BAM!!!
Mr. Tarrou!
Let's get going with our evaluating trig functions using half and double angle identities.
Then part 3 and maybe part 4 we will do some proofs using these half angle and double angle
identities.
Ok, we have the sin(3pi/8) and I want to evaluate that with the unit circle.
I want an exact value answer, no decimals.
I don't care if there is 20 decimals.
I want an exact answer which means NO DECIMALS.
Like square root of 7 is left the square root of 7.
So we are going to take this angle which is not on the unit circle and I am going to double
it.
So I am going to multiply it by 2.
Well that is going to give me the cosine of... well... the 2 and the 8 are going to cancel
out and give me 4.
So I have the cosine of -3pi/4.
But, I can't just willy nilly change the problem because I don't like what the teacher gave
me.
I still need to evaluate the cos(-3pi/8).
So that means I am not going to evaluating -3pi/4... at least not as a final answer.
I am just trying to change the way that this looks.
So I have made this angle twice as big, so now I have introduce a division of 2 to still...
you know... have the same value.
If I take -3pi/4 and then divide it by 2, or flip up the 2/1, I am going to get -3pi/8
again.
At any rate, now I am looking at the cosine of theta divided by 2.
See I have set up a half angle, which means I can use a half angle identity for cosine,
which is plus or minus the square root of 1 plus the cosine of -3pi/4 all over 2.
If you don't know what that identity look in the back of your PreCalculus or Trig book..
or look at beginning part 1 of my video and write yourself a formula sheet.
So we got plus or minus the square root of 1 plus the cosine of -3pi/4...
Now that is a negative rotation which means that we are not going to in a counter-clockwise
direction.
We are going to go in a clockwise direction.
Any radian measure on the unit circle with a denominator of 4 is a multiple of 45 degrees.
So it is going to be 0, -pi/4, -2pi/4, -3pi/4.
So the cosine of -3pi/4 is negative square root of 2 over 2.
So 1 plus negative square root of 2 over 2 all over 2.
So I am going to find a common denominator for the top two fractions.
That is going to be plus or minus the square root...
This 1 is going to become 2/2.
So this is going to be 2 minus the square root of 2 over my common denominator of 2.
That is all over a denominator of 2.
You see once again this happens so many times in PreCalculus.
We have a fraction over a fraction.
So we are going to take that bottom and flip it up.
The 2/1 is going to become 1/2.
Our final answer is going to be plus or minus the square root... almost our final answer...
2 minus the square root of 2 over 4.
Now we are so close to writing the final answer.
So let's take some time to clean house a little bit.
-3pi/8, what quadrant is that in?
Now we are evaluating -3pi/4 which is -1pi/4, -2pi/4, -3pi/4... so this angle that we are
actually using in the identity is in quadrant 3.
But -3pi/8, that is going to be in quadrant 4.
That is less than 90 degrees.
So I need to pick the correct sign for my angle that is in quadrant 4.
And cosine in quadrant 4 is positive.
So this isn't plus or minus just because of the square root...
there is not two answers.
The answer is based on the quadrant that the original angle was in... [announcement being
read over intercom] The sign is based on the quadrant the original angle is in, and again
-3pi/8 is in quadrant 4.
That means that my final answer needs to be positive.
Now also the square root of 4 is equal to what?
2.
So this comes out to be an even number.
So as a final answer I am going to be looking for the square root of 2 minus the square
root of 2 over 2.
BAM!!!
Alright.
Let's do another one.
Alrighty then!
So let's take a look at this one.
We have a diagram.
It is an angle in standard position.
Let's pick a different color for that.
It goes into quadrant 3.
Here is theta.
We have got a radius which is equal to 13 and point on the terminal side that is labelled
(-12,y).
And based on that information I want to find out what the tangent of theta over 2 is.
[to be type T/2] Ok.
Well, what do I need?
If I want to figure out what the tan(T/2) and picking the easiest of the three choices
is in my opinion... tan(T/2) equals 1 minus cosT over sinT.
So I need to take that information and find the cosine and sine of theta which means I
need to know x, y and r.
Well I know r is 13.
I am going to come in here and set up a reference triangle.
