Stony Brook MAT 123 Fall 2015
Lecture 22: Inverse trig functions
November 9, 2015

Start   Alright.
Lets just warm up and have you do that.
Alright lets do this one together and make sure we understand it!
So first, what sine inverse of 2/3 be?
means we have triangles somewhere.
0:30This is in the first quadrant How do we know its in the first quadrant?
2/3 is positive. Ok?
So that means that some angle x, the sine of that angle is 2/3.
Sine of that angle is 2/3.
We can now use pythagorean theorem to find this other side 2 squared plus b squared equal 3 squared.
b is square root of 5. I will let you do that
1:03Now you are on your own.
Ok? Now, we just need to find the tangent of x So we had a problem like this in the first exam.
Except, in the first exam we gave you the actual number.
ok? We did not do this as sine inverse.
We said the sine of some angle was 2/3 and then
1:33we said what is the tangent of that angle.
Now, we just write, sine inverse of 2/3.
Same thing. ok?
Sorry that was cosine Here we go!
Ok lets try this again!.
2:12Lets see if you can do that one!
Alright remember the rules.
If you have an inverse trig function and its about a positive angle you always, a positive number you always go to the first quadrant But if you do the inverse trig function of a negative number,
2:32you are in either the second or fourth quadrant.
And you know where it is.
I am just going to remind you.
Ok, if you do the inverse sine, the inverse cosine
3:01and the inverse tangent of a positive number always will be in quadrant 1 If you are going do the inverse sin or tangent of a negative value, you will go in the fourth quadrant And the cosine will be in the second quadrant.
Ok? So the cosine, sine inverse of -4/11.
3:31So we will be in the fourth quadrant.
If you draw an angle the sine is -4/11 We don't really care about the minus sine.
If you put the minus is not that important you need to know what cosine is.
This is a.
So, a squared plus (-4) squared equal 11 squared.
When you square the -4 you get positive 16
4:00a squared is 105.
a square root of 105.
So the cosine of x will then be: square root of 105 over 11 So far so good?
everybody understands this?
I would expect to see a question like this again in less than a month.
Or exactly a month Exactly a month.
4:31Part 1? No Part II everything we do now is in part II problems.
What if I ask the sine of the tan inverse x?
So what I will do?
5:08I am in the first quadrant I guess I need a different letter.
So lets call this theta And I know that the tangent of theta is x.
So we can make it x/1.
ok? I can make it 2x/2, I can make it pix/pi or just 1x over 1
5:33So if I want to find the hypothenuse.
c is the square root of 1 plus x squared.
And then the sine will be equal to x/ square root of 1 plus x squared.
ok? So do you understand where I get the x and the 1 from?
6:01Ok? tangent of that angle, tan inverse of x has some angle And tangent of theta is x over 1.
Because x over 1 is just x.
So that means that tangent opposite over adjacent is x/1.
Then we use pythagorean theorem to find the missing hypothenuse ok?
And now you you get the sine of that.
6:32You got it? One more time?
tan inverse of x If we have some angle theta, tangent of theta is x.
that would be x/1.
I can make it 2x/2 but I still get x.
I can use pythagorean theorem to find the missing side, the hypotheses.
Sine of that angle is x over the square root of 1 plus x squared.
7:10This shows up in calculus.
What is the sine cosine inverse of x?
So whats the sin cosine inverse x?
We can just assume that is in the first quadrant there is no reason to think that x is not in the first quadrant
7:35The cosine inverse of x. cosine of theta equal to x.
That is an x and that is 1.
Why is the hypothenuse 1?
Because opposite, cosine is adjacent over hypothenuse.
This is x and that will be 1.
An this is x over 1 Ok? I can use 2x/2 if I want to, the would be unnecessary complicated.
8:07Now, If I want to find the missing side?
x squared plus b squared equal 1 squared. b squared is 1 minus x squared.
Would be the square root of 1 minus x squared.
This is the square root of 1 minus x squared.
8:32plus or minus the square root.
So now, if I want to find the sine of theta, its going to be square root of 1 minus x squared over 1.
Which is just the square root of 1 minus x squared.
You don't need 1.
This shows up.
This will show up in calculus They show up on final. You do not know yet [unintelligible] This is something you want to make sure you can do.
9:02You understand it?
Should I do one more?
Who wants one more?
