If cos a = - 1/3 , a belongs to ( pi, 3pi/2 ) calculate sin 2a

sin 2a=sin(a+a)=sin a*cos a + cos a*sin a = 2sin a*cos
a

We have the value for cos a, which is negative. The value
for sin a is negative and it could be found from the fundamental formula of
trigonometry.

sin a= sqrt[1-(cos
a)^2]

sin a=sqrt(1-1/9)

sin
a=2sqrt2/3

sin 2a = 2sin a*cos
a

sin 2a =
2(2sqrt2/3)(-1/3)

sin 2a =
-4sqrt2/9