What is sin x if tan x=2/3 and x is in the set (0,pi)?

The range (0,pi) covers the first and the second quadrant
where the values of the sine function are positive.

We'll
apply, for the beginning, the Pythagorean identity:

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

We'll divide the formula with the value 
(sin x)^2:

(sin x)^2/ (sin x)^2 + (cos x)^2/(sin x)^2 = 1 /
(sin x)^2

But the ratio sin x /cos x= tan x and cos x/sin
x=1/tan x

The formula will
become:

1 + (cotx)^2 = 1/(sin
x)^2

sin x = 1/sqrt[1+(cot
x)^2]

sin x =
1/sqrt[1+(3/2)^2]

sin x=
1/sqrt(1+9/4)

sin x = 2/sqrt13 => sin x =
2sqrt13/13

The requested value for sin x is :
sin x = 2sqrt13/13