SOLUTION: Find the exact value of {{{sin(2theta)}}} if {{{cos(theta) = -sqrt(5)/3}}} and {{{"180°"<theta<"270°"}}}

We draw the approximate graph of the angle  in quadrant III,
since it is between 180° and 270°.



Since , and since ,
we take , the numerator of the cosine and ,
the denominator of the cosine, draw a perpendicular from the terminal
side of  to the x-axis, and label the horizontal side
of the resulting triangle , (negative because it goes
to the left), and the hypotenuse .  The hypotenuse  is 
always taken positive:



Now we use the Pythagorean theorem to find the length of
the vertical side:






 

Since the vertical side of the right triangle goes
down from the x-axis, we label it negative as 



Next we want to find .

We use the identity:



We substitute and 





Edwin