Question #afaf9

1 Answer
Nghi N
Dec 9, 2017

#pi/3; (4pi)/3#

Explanation:

Develop sin (x + 30) = sin (x + pi/6), by using trig identity;
sin (a + b) = sin a.cos b + sin b.cos a
In this case,
#sin x.cos pi/6 + sin pi/6.cos x = 2cos x#
#(sqrt3sin x)/2 + cos x/2 - 2cos x = 0#
#sqrt3sin x - 3cos x = 0#. Divide both sides by sqrt3
#sin x - sqrt3cos x = 0# (1)
Since #sqrt3 = tan (pi/3) = sin (pi/3)/(cos (pi/3))#,
the equation (1) becomes:
#sin x.cos (pi/3) - sin (pi/3)cos x = 0#, or
#sin (x - pi/3) = 0#
Unit circle give 2 solutions:
a. #x - pi/3 = 0# --> #x= pi/3#
b. #x - pi/3 = pi #--> #x = pi + pi/3 = (4pi)/3#
Check.
#x = pi/3# --> #sin (pi/3 + pi/6) = sin (pi/2) = 1# -->
#cos x = cos (pi/3) = 1/2# --> #2cos x = 1# -->
#sin (x + 30) = 2cos x = 1#. Proved

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