Is there an analytic solution to this equation? $a+b\sin(x)+c\tan(x)=0$

Question

Here is the equation which should be solved for $x$:

$$ a+b\sin(x)+c\tan(x)=0 $$

where $a$, $b$, and $c$ are constants.

Is there an analytic solution, or at least an algorithm to approximate it for any given set of constants?

Answer 1

Using the universal substitution $t=\tan(x/2)$ gives $$ a+b\cdot\frac{2t}{1+t^2}+c\cdot\frac{2t}{1-t^2}=0 $$ So you just need to clear denominators and solve the quartic. (OK, the last link is a joke. You should solve it using the algorithm from Galois theory instead of a single nasty formula.)