Wolfram generating different graphs for equivalent equations

Question

Wolfram generates this funky graph for eˣ = sin(x·y):

Whereas for the equivalent arcsin(eˣ)/x = y we get this rather boring graph:

Are they both correct?

Answer 1

It's Wolfram Alpha making a bad guess at a sensible range to show, which is not obvious from the diagram. Clicking on the Open Code button reveals that it's interpreting the input as

ContourPlot[E^x == Sin[x y], {x, -151, 37}, {y, -95, 26}]

and because there are so many solutions to the equation in this region, it can't correctly guess how to join them up. If you ask for

plot e^x = sin(xy) for -1<x<1

you get a reasonably respectable-looking graph:

and specifying an appropriate range for y as well gives an even better one:

Of course, $\arcsin$ is not a true inverse to $\sin$, so the other graph only picks out one of these lines.