Difficulty in solving transcendental equation

Question

Let $A,B,C$, and $D$ be positive constants. What's the most concise way to express $x$ in the equation below?

$$ A = B\arctan(x/C)+Dx,$$

where $0<x<1$ and we know that $C=\cos(30^\circ)$.

Answer 1

Equations which mix polynomial and trigonometric terms do not show explicit solutions (this is already the case for $x=\cos(x)$). As Henning Makholm commented, only numerical methods could do it.

I suppose that the best I could propose (taking into account the fact that $C=\cos(\frac \pi 6)$) is to use a low order Taylor expansion for the term $\tan^{-1}(\frac x C)$. Built at $x=\frac 12$, we can obtain $$\tan^{-1}(\frac x C)= \frac{\pi }{6}+\frac{\sqrt{3}}{2} \left(x-\frac{1}{2}\right)-\frac{\sqrt{3}}{4} \left(x-\frac{1}{2}\right)^2+O\left(\left(x-\frac{1}{2}\right)^3\right)$$ This will let you solving a quadratic equation in $y=x-\frac 12$ $$\frac{1}{6} (-6 A+\pi B+3 D)+y \left(\frac{B\sqrt{3} }{2}+D\right)-\frac{B\sqrt{3}}{4} y^2=0$$ Any other approximation would lead to polynomials of degree $>3$ and would not be very useful.

This must be handled with a lot of care.