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!