How to get the exact value of $\sin(x)$ if $\sin(2x) = \frac{24}{25}$ How to get the exact value of $\sin(x)$ if $\sin(2x) = \frac{24}{25}$ ?
I checked various trigonometric identities, but I am unable to derive $\sin(x)$ based on the given information.
For instance:
$\sin(2x) = 2 \sin(x) \cos(x)$

Refer to the diagram below.
$AD$ is the angle bisector of the right-angled triangle $\Delta ABC$.
Given $BC=24$ and $AC=25$.
Let $\angle DAB = x$.
From $\Delta ABC$ we see that $\sin(2x) = \dfrac{24}{25}$.
Now, by angle bisector theorem, $BD:DC = 7:25$.
Therefore, $BD = \dfrac{7}{7+25} \times 24 = \dfrac{21}4$.
Observing that $AB:BD = 4:3$, we see that $AD = \dfrac{35}4$
Therefore, $\sin x = \dfrac {BD} {DA} = \dfrac 3 5$.
The other possible $x$ would be $x+180^\circ$, since the period of sine is $360^\circ$, making $\sin x = -\dfrac35$.
In conclusion, $\sin x = \pm \dfrac 35$.

