Find the exact value of sin θ/2 if cos θ = 2/3 and 270° < θ < 360°.

Use the formula 
             __________
sin(θ/2) = ±Ö(1-cosθ)/2

Where the sign of the sine is taken according to which quadrant x
is in.

It is unfortunate that the words "sine" and "sign" are homophones,
because it sounds funny and is often confusing to speak of "the
sign of the sine".  Some teachers use "positiveness or
negativeness of the sine" instead of "sign of the sine".

Anyway, substitute 2/3 for cosθ in the formula:

             __________
sin(θ/2) = ±Ö(1-2/3)/2 

Change the last "/" to a "÷"

             _______ 
sin(θ/2) = ±Ö(1/3)÷2 

Change the 2 to 2/1
             ___________
sin(θ/2) = ±Ö(1/3)÷(2/1)

Invert and multiply to perform the divion under
the radical:
             ___________   
sin(θ/2) = ±Ö(1/3)×(1/2) 
             ___   
sin(θ/2) = ±Ö1/6

Now we have to determine whether to use the + or
the - .

We are givem:

270° < θ < 360°

so we whii divide all three sides of that by 2

(270/2)° < θ/2 < (360/2)°

 135° < θ/2 < 180°

This tells us that θ/2 is in quadrant II.
The sine is positive in the second quadrant
Therefore we know that  sin(θ/2) is positive,
So
            ___
sin(θ/2) = Ö1/6

Your teacher may want you to rationalize the
denominator and get
            _   
sin(θ/2) = Ö6/6
 
Edwin