What do you call an equation that is impossible to algebraically solve? For example... 

$$3.5=\sin{(x)}+\frac{\cos{(x)}}{\ln{(x)}+\sin{(x)}}$$

So your options to solving such a thing are iteration and other numerical methods. Does an equation that's not solvable algebraically have a specific name?
These are called transcendental equations. When you have a polynomial equation that you cannot solve, then you say that the equation "is not solvable by radicals."
These are called transcendental equations. When you have a polynomial equation that you cannot solve, then you say that the equation "is not solvable by radicals."