 --> 00:19:43.540
Some of you got this?
00:19:44.640 --> 00:19:45.140
sure!
00:19:49.300 --> 00:19:52.820
So notice, I am doing cosine inverse of negative 1/2.
00:19:53.380 --> 00:19:57.920
Cosine inverse of negative 1/2 in an angle in the second quadrant.
00:20:00.920 --> 00:20:07.880
So what if I have cosine inverse of cosine 4pi/3?
00:20:11.640 --> 00:20:15.000
Well, lets see! 4pi/3
00:20:20.160 --> 00:20:26.900
down here is 240 degrees.
00:20:27.000 --> 00:20:31.700
So this is 60 degrees, pi/3 again
00:20:34.240 --> 00:20:34.740
ok?
00:20:37.060 --> 00:20:38.420
Still cant write it!
00:20:42.780 --> 00:20:46.860
Ok? So the cosine of pi/3 is negative 1/2.
00:20:47.920 --> 00:20:51.240
So this is cosine inverse negative 1/2.
00:20:51.240 --> 00:20:56.300
Which means you end up back in the second quadrant
00:20:56.300 --> 00:20:59.800
because in the second quadrant we have negative values.
00:21:00.340 --> 00:21:05.420
ok? Were are we lost?
00:21:05.860 --> 00:21:10.500
Why do we get different answers something and same answer other times.
00:21:10.500 --> 00:21:13.980
if we start off in the first quadrant.
00:21:14.200 --> 00:21:16.440
You finish in the first quadrant.
00:21:17.140 --> 00:21:19.980
For all these, for all the ones that look like this.
00:21:20.000 --> 00:21:22.880
Inverse trig the first quadrant angle,
00:21:22.880 --> 00:21:24.900
you end back in the first quadrant angle.
00:21:26.460 --> 00:21:30.860
ok? If you do inverse trig of trig that is not in the first quadrant,
00:21:30.860 --> 00:21:34.040
figure out where you going to be [unintelligible]
00:21:34.040 --> 00:21:37.320
Just remember inverse sine and inverse cosine and inverse tangent.
00:21:37.320 --> 00:21:43.740
All we get answers if you are in the first and fourth quadrant or first quadrant and second quadrant.
00:21:44.840 --> 00:21:47.780
ok? You just need to figure out where you are
00:21:47.780 --> 00:21:49.460
So lets practice a couple of these.
00:21:50.460 --> 00:21:58.800
So you guys get the hang of it! [unintelligible]
00:23:14.180 --> 00:23:16.340
Here we go, they are five of these
00:23:18.080 --> 00:23:20.560
sin pi/3 is radical 3/2.
00:23:21.540 --> 00:23:25.900
So, that becomes sine inverse of radical 3/2.
00:23:25.900 --> 00:23:34.840
Radical 3/2 is positive, so the answer has to be in the first quadrant. pi/3.
00:23:37.480 --> 00:23:43.040
ok? Yes! got it? You are good?
00:23:46.240 --> 00:23:49.880
Well, this is positive
00:23:49.880 --> 00:23:54.620
Look at the chart, if its positive you always use the first quadrant angle.
00:23:58.840 --> 00:24:03.220
ok! Sin of 4pi/3 is 240 degrees.
00:24:06.400 --> 00:24:08.000
Somewhere around here.
00:24:10.060 --> 00:24:12.780
That is going to be the same thing as pi/3.
00:24:13.640 --> 00:24:16.340
Same values but negative because ok?
00:24:22.760 --> 00:24:25.480
All Seawolves Taste Chocolate.
00:24:31.400 --> 00:24:34.440
ok? So thats going to be negative radical 3/2.
00:24:39.360 --> 00:24:44.400
But, if I had a negative quantity in there I am going to use a fourth quadrant.
00:24:45.420 --> 00:24:50.220
So where in the fourth quadrant do I get sine of negative radical 3 over 2?
00:24:51.080 --> 00:24:53.860
fourth quadrant is negative pi/3.
00:24:58.800 --> 00:25:01.640
ok? I go down pi/3.
