the solution to minimizing a sine function in the domain $x^2 \le3$

I'm reading a book where an example looks at the following optimization problem:

$$\min_{x} \sin(x)$$ s.t. $$x^2 \le 3 $$

When I write out the Lagrangian:

$$L = sin(x) + \mu(x^2-3) $$, where $\mu$ is the Lagrange multiplier and is $\ge 0$. Taking the partial derivative wrt $x$ : $cos(x)+2\mu x$ and using the KKT-conditions, I get two possible solutions for $x$, namely {$ \approx-1.57, \approx 1.73$}. Checking both solutions shows the global optimum lays at $x=-1.57$ with a value of $-1$.

If I verify this via WolframAlpha, I seem to be correct: https://www.wolframalpha.com/input/?i=minimize+sin%28x%29+on+x%5E2%3C%3D3

But in the book it says that the optimum lays at $x = -\sqrt{3}$.

What am I missing?

We have $-1 \le \sin x \le 1$ for all $x$. Since $ \pi/2 < \sqrt{3}$ and $\sin (-\pi/2)=-1$, we get

$$ \min\{ \sin x: x^2 \le 3\}= -1.$$

Your book is wrong.