# Calculate sin2a if sina+cosa=1/3

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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**

We'll raise the square the given constraint:

(sina+cosa)^2=(1/3)^2

We'll expand the binomial:

(sina)^2 + 2sina*cosa + (cosa)^2 =1/9

But (sina)^2+ (cosa)^2= 1 ( Pythagorean identity )

2sina*cosa+1=1/9

2sina*cosa = 1/9 - 1

We'll apply double angle identity:

sin2a=2sina*cosa

sin2a=1/9 - 1

sin2a = -8/9

**The value of sin2a is sin2a = -8/9.**