<s>[INST] Solve $x+\sin(x) = \frac\pi2$ I was recently doing a high school geometry problem.
I believe that the intermediate steps are less than relevant, but if needed, I can post them too.
I ended up with the equation $$x+\sin(x)=\frac\pi2$$
I have no idea how to solve it. Any help would be appreciated.
 [/INST] As said in comments, there is no closed form but you can have good approximations since you noticed that the solution is close to $\frac \pi 4$.
Expanding as series
$$x+\sin(x)=\left(\frac{1}{\sqrt{2}}+\frac{\pi }{4}\right)+\left(1+\frac{1}{\sqrt{2}}\right) \left(x-\frac{\pi }{4}\right)+\frac{1}{\sqrt{2}}\sum_{n=2}^\infty \frac{\sin \left(\frac{\pi  n}{2}\right)+\cos \left(\frac{\pi  n}{2}\right)}{n!}\left(x-\frac{\pi }{4}\right)^n$$
Truncating to some order and using series reversion,
$$x=\frac{\pi }{4}+t+\left(\frac{1}{\sqrt{2}}-\frac{1}{2}\right) t^2+\frac{8-5 \sqrt{2}}{6}
    t^3+\frac{27 \sqrt{2}-37}{12}
   t^4+\frac{518-363 \sqrt{2}}{60}  t^5+O\left(t^{6}\right)$$ where $t=\frac{\pi -2 \sqrt{2}}{2 \left(2+\sqrt{2}\right)}$.
Using these terms only, this gives,as an approximation $x=\color{red}{0.83171119}30$ while the solution, given using Newton method, is
$x=\color{red}{0.83171119360}$
Edit
What you could also do is performing one single iteration of Halley method (instead of Newton method) and face the problem of solving for $x$ the linear equation
$$\frac \pi 2=\frac {\frac{1}{4} \left(2 \sqrt{2}+\pi \right)+\frac{\left(28+16 \sqrt{2}+\sqrt{2} \pi
   \right) }{8
   \left(2+\sqrt{2}\right)}\left(x-\frac{\pi }{4}\right) } {1+\frac{1}{\sqrt{2}
   \left(2+\sqrt{2}\right)}\left(x-\frac{\pi }{4}\right) }$$ which gives as an estimate
$$x=\frac \pi 4 +\frac{2 \left(2+\sqrt{2}\right) \pi -8 \left(1+\sqrt{2}\right)}{4 \left(7+4
   \sqrt{2}\right)-\sqrt{2} \pi }=0.831700\cdots$$
Using Householder method
$$x=\frac \pi 4 +\frac{3 \left(120+88 \sqrt{2}-16 \left(3+2 \sqrt{2}\right) \pi
   +\left(1+\sqrt{2}\right) \pi ^2\right)}{-8 \left(119+83 \sqrt{2}\right)+8
   \left(5+4 \sqrt{2}\right) \pi +\left(1+\sqrt{2}\right) \pi ^2}$$ which is $0.83171110\cdots$
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