Dear Uncle Colin,

I need to find the largest solution to $e^x + \sin(x)=0$ and I don’t really know where to start. Any ideas?

- Some Options Look Virtually Equal

Hi, SOLVE, and thanks for your message!

That is something we in the trade call ‘not a very nice equation to solve’. It’s quite unusual for mixtures of exponentials and trig functions to have simple roots, and this, I’m afraid, is one of the awkward cases.

Finding bounds

We can put some bounds on it, though: we know that when $x=0$, $e^x=1$ and $\sin(x)=0$, and a sketch shows that there can be no positive solutions.

Similarly, the largest solution must be larger that $x=-\piby 2$, because the left-hand expression is negative there, and we have a sign change between there and $x=0$.

Getting closer

At this point, making an estimate of $x=-\piby 4$ and sticking it into your calculator (the new Classwiz will solve it numerically just fine) or your favourite numerical method (Newton-Raphson for the win!) will give you an answer.

Wolfram|Alpha puts it at -0.5885…, and doesn’t suggest any nice exact form for the answer.

Hope that helps!

- Uncle Colin

* Edited 2018-01-12 to clarify what kind of case it was. Thanks, Adam!