Calculate sin2a if sina+cosa=1/3

We have sin a + cos a = 1/3

Use the
fact that sin 2a = 2*(sin a)(cos a)

sin a + cos a =
1/3

square the right and left hand
sides

(sin a + cos a)^2 =
(1/3)^2

=> (sin a)^2 + (cos a)^2 + 2*(sin a)(cos a) =
1/9

(sin a)^2 + (cos a)^2 = 1

=>
1 + 2*(sin a)(cos a) = 1/9

=> 2*(sin a)(cos a) = 1/9 -
1

=> 2*(sin a)(cos a) =
-8/9

=> sin 2a =
-8/9

The required value of sin 2a =
-8/9