Question 1155366
i believe this will have 5 roots between 0 and 360 degrees.
the roots will be at 0, 180, 360 degrees for tan(x) = 0, and 210, 330 degrees for sin(x) = -1/2.
to confirm, graph the equation as shown below:


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to solve algebraically, do the following.


start with 2 * sin(x) * tan(x) = -tan(x)
add tan(x) to both sides of the equation to get:
2 * sin(x) * tan(x) + tan(x) = 0
factor out tan(x) to get:
tan(x) * (2 * sin(x) + 1) = 0
this is true when tan(x) = 0 or sin(x) + 1 = 0 
when tan(x) = 0, x = 0 or 180 or 360 degrees.
when 2 * sin(x) + 1 = 0, solve for sin(x) to get:
sin(x) = -1/2
solve for sin(x) = plus 1/2 to get x = 30 degrees.
that's in the first quadrant where all trig functions are positive and where all reference angles reside.
sine is negative in the third and fourth quadrants.
in the third quadrant 180 + 30 = 210 degrees.
in the fourth quadrant 360 - 30 = 330 degrees.
confirm by solving for sin in all 4 quadrants to see that the sine is only negative in the third and fourth quadrant.
30 degrees in the second quadrant = 180 - 30 = 150.
30 degrees in the third quadrant = 180 + 30 = 210.
30 degrees in the fourth quadrant = 360 - 30 = 330 degrees.
sin(30) = 1/2
sin(150) = 1/2
sin(210) = -1/2
sin(330) = -1/2
the sine is -1/2 at 210 and 330 degrees only in the interval between 0 and 360 degrees.


bottom line:
5 roots.
0, 180, 360 for tan(x0 = 0
210 and 330 for sin(x0 = -1/2