Oh thats not bad!
This is very a few of you!
Because there aren't anymore.
How many of you think that the Monday before thanksgiving is not good for class?
I know many of you have already tickets.
What is the sine, what is the sine sine inverse of 1/2?
9:30Why is it 1/2?
Ok, this says give me and angle whose sine is 1/2 and take the sine of it.
But if you work it out, you say alright sin x is 1/2.
You rewrite this as sin x, this will be 1/2 and then sin x is 1/2.
They sort of cancel them out.
Because they are inverse functions.
10:01ok? What if I said.
What if we do sine inverse of sine of pi/6?
[unintelligible] What is the sine of pi/6?
10:33its 1/2 ok?
Sine 30, pi/6 is 30.
sine of pi/6 sine of 30 is 1/2.
Sine inverse of 1/2 is pi/6.
Because remember what this would be. sine of pi/6 is 1/2 Sine inverse of 1/2 is what angle the sine is pi/6.
11:03ok? Got is? Good.
What is the sine inverse of sine 5pi/ 6?
You want to say 5pi/6.
That would be incorrect.
ok? Think about that for a couple more seconds
11:33What is the sine of 5pi/6?
Where is 5pi/6?
Its in the second quadrant.
whats the reference angle close. 150 oK?
12:02So 150 gives you a reference angle of how much?
30. So whats the sine of 30? 1/2.
So pi/6 is here the reference angle is then pi/6.
The sine of pi/6 is 1/2.
So this is becomes the sine inverse of 1/2.
And what it the sine inverse of 1/2? pi/6.
12:31Ok? So you do not end up back where you started.
The reason is remember when we you do inverse sine If you do the inverse sine of positive quantity we always use the first quadrant angle.
ok? So even though we get a different angle.
We took the sine and got a positive number The inverse sine of a positive number is in the first quadrant.
You have to decide what angle you want to use.
Even inverse sine of 1/2.
13:00There are infinite number of angle that are sine equal to 1/2 We always use the first quadrant. ok?
There is a lot of types of trouble with this!
I am doing it again.
The sine of 5pi/6 is 1/2.
the inverse sine of 1/2 you alway use the first quadrant angle You don't use any other quadrant.
ok? If this was negative 1/2, you were only use the fourth quadrant, negative pi/6.
ok! Lets do another one of these.
13:45Ok! The sine of pi/4 is radical 2 over 2 or So you get sine inverse of radical 2 over 2. Ok?
Sine of what angle is radical 2 over 2
14:00For those who memorized it, 45 degrees or also known as pi/4.
So notice, you right back where you started!
So, what if I asked for sine inverse, sine of 9pi/4?
Ok? You want to say it is 9pi/4 but you cannot!
14:31Because you have to get first quadrant answer.
So, what is sine 9pi/4?
Well, thats 9pi/4 is the same thing as pi/4.
Since you go all the way around to 8pi/4 (2pi) and then you go another pi/4.
Do with degrees if it bothers you!
Ok? So the sine of 9pi/4 has the same value as sin of pi/4.
This is still sine inverse radical 2 over 2.
15:04which is equal to pi/4.
ok? Yes!
If start off and you are doing the inverse trig function of a first quadrant angle you will end up right back where you started.
If you are not in the first quadrant angle and positive you have to
15:30[unintelligible] in the first quadrant. ok?
ok? so we practice some more of these until we grasp it.
cosine inverse of cosine pi/3.
Cosine of pi/3 is 1/2.
So this is cosine inverse of 1/2.
16:08And cosine inverse of 1/2 is in fact pi/3 Because we just wanted to know the sine of what angle is 1/2?
Last, the way you do these you work them out.
16:33What is the cosine of 5pi/3?
You all got this on the first exam If you are not sure whats 5pi/3 is, you turn it into degrees.
That is 300 degrees.
ok? 300 degrees will get you 60 as the reference angle.
5pi/3 is down here.
Cosine is positive down here. pi/3 left over
17:00So you will get 1/2.
Cosine inverse of 1/2 is back to pi/3.
ok? This looks messy but it isn't.
What you have to do is to find what the middle, what the angle in the middle first.
Then, you take the angle of that value in the middle and then you do the inverse trig function to make sure you get the right quadrant.
17:33What if I have, well lets see!
cosine inverse cosine 2pi/3.
