`tan x= -1/sqrt3`
Isolate angle x.
`x=tan^(-1) (-1/sqrt3)`
Using a calculator to compute, it yields a value:
`x=-30^o`
Since it states to find the principal solution of the given equation, it restricts the values of the angle from `0^o` to `360^o` . To determine the values of x in this interval, take note that tangent has negative values in Quadrants II and IV.
So to get the equivalent values of x in that interval, add `-30^o` to the quadrantal angles on the horizontal axis.
`x_1= 180 + (-30^o) = 150^o`
`x_3= 360 + (-30^o)=330^o`
Therefore, the principal solutions of the equation `tan x =-1/sqrt3` are `x={150^o, 330^o}` .
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