Since a is in the third quadrant, both sina and cosa are negative.
Now, sin(2a) can be rewritten as 2sin(a)cos(a).
Thus, we can find sin(a) by using Pythagoras' Theorem.
[tex]cos(a) = -\frac{4}{5}[/tex]
Now, by focusing on the positive value, we can see that 5 is the hypotenuse.
Let the missing side be x.
x² + 16 = 25
x² = 9
x = 3, because x > 0
[tex]\text{Thus, } sin(a) = -\frac{3}{5}[/tex]
Substituting these into the equation, we get:
[tex]2sin(a) cos(a) = 2 \cdot -\frac{3}{5} \cdot -\frac{4}{5}[/tex]
[tex]2sin(a) cos(a) = \frac{24}{25}[/tex]
[tex]\therefore sin(2a) = \frac{24}{25}[/tex]
