I would appreciate help in solving this equation: $$2x =\sin 2x + \frac{\pi}{2}$$ I am aware that instead of $2x$ in $\sin(2x)$ I could put the whole right part of the equation, and then again and again the same thing, till infinity. I know the solution exists (from photomath app). How is this type of equation called and how to find the solution?
Thanks for any help!
By introducing the variable $u = 2x - \frac{\pi}{2}$ the equation simplifies to $\cos u = u$, a well known equation whose only real solution $\alpha$ is known as the Dottie number and no closed form is known for it.
So the solution of your equation is $\frac{1}{2}(\alpha+\frac{\pi}{2}) \approx 1.1549$
This kind of equations are known as transcendental equations and their solutions usually don't have closed forms in terms of elementary functions.
There is no closed form, in terms of elementary functions, for this type of equation. Numerical solution will at least give you a value.
Newton's method gives $$ \begin{align} x_{n+1} &=x_n-\frac{2x_n-\sin(2x_n)-\frac\pi2}{2-2\cos(2x_n)}\\ &=\frac{\frac\pi2+\sin(2x_n)-2x_n\cos(2x_n)}{2-2\cos(2x_n)}\\ \end{align} $$ which converges fairly rapidly.
$$ \begin{array}{r|l} n&\text{value}\\\hline 0&1\\ 1&1.1695070529519190789\\ 2&1.1550315982863453148\\ 3&1.1549407336507333597\\ 4&1.1549407300050286360\\ 5&1.1549407300050286300\\ 6&1.1549407300050286300\\ \end{array} $$
As @jjagmath answered, you can try to solve $$u=\cos(u)\qquad \text{where} \qquad u = 2x - \frac{\pi}{2}$$ First, you can have an approximation using the $1,400$ years old approximation $$\cos(u) \simeq\frac{\pi ^2-4u^2}{\pi ^2+u^2}\qquad \text{ for}\qquad -\frac \pi 2 \leq u\leq\frac \pi 2$$ and solve the cubic equation $$u^3+4 u^2+\pi ^2 u-\pi ^2=0$$ which has only one real root since $\Delta=256 \pi ^2-83 \pi ^4-4 \pi ^6 <0$. Using the hyperbolic method for this case $$u=-\frac{2}{3} \left(2+\sqrt{3 \pi ^2-16} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{128-63 \pi ^2}{2 \left(3 \pi ^2-16\right)^{3/2}}\right)\right)\right)\sim 0.738305$$ while the "exact" solution (Dottie number) is $0.739085$. This gives, as an approximation, $x=1.15455$.
