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.

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 $0.83171110\cdots$
