Let us work with the Half Angle Identities.

For example, if sine of theta is

negative three-fifths and theta is between three pi halves and two pi,

we will find the exact value of sine of theta over 2.

Let us recall the half-angle identity for sine.

We have that sine of theta over 2 is equal to plus or minus

the square root of 1 minus cosine theta all divided by 2.

Now, we're given that sine of theta is

negative three-fifths and that theta is between three pi halves and two pi.

Therefore, we can draw the following triangle.

Remembering that sine of theta is equal to y divided by r,

and if this is negative three-fifths,

notice down here we put the negative with the y because y's are

negative in quadrant four and r is always positive.

And remember that cosine of theta is equal

to x divided by r. And so if we can find x here,

then we'll be able to determine what cosine of theta is,

which we then can use over here on the left to determine sine of theta over 2.

And to find x,

we can use Pythagorean theorem to help us.

Namely, we have x squared plus y squared is equal to r squared.

Therefore, x squared plus negative 3 squared is equal to

5 squared or x squared plus 9 is equal

to 25 or x squared is equal to 25 minus 9,

which is 16, which means that x is equal to plus or

minus the square root of 16 or x is equal to plus or minus 4.

But, x is in quadrant four,

and in quadrant four,

x is greater than zero.

Therefore, we are going to choose the positive value here.

And therefore cosine of theta is equal to positive 4 divided by 5,

which we can plug into our half-angle formula over here.

But, there is one more issue with this formula.

Is not there? What is this plus or minus mean?

Does that mean that there are two solutions here?

No. We are going to choose either the positive or the

negative depending upon the quadrant that theta over 2 lies.

That is if sine of theta over two in that quadrant is positive,

we will choose the positive,

and if it is negative, we will choose the negative.

Now, remember that we are given that theta lies between three pi halves and two pi.

However, we do not want to know where theta lies.

We want to know where theta over 2 lies.

So, let us divide everything by two then.

So, 3 pi divided by 2 all divided by 2 is three pi

fourth and 2 pi divided by 2 is pi.

So, therefore, theta over 2 lies between three pi fourth and pi.

That is, if this

is three pi fourth and this is pi that means theta over 2 is somewhere in here,

and since sines are positive in quadrant two,

sine of theta over 2 is greater than zero.

Therefore, we are going to choose the positive value up here in our formula.

Now, there is a common mistake that students make that should be pointed out here.

Looking over at the figure on our right,

we see that theta is in quadrant four and students will think because theta is

in quadrant four and sines are negative in

quadrant four that they should choose the negative value here.

Do not look at where theta lies,

look at where theta over 2 lies.

Alright.

So, then this is equal to,

we're choosing the positive with square root 1 minus,

we found that cosine theta was four-fifths, still have all divided by 2,

which is equal to the square root of 1 minus four-fifths,

which one-fifth divided by 2,

which is the square root of 1 over 10 or rationalizing,

we get square root of 10 over 10, which would be our answer.

And this is how we work with these half-angle identities.

Thank you.

We will see you next time.