Solve for x by factoring where pi/2 <= x <= pi ? 3 csc x cot^2 x = csc x

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Answer 1 · 2018-05-25T17:15:04

#x=2/3*pi#

writing your equation in the form
#csc(x)(3cot^2(x)-1)=0#
so we get
#csc(x)=0#
or
#cot(x)=pm1/sqrt(3)#

Answer 2 · 2018-05-25T21:09:13

#x = (2pi)/3#

#3csc x.cot^2 x - csc x= 0#

#(3/sin x)(cos^2 x)/(sin^2 x) - 1/sin x = 0#

#3cos^2 x - sin^2 x = 0# (condition #sin x != 0#)

#3cos^2 x - 1 + cos^2 x = 0#

#4cos^2 x = 1# --> #cos^2 x = 1/4#

#cos x = +- 1/2#

Trig table and unit circle give 4 solutions of x:

a.# cos x = 1/2# --> #x = +- pi/3#

b. #cos x = - 1/2# --> #x = +- (2pi)/3#

Inside the interval #(pi/2, pi)#, there is one answer:
#x = (2pi)/3#