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1 Answer
Nghi N
Dec 8, 2017

pi/3, pi, (5pi)/3

Explanation:

#3cos (x/2) - 3sin x = 0#.
Replace sin x by #2sin (x/2).cos (x/2)#
#3cos (x/2) - 6sin (x/2).cos (x/2) = 0#
#3cos (x/2)(1 - 2sin (x/2)) = 0#
Either one of the factors should be zero.

A. #cos (x/2) = 0#. Unit circle gives 2 solutions:
#x/2 = pi/2# and #x/2 = (3pi)/2# -->
a.# x/2 = pi/2# --> #x = pi#
b. x/2 = (3pi)/2 --> #x = 3pi# (rejected because outside of range)
B. #1 - 2sin (x/2) = 0 #
#sin (x/2) = 1/2#
Trig table and unit circle give 2 solutions:
#x/2 = pi/6# and #x/2 = (5pi)/6 #-->
a. #x/2 = pi/6# --> #x = pi/3#
b. #x/2 = (5pi)/6# --> #x = (5pi)/3#
Answers for #[0, 2pi)#:
#pi/3, pi, (5pi)/3#

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