Question

sin(7/2) (x)=√3/2

would one of the answers be:
x=4pi/21 + (4n/7)(pi)

thank you

Answer 1

not clear what
sin(7/2) (x)=√3/2 means
do we have
sin ( (7/2)x ) = √3/2 ? I will assume that
I know that sin π/3 = √3/2

Let me switch to degrees ...
I know that sin 60° = √3/2
then (7/2)x = 60 or (7/2)x = 120° , since the sine is + in quads I or II
7x = 120 or 7x = 240
x = 120/7 ° or x = 240/7 °

back to radians: 120° = 2π/3
so
x = 2π/21 or x = 4π/21

The period of our sine curve is 2π/(7/2) = 4π/7

so we have
x = 2π/21 + n(4π/7) , 4π/21 + n(4π/7)

you had one of these