# Why is it impossible to solve y explicitly in this equation?

## Main Question or Discussion Point

$$y+siny=x$$

How would you even graph a function like this if you can't solve for y explicitly?

eumyang
Homework Helper
$$y+siny=x$$

How would you even graph a function like this if you can't solve for y explicitly?
Make a table of values. Pick some y-values and solve each of them for x.

Why is it impossible to solve y explicitly in this equation?

SteamKing
Staff Emeritus
Homework Helper
The presence of sin y. Trig functions are transcendental and cannot be solved using algebraic methods alone.

The presence of sin y. Trig functions are transcendental and cannot be solved using algebraic methods alone.
With this in mind (and I agree) one cannot solve this either,
$$\sin{y} = x$$
We have to invent a special function, the inverse sine, to solve it. By that same logic, we can invent a special function to solve the original equation. But it would be no more "solved" than using the inverse sine function to solve my equation above.

rbj
you can graph $x$ as a function of $y$ explicitly, then turn the graph by 90° and flip (mirror image) the rotated graph.

i do not think there is a closed form solution to

$$y + a \sin(y) = x$$

but you can still solve it numerically, as long as $|a| \le 1$. there are places in the $y=f(x)$ function where the slope is infinite, but only on a single point. sorta like the real function $y = x^{1/3}$.

Bacle2
Have you tried to apply the explicit function theorem to see if a local solution for y is
possible?

HallsofIvy
$$y+siny=x$$