00:25:04.380 --> 00:25:07.940
The sine inverse of the sine 7pi/3
00:25:09.080 --> 00:25:11.580
Except 7pi/3 had to be right there
00:25:13.340 --> 00:25:19.260
7pi/3 is in the first quadrant
00:25:22.560 --> 00:25:23.060
ok?
00:25:23.440 --> 00:25:27.420
7pi/3 is 420 degrees.
00:25:27.560 --> 00:25:29.480
That is the same as 60 degrees
00:25:29.640 --> 00:25:36.100
ok? So, this would be sine inverse radical 3/2.
00:25:38.240 --> 00:25:42.640
Sine inverse of radical 3/2 said sine of what angle is radical 3/2?
00:25:43.280 --> 00:25:45.140
60 degrees or pi/3.
00:25:47.960 --> 00:25:48.460
ok?
00:25:49.260 --> 00:25:51.260
00:25:55.700 --> 00:25:57.540
Take a time out for a second!
00:25:58.840 --> 00:26:00.860
lets work in degrees
00:26:00.960 --> 00:26:04.080
Lets see if it is easier for you to visualize it!
00:26:04.500 --> 00:26:09.460
sine inverse sine of 60 degrees.
00:26:10.580 --> 00:26:13.860
You say the sine of 60 degrees is square root of 3/2
00:26:18.560 --> 00:26:20.160
That is equal 60 degrees
00:26:20.160 --> 00:26:21.220
Now it is simple! right?
00:26:21.220 --> 00:26:22.320
That is what we had up there.
00:26:23.020 --> 00:26:29.640
Sine inverse sine 420 degrees ok?
00:26:32.420 --> 00:26:37.200
Sine of 420 degrees is the same think as sine of 60 degrees
00:26:38.240 --> 00:26:40.400
Sine inverse of radical 3/2
00:26:43.360 --> 00:26:47.360
Remember the sine inverse of radical 3/2 is 60 degrees.
00:26:48.120 --> 00:26:52.600
ok? So the fact that this was 420 at the beginning, it did not matter.
00:26:52.600 --> 00:26:56.440
We get back to 60 degrees because you get this formula where I asked you.
00:26:56.440 --> 00:26:57.880
What is the inverse sine?
00:26:58.100 --> 00:27:01.300
The inverse sine has to be in the first quadrant.
00:27:02.140 --> 00:27:06.120
Ok? Cannot go around. Got it?
00:27:06.120 --> 00:27:08.640
So I am doing [unintelligible] instead of degrees
00:27:09.060 --> 00:27:12.200
So the cosine inverse of cosine 5pi/6.
00:27:17.920 --> 00:27:23.020
ok? That is saying cosine inverse of cosine of 150 degrees.
00:27:23.880 --> 00:27:25.800
Where is sine of 150 degrees?
00:27:27.440 --> 00:27:29.680
Over here in the second quadrant.
00:27:33.200 --> 00:27:33.700
ok?
00:27:34.920 --> 00:27:37.900
So the reference angle is 30 degrees
00:27:41.040 --> 00:27:45.480
Cosine of 150 is negative radical 3 over 2. ok?
00:27:53.040 --> 00:27:57.220
If you do cosine inverse of negative value I am going to use a second quadrant angle
00:27:57.880 --> 00:28:02.140
I will came right back at 150 degrees or 5pi/6.
00:28:12.920 --> 00:28:18.160
Suppose instead I had done
00:28:28.120 --> 00:28:32.740
cosine of 7pi/6. 7pi/6 is 210 degrees
00:28:38.660 --> 00:28:40.260
which is in the third quadrant.
00:28:41.160 --> 00:28:43.560
That is the same value as 30 degrees.
00:28:44.160 --> 00:28:49.400
This will be the cosine inverse of negative radical 3/2.
00:28:49.400 --> 00:28:53.500
The cosine inverse of negative radical 3/2. I use the second quadrant angle
00:28:54.240 --> 00:28:58.100
150 degrees or 5pi/6.
00:28:59.100 --> 00:29:02.680
ok?Are you starting to go the hang of it?
00:29:03.180 --> 00:29:07.880
[NO] No? yes? Maybe?
