end value has z

Here is my code:

syms cos3 sin3 q theta3
B = ((((((3*cos3) + 10)-(3*sin3))*12)^2)/(576))+(((3*sin3-3*cos3)^2)-((3*cos3+10)^2)-((-3*sin3)^2));
B1 = subs(B,cos3,(1-q^2)/(1+q^2));
B2 = subs(B1,sin3,(2*q/(1+(q^2))));
Q = subs(B2,q,tan(theta3/2));
theta_3 = solve(Q, theta3, 'MaxDegree',1)

have tried

theta_3 = solve(Q, theta3)

returning

theta_3 =
2*atan(root(z^4 - (20*z^3)/37 + (194*z^2)/37 + (100*z)/37 + 157/37, z, 1))

please help since I dont know what to do.

Answer 1

[Walter Roberson]

theta_3 = simplify( solve(Q, theta3, 'MaxDegree',4) );

You will get four closed-form solutions that look similar t

 -2*atan(((-1)^(3/4)*805^(3/4)*814^(1/4)*(50156 + 1500747^(1/2)*45i)^(1/4)*(- 50156 - 1500747^(1/2)*45i)^(1/6)*(3*4366^(1/2)*(- 65527*37^(1/2)*(65527 + 37*(- 50156 + 1500747^(1/2)*45i)^(2/3) - 1757*(- 50156 + 1500747^(1/2)*45i)^(1/3))^(1/2) + 38940*222^(1/2)*(- 50156 + 1500747^(1/2)*45i)^(1/2) - 3514*37^(1/2)*(- 50156 + 1500747^(1/2)*45i)^(1/3)*(65527 + 37*(- 50156 + 1500747^(1/2)*45i)^(2/3) - 1757*(- 50156 + 1500747^(1/2)*45i)^(1/3))^(1/2) - 37*37^(1/2)*(- 50156 + 1500747^(1/2)*45i)^(2/3)*(65527 + 37*(- 50156 + 1500747^(1/2)*45i)^(2/3) - 1757*(- 50156 + 1500747^(1/2)*45i)^(1/3))^(1/2))^(1/2) - 37^(3/4)*177^(1/2)*(6^(1/2)*(65527 + 37*(- 50156 + 1500747^(1/2)*45i)^(2/3) - 1757*(- 50156 + 1500747^(1/2)*45i)^(1/3))^(3/4) - 15*(- 50156 + 1500747^(1/2)*45i)^(1/6)*(65527 + 37*(- 50156 + 1500747^(1/2)*45i)^(2/3) - 1757*(- 50156 + 1500747^(1/2)*45i)^(1/3))^(1/4)))*(1500747^(1/2)*195027i + 44170*(- 50156 + 1500747^(1/2)*45i)^(2/3) + 37*(- 50156 + 1500747^(1/2)*45i)^(1/3)*(83725 + 1500747^(1/2)*111i) + 68879755)^(1/4))/22800012840990)

If you are able to make sense of that, then your visualization is better than mine!

I would suggest to you that most of the time you have no interest in the exact expression for the root of a quartic or even a cubic: most of the time you should use vpasolve() instead of solve() to get numeric approximations.