Find the exact value of sin 2θ if
cosθ = -(√5)/3 and 180° < θ < 270°. 
Thank You 

We need to use the formula

sin(2θ) = 2sinθ·cosθ

We know cosθ but not sinθ.  We have to find sinθ.

We are given 180° < θ < 270° so we know θ is in 
quadrant III.  So we draw the picture of θ in the
third quadrant

    _ |
  -√5 |
-------------
  |  /|
  | /3|
  |/  |

Since the cosine is x/r or adjacent/hypotenuse, we
                       _             _ 
put the numerator of -√5/3, namely -√5 the on the x-side
(or adjacent side) and 3 on the r, the radius vector (or
hypotenuse).

We use the Pythagorean theorem to find the y-side (or the
opposite side).
      _______     ____________      ___     _
y = ±√r² - x² = ±√(3)² - (-√5)² = ±√9-5 = ±√4 = ±2

We know to take the negative sign since y goes down
from the x-axis, so we have

    _ |
  -√5 |
-------------
  |  /|
-2| /3|
  |/  |
                        
Now we know sinθ = -2/3

So
sin(2θ) = 2sinθ·cosθ 
                     _ 
sin(2θ) = 2(-2/3)·(-√5/3)
                     _ 
sin(2θ) = 2(-2/3)·(-√5/3)
            _ 
sin(2θ) = 4√5/9

Edwin
AnlytcPhil@aol.com