<s>[INST] Solution for $(a\cdot x+b)\cdot \sin(x)+c\cdot \cos(x)=0$? As part of an engineering problem, I've been trying for a generic solution for $(a\cdot x+b)\cdot \sin(x)+c\cdot \cos(x)=0$
Here's something I tried:
$(a⋅x+b)⋅\tan(x)+c=0$, when $\cos(x)\neq 0$
$$\tan(x)=\frac{-c}{a⋅x+b}$$
It sort of looking like this, when, say, $a=2, b=3, c=5$:
Any chance that can be solved without using numerical methods?
EDIT: I was looking for solutions in the form of $x \approx f(a,b,c) $, preferably finding solution in any range, because $a$, $b$, and $c$ can take wierd values. It's used for a program where it will not be trivial to solve problem procedurally.
 [/INST] Besides numerical solutions for particular parameter values, you could try series solutions.  Thus when $c$ is small compared to $a$ and $b$, the solution near $x=0$ is
$$ -\frac{b}{a} + \cot\left(\frac{b}{a}\right) \frac{c}{a} + \cot\left(\frac{b}{a}\right) \csc^2 \left(\frac{b}{a}\right) \frac{c^2}{a^2}
+  \left(\cot^3\left(\frac{b}{a}\right) + \cot\left(\frac{b}{a}\right) \sec^4  \left(\frac{b}{a}\right)\right) \frac{c^3}{a^3} + \ldots $$
</s>