Reference angles are less than 90 and they measure the number of degrees or radians between
the terminal side and the x axis.
So we have a hypotenuse of 3 or a radius of 3.
We have an x value of -12 and the Pythagorean Theorem...
-12 squared plus y squared equals r squared, or 13 squared.
That is going to give me the answer of 5.
-5 because down.
That is one of our Pythagorean Triples.
Ok.
Awesome.
Now that I have got my reference triangle I am practically half way done with this question.
So we have 1 minus cosine of theta or -12/13... all over the sine of theta...
sine... y/r... -5/13.
Ok, two negatives cancel out to make a positive.
We need a common denominator with that 13, so 13 over 13.
And we get 13 plus 12 is 25 over the common denominator of 13... over -5/13.
And whenever you have a fraction over a fraction, the denominators always cancel out (when the
same)... so when we flip the denominator up...
I am going to just cross these out because they are EXACTLY the same.
But this does flip up like I showed you in the previous example.
I am not move up, but flip up.
Our final answer is going to be 25/-5.
So the tangent... these don't usually come out to be integers... the tangent of T/2 ends
up being -5.
Sweet.
Ok.
Let's talk for a second by the way, what quadrant is this half angle in?
Well, the original angle of theta is clearly in quadrant 3.
So that means that theta is bigger than pi... greater than pi and less than 3pi/2.
(equal signs should not be there) That means this is in quadrant 3 like we just said.
We are finding the angle measure that is half the size.
That means that T/2... is going to be.... well what is half of pi?
pi/2 And what is 1/2 of 3pi/2?
If I take half of 3pi/2, I get 3pi/4.
So my original angle is in quadrant 3, but my half angle is in quadrant 2.
Now, we just got a trig ratio... a tangent ratio... which was negative five.
Tangent came out to be a negative number.
Just making sure I got the right answer.
At least, is it even reasonable that is the right answer.
Well if I am in quadrant 2, tangent with the negative x and positive y tangent is negative
in quadrant 2.
My answer is supposed to be negative in quadrant 2 so it looks like we got the right answer.
Let's move on to the next example.
WHOOO!
Alright.
How about the sine of alpha divided by 2.
Let's see how much time I have left.
Yeah, we can do that.
Ok.
So the sin(A/2).
The half angle identity for sine is plus or minus the square root of 1 minus the cosine
of theta all over 2.
And here is what the identity is asking for.
Cosine of theta is as we have stated over here -12/13.
Let me just take that out and put in -12/13.
So we need to once again have common denominator.
These two negatives are also going to cancel out.
So this becomes 13/13, except that looks like a v. And we get plus or minus the square root
of 13 plus 12 which is 25 over 13 divided by 2 over 1.
So that 2/1 is going to get flipped up and it is going to become 1/2.
And we have plus or minus the square root of 23/26.
Now that is reducible.
So I will do that in a second.
But quadrant 3, we just discussed how our half angle is in quadrant in 2 and sine is
positive in quadrant 2.
So I keep writing plus or minus, but really since we are in quadrant 2 the answer is going
to be positive.
You only know that by evaluating what quadrant you are in.
So just make sure you pay attention to that.
And let's see what we have got here.
The square root of 25 is clearly 5.
For 26 we have got what?
6 times um... actually there is no perfect square in 26 is there.
So this is going to be 5 over the square root of 26.
Which is going to be 5 times square root of 26 over 26 after you rationalize the denominator.
That is the sine of the half angle.
Alright!
Yep.
Just making sure.
BAM!
Got it done.
Alright.... one more example really really quick and we will call it a day...for evaluating
trig functions.
In the next video we will be doing proofs.
The last example is like sum and difference of angles, when they give you something with
numbers it is probably because it does not work ok.
Evaluate this using no calculator... or give the answer in exact form.
That is not going to happen, because 112.5 is not on the unit circle.
At least it will not happen in this form.
But if you recognize that cosine squared minus sine squared is your double angle for cosine,
this is the cosine of 2 times 112.5.... which is the cosine 225 degrees.
Well the cosine of 225 is negative square root of 2 over 2.
BAM!
I am Mr. Tarrou.
Go Do Your Homework.
Or watch the video about proofs next:)