# how to solve equations

I am trying to find the values of L and \theta, but mathematica is unable to produce output. My code is given below, where the value of k is fixed, i.e., 0.75, but the value of B can be positive or negative.

Solve[{
1/r^4 2 (r^2 + (L - B r^k +
B r^k Cos[\[Theta]])^2 Csc[\[Theta]]^2 - (-2 + r) (L -
B r^k + B r^k Cos[\[Theta]]) (L + B (-1 + k) r^k -
B (-1 + k) r^k Cos[\[Theta]]) Csc[\[Theta]]^2) == 0,
-(1/(r^3))
2 (-2 + r) (L - B r^k +
B r^k Cos[\[Theta]]) (B r^
k + (L - B r^k) Cos[\[Theta]]) Csc[\[Theta]]^3 == 0},
{L, \[Theta]}]

One approach is to introduce variables $$x$$ and $$y$$ in place of $$\cos\theta$$ and $$\sin\theta$$, like this

eq1 = 1/r^4 2 (r^2 + (L - B r^k +
B r^k Cos[θ])^2 Csc[θ]^2 - (-2 + r) (L -
B r^k + B r^k Cos[θ]) (L + B (-1 + k) r^k -
B (-1 + k) r^k Cos[θ]) Csc[θ]^2) == 0;

eq2 = -(1/(r^3)) 2 (-2 + r) (L - B r^k +
B r^k Cos[θ]) (B r^
k + (L - B r^k) Cos[θ]) Csc[θ]^3 == 0;

eq3 = x^2 + y^2 == 1;

eqns = {eq1, eq2, eq3} /. Cos[θ] -> x /. Csc[θ] -> (1/y);

s = Solve[eqns, {L, x, y}] // Simplify;

Then use ArcTan[x,y] to recover solutions for $$\theta$$.

• Another alternative would be to use the Weierstrass substitution θ -> 2 ArcTan[u]. – J. M. will be back soon Sep 16 '19 at 12:53