What are all the solutions to $\sin(x)^{\sin(x)} = 0.7$?

Question

I could only find $x = \sin^{-1}(e^{W(\ln 0.7)}) + 2πk$ and $x = \left(π -\sin^{-1}(e^{W(\ln 0.7)})\right) + 2πk$ where $k = (-1,0,1,2,3,4,5...)$, $W$ being the lambert $W$ function.

Basically I did:

1. $\sin(x)^{\sin(x)} = 0.7$
2. $\sin(x)\ln(\sin(x)) = \ln(0.7)$
3. $\ln(\sin(x))e^{\ln(\sin(x))} = \ln(0.7)$
4. $\ln(\sin(x)) = W(\ln(0.7))$
5. $\sin(x) = e^{W(\ln(0.7))}$

giving the solutions given above.

But the problem is that there are 2 more solutions that I can't find (total of 4 solutions for a $2π$ cycle). I'm just really interested in what I'm doing wrong here, was just learning about the lambert $W$ function.

Answer 1

There are two branches of the $W$ function that have real values, the $0$ branch and the $-1$ branch. Wikipedia has a plot of these together. (Exactly the same thing happens with $x^2 = 4$. The square root function gets the solution $x = 2$, but you have to know another solution is given by the "other branch" of the square root function, $x = -2$.)

Additionally, $\sin(x) = U$ has two solution families $$ x \in \{ \arcsin(U) + 2\pi k : k \in \Bbb{Z} \} $$ and $$ x \in \{ \pi - \arcsin(U) + 2\pi k : k \in \Bbb{Z} \} \text{.} $$

So, in the interval $[0,2\pi]$, the four solutions are \begin{align*} \arcsin \left( \mathrm{e}^{W_0(\ln 0.7)} \right) &= 0.4806{\dots} \\ \pi - \arcsin \left( \mathrm{e}^{W_0(\ln 0.7)} \right) &= 2.6609{\dots} \\ \arcsin \left( \mathrm{e}^{W_{-1}(\ln 0.7)} \right) &= 0.2847{\dots} \\ \pi - \arcsin \left( \mathrm{e}^{W_{-1}(\ln 0.7)} \right) &= 2.8568{\dots} \end{align*} This is all four solutions in the first period of sine.

(The plot does not cover $(\pi, 2\pi)$ because noninteger powers of negative numbers are only rarely real. Over most of this interval, $\sin^{\sin x}x$ is complex and not real.)

Answer 2

The Lambert W function $W(x)$ is defined as the inverse function of $f(x) = xe^x$. However, notice that this function is not actually injective, so it doesn’t have an inverse. That is, for some values of $W$, there exist $2$ solutions to the equation

$$xe^x = W$$

For example, if $W = -\ln 2 / 2$, then $x$ could be either $x=-\ln 2$ or $x=-2\ln 2$.

In your solution, you find yourself with the following equation:

$$\ln(\sin x) e^{\ln(\sin x)} = \ln (0.7)$$

From this, you cannot validly deduce that

$$\ln(\sin x) = W(\ln(0.7))$$

because there are actually two values of $y$ solving the equation $ye^y = \ln(0.7)$, and they are $y\approx -0.77$ and $y\approx -1.27$.