Where is 2pi/3 located? Second quadrant.
18:01Also known as 120 degrees.
So this is 60 degrees (pi/3) So that is negative 1/2.
Cosine inverse of negative 1/2 is 2pi/3.
You are in the second quadrant.
Back to were you started a few minutes ago. ok?
You have to use the second quadrant because its a negative value.
18:30ok? It is not a positive value its a negative value.
That is negative 1/2.
So remember inverse trig functions, inverse cosine of a negative values you use the second quadrant angle.
You want to try it again?
19:05ok! if x is a positive number, for all the inverse trig function you always use first quadrant angle ok? if its a negative number, inverse sine and inverse tan use the fourth quadrant.
Inverse cosine use second quadrant.
19:34Everybody see that ok?
You guys can see that?
Some of you got this?
So notice, I am doing cosine inverse of negative 1/2.
Cosine inverse of negative 1/2 in an angle in the second quadrant.
20:00So what if I have cosine inverse of cosine 4pi/3?
Well, lets see! 4pi/3 down here is 240 degrees.
So this is 60 degrees, pi/3 again
Still cant write it!
Ok? So the cosine of pi/3 is negative 1/2.
So this is cosine inverse negative 1/2.
Which means you end up back in the second quadrant because in the second quadrant we have negative values.
21:00ok? Were are we lost?
Why do we get different answers something and same answer other times.
if we start off in the first quadrant.
You finish in the first quadrant.
For all these, for all the ones that look like this.
Inverse trig the first quadrant angle, you end back in the first quadrant angle.
ok? If you do inverse trig of trig that is not in the first quadrant,
21:30figure out where you going to be [unintelligible] Just remember inverse sine and inverse cosine and inverse tangent.
All we get answers if you are in the first and fourth quadrant or first quadrant and second quadrant.
ok? You just need to figure out where you are So lets practice a couple of these.
So you guys get the hang of it! [unintelligible]
23:14Here we go, they are five of these sin pi/3 is radical 3/2.
So, that becomes sine inverse of radical 3/2.
Radical 3/2 is positive, so the answer has to be in the first quadrant. pi/3.
23:37ok? Yes! got it? You are good?
Well, this is positive Look at the chart, if its positive you always use the first quadrant angle.
ok! Sin of 4pi/3 is 240 degrees.
24:06Somewhere around here.
That is going to be the same thing as pi/3.
Same values but negative because ok?
All Seawolves Taste Chocolate.
24:31ok? So thats going to be negative radical 3/2.
But, if I had a negative quantity in there I am going to use a fourth quadrant.
So where in the fourth quadrant do I get sine of negative radical 3 over 2?
fourth quadrant is negative pi/3.
ok? I go down pi/3.
25:04The sine inverse of the sine 7pi/3 Except 7pi/3 had to be right there 7pi/3 is in the first quadrant ok?
7pi/3 is 420 degrees.
That is the same as 60 degrees ok? So, this would be sine inverse radical 3/2.
25:38Sine inverse of radical 3/2 said sine of what angle is radical 3/2?
60 degrees or pi/3.
Take a time out for a second!
lets work in degrees
26:00Lets see if it is easier for you to visualize it!
sine inverse sine of 60 degrees.
You say the sine of 60 degrees is square root of 3/2 That is equal 60 degrees Now it is simple! right?
That is what we had up there.
Sine inverse sine 420 degrees ok?
26:32Sine of 420 degrees is the same think as sine of 60 degrees Sine inverse of radical 3/2 Remember the sine inverse of radical 3/2 is 60 degrees.
ok? So the fact that this was 420 at the beginning, it did not matter.
We get back to 60 degrees because you get this formula where I asked you.
What is the inverse sine?
The inverse sine has to be in the first quadrant.
27:02Ok? Cannot go around. Got it?
So I am doing [unintelligible] instead of degrees So the cosine inverse of cosine 5pi/6.
ok? That is saying cosine inverse of cosine of 150 degrees.
Where is sine of 150 degrees?
Over here in the second quadrant.
So the reference angle is 30 degrees Cosine of 150 is negative radical 3 over 2. ok?
If you do cosine inverse of negative value I am going to use a second quadrant angle I will came right back at 150 degrees or 5pi/6.
28:12Suppose instead I had done cosine of 7pi/6. 7pi/6 is 210 degrees
28:38which is in the third quadrant.
