We have: #frac(5 csc(x))(3) = frac(9)(4)#; #0 < x < 2 pi#
#Rightarrow 5 csc(x) = frac(27)(4)#
#Rightarrow csc(x) = frac(27)(20)#
#csc(x)# is the reciprocal of #sin(x)#, namely #csc(x) = frac(1)(sin(x))#:
#Rightarrow frac(1)(sin(x)) = frac(27)(20)#
#Rightarrow sin(x) = frac(20)(27)#
Let the reference angle be #x = arcsin(frac(20)(27)) = 0.834172325#.
Then, the value of #sin(x)# is #frac(20)(27)#, which is a positive value.
So, the angles #x# are located in the first and second quadrants:
#Rightarrow x = 0.834172325, pi - 0.834172325#
#Rightarrow x = 0.834172325, 2.307420329#
#therefore x approx 0.83, 2.31#
Therefore, the solutions to the equation, rounded to the nearest hundredth of a radian, are #x = 0.83# and #x = 2.31#.