10.
Given $2 \cos ^{2} x+3 \sin x-3=0, x \in\left[0, \frac{\pi}{2}\right]$
What are the exact values of $x ? \mathrm{An}$ algebralc solution is required.

Step 1 ：Given $2 \cos ^{2} x+3 \sin x-3=0, x \in\left[0, \frac{\pi}{2}\right]$, we need to find the exact values of $x$ that satisfy the given equation in the interval $\left[0, \frac{\pi}{2}\right]$.

Step 2 ：Using the Pythagorean identity, we can rewrite the given equation as: $2(1 - \sin^2 x) + 3 \sin x - 3 = 0$

Step 3 ：Solving the quadratic equation for $\sin x$, we get the solutions $\sin x = \frac{1}{2}$ and $\sin x = 1$

Step 4 ：Finding the corresponding values of $x$ in the interval $\left[0, \frac{\pi}{2}\right]$, we get $x = \frac{\pi}{6}$ and $x = \frac{\pi}{2}$

Step 5 ：\(\boxed{x = \frac{\pi}{6}, \frac{\pi}{2}}\)