# sin 2 theta + sin theta - tan theta = 0 #
# 2 sin theta cos theta + sin theta - sin theta / cos theta = 0 #
We can assume #cos theta ne 0#. Let's multiply to clear the fraction.
# 2 sin theta cos^2 theta + sin theta cos theta - sin theta = 0 #
# sin theta ( 2 cos ^2 theta + cos theta - 1 ) = 0#
# sin theta = 0 # or #2 cos ^2 theta + cos theta - 1 = 0#
# theta = 0^circ or 180^circ or 360^circ# from the first one.
# ( 2 cos theta - 1)(cos theta + 1) = 0 #
#cos theta = 1/2 or cos theta = -1 #
The first is trig's go-to triangle. #theta =60^circ or 300^circ#
The second is #theta=180^circ# which we already had.
Check:
#theta =0 quad sin 0 + sin 0 - tan 0 = 0 quad sqrt #
#theta=60^circ quad sin 120+ sin 60-tan60=\sqrt{3}/2+\sqrt{3}/2- \sqrt{3} = 0 quad sqrt #
#theta=180^circ quad sin 360 + sin 180 - tan 180= 0+0-0=0 quad sqrt#
#theta =300^circ quad sin 600 + sin 300 - tan 300 ##= sin(-120)+sin(-60)-tan(-60) ##= (-\sqrt{3}/2) + (-\sqrt{3}/2) - (-\sqrt{3}) = 0 quad sqrt#
