Solve trigonometric equation $2x=\sin(2x)+\pi/2$

Question

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!

Answer 1

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.

Answer 2

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.

$$ \begin{array}{r|l} n&\text{value}\\\hline 0&1\\ 1&1.1695070529519190789\\ 2&1.1550315982863453148\\ 3&1.1549407336507333597\\ 4&1.1549407300050286360\\ 5&1.1549407300050286300\\ 6&1.1549407300050286300\\ \end{array} $$

Answer 3

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, $x=1.15455$.