That is the same value as 30 degrees.
This will be the cosine inverse of negative radical 3/2.
The cosine inverse of negative radical 3/2. I use the second quadrant angle 150 degrees or 5pi/6.
ok?Are you starting to go the hang of it?
29:03[NO] No? yes? Maybe?
Need to work! What is the cosine of pi/3?
1/2 you know that the the cosine of pi/3 is 1/2 What is the cosine of 1/2?
cosine inverse of 1/2 is pi/3.
Alright, [unintelligible] do some more problems so you keep quite.
30:24Ok there is some more entertaining.
Alright cosine 11pi/6.
30:33Now you are going to do this in radiants before was in degrees.
Cosine of 11pi/6. 11pi/6 is in the fourth quadrant.
here is a clue, maybe it will help you straighten it out a little bit.
11pi/6 is in the fourth quadrant. ok?
so if we have our All Students Take Candy. ok?
So, cosine is going to be positive down here.
31:02In fourth quadrant. right?
isn't the cosine positive in the fourth quadrant?
So when we get our answer, we will be doing inverse cosine of positive number.
You have to get an answer that is in the first quadrant.
Its going to be a positive number now. Ok?
So the cosine of 11pi/6 is radical 3/2.
If you can do that I will be very happy.
31:31ok? Because the cosine of 11pi/6 is radical 3/2, the inverse cosine of radial 3/2 is then pi/6.
Back to the first quadrant. ok?
Cosine of what angle is radical 3/2?
cosine of... pi/6. [unintelligible] ok?
32:03tan inverse of tan of pi/4.
Where is pi/4? It is in the first quadrant.
So if you see that this is in the first quadrant, this is the same.
Because tangent pi/4 (1) tangent inverse 1 is pi/4.
ok? So first quadrant angle stuff is very easy.
32:30They will just always cancels.
If you are not in the first quadrant it gets messy.
Tan inverse of the tan of 3pi/4.
What is the tangent 3pi/4?
We are in the second quadrant Is tangent positive or negative in the second quadrant?
Negative! And if we do inverse tan of negative what are you going to end up?
in the fourth quadrant. ok?
So this is going to be tangent inverse -1 because tangent of 3pi/4 is -1.
33:03tan inverse of -1 is negative pi/4.
ok? Sine of 7pi/6 is in the third quadrant.
So if you are in the third quadrant you will get a negative answer.
And the sine inverse of a negative answer is in the fourth quadrant.
So what you really want to do, you really just want to say: when is the fourth quadrant equivalent to 7pi/6?
33:34That is really what is going to happen. ok?
this will be the sine inverse of negative1/2.
And that will take you to negative pi/6.
So really what you want to do [student] because same question, you want to go down from the x-axis.
So you don't use 11pi/6 going this way, but you use negative pi/6.
34:03[unintelligible] instead of going around over here, we go down there. ok?
You might end up in the same place but you are using a negative angle.
its a technical thing. It has to to with the fact that when you suing the graph, you are using this angle. ok?
pi/6 is that angle ok?
34:35We are really only using this piece of inverse sine graph. Ok?
Lets do a couple more.
Lets make sure we get this.
I am not going to do it again.
All we really are asking ok? [unintelligible]390 degrees where?
35:03[First quadrant] is the same as what angle in the first quadrant?
Really all we are asking is that ( 30 degrees).
ok? Sine inverse of sine 210 degrees.
210 degrees is where?
which quadrant? third!
Is sine positive or negative in the third quadrant?
Is negative.
So let it be fourth quadrant equivalent of 210.
Negative 30 degrees ok?
35:34Lets see cosine inverse cosine of 750 degrees.
Im sorry 7 hundred and [unintelligible] 7 once, twice, and another 30 degrees So the equivalent is just 30 degrees.
This is really what you are trying to do. ok?
Of course you can do this in radians.
36:02So If I said the tangent inverse of tan of 15pi/4.
You saying yourself where is 15pi/4?
Well lets see!
Thats 8pi/4.
Another seven.
I am down here in the fourth quadrant.
So that is the same as negative pi/4.
36:32Really what you are doing.
[unintelligible] ok? [unintelligible] what you really want to do is you just want to say, what is the first, or second, or fourth quadrant equivalence of these angles ok?