Find all solutions of the equation 3 sin2 x 7 sin x 2 0 i

Find all solutions of the equation 3 sin^2 x - 7 sin x + 2 = 0 in the interval [0, 2 pi). The answer is x_1 = and x_2 = with x_1

Step-1:

Given equation 3sin²x-7sinx+2=0

The interval in which to solve is [0, 2*\\pi]

Step-2: Let sinx = y

So the given equation becomes

3y²-7y+2 = 0

Factorizing gives

3y²-6y-y+2 = 0

3y(y-2)-1(y-2)=0

(3y-1)(y-2)=0

So roots are y = 1/3 or y = 2

Step-3:

Roots found are y = 1/3 or y = 2

so sinx = 1/3 or sinx = 2

both are positive values

---------------------------

So the roots are

sinx = 1/3

x = sin^-1(1/3) = 19.5^o= x1

and x = sin^-1(2) this does not exist in [0, 2pi]

hence another root is x2 = 180^o-19.5^o = 160.5^o