00:29:08.820 --> 00:29:12.360
Need to work! What is the cosine of pi/3?
00:29:12.820 --> 00:29:13.320
1/2
00:29:13.780 --> 00:29:16.500
you know that the the cosine of pi/3 is 1/2
00:29:19.620 --> 00:29:21.300
What is the cosine of 1/2?
00:29:21.560 --> 00:29:24.260
cosine inverse of 1/2 is pi/3.
00:29:25.540 --> 00:29:27.640
Alright, [unintelligible]
00:29:27.640 --> 00:29:29.460
do some more problems so you keep quite.
00:30:24.060 --> 00:30:26.460
Ok there is some more entertaining.
00:30:29.120 --> 00:30:33.020
Alright cosine 11pi/6.
00:30:33.020 --> 00:30:36.200
Now you are going to do this in radiants before was in degrees.
00:30:37.220 --> 00:30:40.660
Cosine of 11pi/6. 11pi/6 is in the fourth quadrant.
00:30:42.700 --> 00:30:46.860
here is a clue, maybe it will help you straighten it out a little bit.
00:30:47.440 --> 00:30:51.660
11pi/6 is in the fourth quadrant. ok?
00:30:52.120 --> 00:30:59.160
so if we have our All Students Take Candy. ok?
00:30:59.720 --> 00:31:02.460
So, cosine is going to be positive down here.
00:31:02.460 --> 00:31:04.680
In fourth quadrant. right?
00:31:04.680 --> 00:31:07.500
isn't the cosine positive in the fourth quadrant?
00:31:07.500 --> 00:31:12.520
So when we get our answer, we will be doing inverse cosine of positive number.
00:31:12.520 --> 00:31:15.320
You have to get an answer that is in the first quadrant.
00:31:15.320 --> 00:31:17.880
Its going to be a positive number now. Ok?
00:31:18.600 --> 00:31:22.080
So the cosine of 11pi/6 is radical 3/2.
00:31:22.300 --> 00:31:24.480
If you can do that I will be very happy.
00:31:31.900 --> 00:31:36.300
ok? Because the cosine of 11pi/6 is radical 3/2,
00:31:36.820 --> 00:31:41.100
the inverse cosine of radial 3/2 is then pi/6.
00:31:41.400 --> 00:31:44.540
Back to the first quadrant. ok?
00:31:53.940 --> 00:31:56.340
Cosine of what angle is radical 3/2?
00:31:56.620 --> 00:31:57.960
cosine of...
00:31:59.160 --> 00:32:02.840
pi/6. [unintelligible] ok?
00:32:03.680 --> 00:32:06.220
tan inverse of tan of pi/4.
00:32:06.220 --> 00:32:08.920
Where is pi/4? It is in the first quadrant.
00:32:10.060 --> 00:32:13.740
So if you see that this is in the first quadrant, this is the same.
00:32:14.200 --> 00:32:25.680
Because tangent pi/4 (1) tangent inverse 1 is pi/4.
00:32:27.140 --> 00:32:30.260
ok? So first quadrant angle stuff is very easy.
00:32:30.260 --> 00:32:31.620
They will just always cancels.
00:32:31.620 --> 00:32:34.240
If you are not in the first quadrant it gets messy.
00:32:35.300 --> 00:32:37.900
Tan inverse of the tan of 3pi/4.
00:32:37.920 --> 00:32:39.600
What is the tangent 3pi/4?
00:32:39.600 --> 00:32:40.920
We are in the second quadrant
00:32:45.500 --> 00:32:47.920
Is tangent positive or negative in the second quadrant?
00:32:47.920 --> 00:32:51.800
Negative! And if we do inverse tan of negative what are you going to end up?
00:32:52.000 --> 00:32:54.420
in the fourth quadrant. ok?
00:32:54.940 --> 00:33:02.100
So this is going to be tangent inverse -1 because tangent of 3pi/4 is -1.
00:33:03.220 --> 00:33:07.260
tan inverse of -1 is negative pi/4.
00:33:10.600 --> 00:33:17.980
ok? Sine of 7pi/6 is in the third quadrant.
00:33:17.980 --> 00:33:21.220
So if you are in the third quadrant you will get a negative answer.
