how to get simplified result of solving equation

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First, be sure to define your variable using var('t').

To find a numerical solution, you can plot the function to help identify where the roots are.

plot(-2*sqrt(3)*sin(t)^2+2*cos(t)*sin(t)+sqrt(3),(t,-10,10))

For example, to get the first positive root, you can now use find_root to find the root between 0 and 3.

find_root(-2*sqrt(3)*sin(t)^2+2*cos(t)*sin(t)+sqrt(3),-3,3)

which gives 2.6179938779914944.

For an analytic solution, you can do the following:

solve(-2*sqrt(3)*sin(x)^2+2*cos(x)*sin(x)+sqrt(3)==0, x,to_poly_solve ='force')

This gives: [x == 1/3*pi + pi*z1, x == -1/6*pi + pi*z2]

The z1 and z2 can be any integers.

(Interestingly, I could not get the solve to work with t as the variable. I'm not sure why.)