#sin(x+\frac{\pi }{6})=2cos(x)#
Manipulate the left hand side by using the identity:
#\sin (s+t)=\cos (s)\sin (t)+\cos (t)\sin (s)#
#\cos (x)\sin (\frac{\pi }{6})+\cos (\frac{\pi }{6})\sin (x)=2cos(x)#
Put the values #\sin (\frac{\pi }{6})=\frac{1}{2}# #\color(red){and}# #\cos (\frac{\pi }{6})=\frac{\sqrt{3}}{2}#
#\cos (x)\sin (\frac{\pi }{6})+\cos (\frac{\pi }{6})\sin (x)=2cos(x)#
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#\frac{1}{2}\cos (x)+\frac{\sqrt{3}}{2}\sin (x)=2cos(x)#
Subtract #2cos(x)# from both sides and then simplify to get:
#\sqrt{3}\sin (x)-3\cos (x)=0#
Dividing both sides with #cos(x)# will give us:
#\frac{\sqrt{3}\sin (x)-3\cos (x)}{\cos (x)}=\frac{0}{\cos (x)}#
Simplify:
#\tan (x)=\sqrt{3}#
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General solutions for the above equation are #x=\frac{\pi }{3}+\pi n#
Solutions within the specified range are:
#x=\frac{\pi }{3}##" "##\color(blue){and}##" "# #x=\frac{4\pi }{3}#