00:33:21.220 --> 00:33:24.940
And the sine inverse of a negative answer is in the fourth quadrant.
00:33:25.660 --> 00:33:29.100
So what you really want to do, you really just want to say:
00:33:29.420 --> 00:33:34.060
when is the fourth quadrant equivalent to 7pi/6?
00:33:34.060 --> 00:33:36.760
That is really what is going to happen. ok?
00:33:37.520 --> 00:33:43.700
this will be the sine inverse of negative1/2.
00:33:44.480 --> 00:33:47.120
And that will take you to negative pi/6.
00:33:47.680 --> 00:33:55.840
So really what you want to do [student] because same question,
00:33:56.160 --> 00:33:58.760
you want to go down from the x-axis.
00:33:58.900 --> 00:34:03.660
So you don't use 11pi/6 going this way, but you use negative pi/6.
00:34:03.660 --> 00:34:09.160
[unintelligible]
00:34:10.220 --> 00:34:15.000
instead of going around over here, we go down there. ok?
00:34:15.000 --> 00:34:17.520
You might end up in the same place but you are using a negative angle.
00:34:18.440 --> 00:34:24.260
its a technical thing. It has to to with the fact that when you suing the graph,
00:34:26.040 --> 00:34:29.520
you are using this angle. ok?
00:34:29.800 --> 00:34:34.180
pi/6 is that angle ok?
00:34:35.740 --> 00:34:43.520
We are really only using this piece of inverse sine graph. Ok?
00:34:44.000 --> 00:34:45.800
Lets do a couple more.
00:34:46.020 --> 00:34:47.480
Lets make sure we get this.
00:34:47.480 --> 00:34:48.520
I am not going to do it again.
00:34:56.320 --> 00:35:02.140
All we really are asking ok? [unintelligible]390 degrees where?
00:35:03.460 --> 00:35:06.600
[First quadrant] is the same as what angle in the first quadrant?
00:35:06.800 --> 00:35:09.840
Really all we are asking is that ( 30 degrees).
00:35:10.460 --> 00:35:16.460
ok? Sine inverse of sine 210 degrees.
00:35:16.900 --> 00:35:18.340
210 degrees is where?
00:35:18.760 --> 00:35:20.900
which quadrant? third!
00:35:20.900 --> 00:35:23.520
Is sine positive or negative in the third quadrant?
00:35:23.760 --> 00:35:24.640
Is negative.
00:35:24.940 --> 00:35:29.540
So let it be fourth quadrant equivalent of 210.
00:35:29.880 --> 00:35:33.320
Negative 30 degrees ok?
00:35:34.040 --> 00:35:42.780
Lets see cosine inverse cosine of 750 degrees.
00:35:43.240 --> 00:35:48.300
Im sorry 7 hundred and [unintelligible] 7
00:35:48.360 --> 00:35:52.820
once, twice, and another 30 degrees
00:35:53.380 --> 00:35:56.560
So the equivalent is just 30 degrees.
00:35:56.560 --> 00:35:58.760
This is really what you are trying to do. ok?
00:35:58.760 --> 00:36:00.760
Of course you can do this in radians.
00:36:02.140 --> 00:36:18.260
So If I said the tangent inverse of tan of 15pi/4.
00:36:19.440 --> 00:36:22.200
You saying yourself where is 15pi/4?
00:36:22.340 --> 00:36:23.300
Well lets see!
00:36:23.580 --> 00:36:25.140
Thats 8pi/4.
00:36:25.920 --> 00:36:26.960
Another seven.
00:36:26.960 --> 00:36:28.680
I am down here in the fourth quadrant.
00:36:29.180 --> 00:36:32.700
So that is the same as negative pi/4.
00:36:32.920 --> 00:36:34.680
Really what you are doing.
00:36:34.720 --> 00:36:38.060
[unintelligible]
00:36:38.180 --> 00:36:43.400
ok? [unintelligible] what you really want to do is you just want to say,
00:36:43.760 --> 00:36:48.240
what is the first, or second, or fourth quadrant equivalence of these angles
00:36:49.040 --> 00:36:49.540
ok?