Can someone help please Find all solutions of the equation 3

Can someone help please?

Find all solutions of the equation (3sin (x) + 5) cos(x/2) = 0 in the interval [0, 2 pi). Show your work.

(3sinx +5)cos(x/2) =0

3sinx +5 =0 ; cos(x/2) =0

3sinx = -5

sinx = -5/3 < -1 ; (-1<=sinx < = 1)

So no solution

Now take cosx/2 =0

x/2 = pi/2 +2pi*k -----> x = pi + 4pi*k

x/2 = 3pi/2 +2pi*k -----> x = 3pi + 4pi*k

we have to find solution in the interval of [ 0, 2pi )

So, k =0 ; x = pi , 3pi .....

So, only x = pi lies in the interval of [ 0, 2pi )

Now plug k =0