How do you solve (5cscx)/3 = 9/4 for 0 < x < 2pi rounded to the nearest hundredth of a radian ?

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Answer 1 · 2018-04-10T21:32:27

#x = 0.83, 2.31#

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#.