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:
$\sin(2x) = 2 \sin(x) \cos(x)$

$$\sin(2x)=\frac{24}{25}$$
The identities you want are:
$$\sin(2x)=2\sin(x)\cos(x) \space [I]$$
and
$$\sin^2(x)+\cos^2(x)=1\space[II]$$
Square $[I]$ to get:
$$4\sin^2(x)\cos^2(x)=\frac{576}{625}\to\sin^2x(1-\sin^2x)=\frac{144}{625}$$
$$\to\sin^4(x)-\sin^2x+\frac{144}{625}=0$$
Let $\sin^2(x)=\sigma$. Your equation becomes:
$\sigma^2-\sigma+\frac{144}{625}=0$
Solve via Quadratic Formula:
$$\sigma=\frac{1\pm\sqrt{1-\frac{576}{625}}}{2}=\frac 12\pm\frac{\sqrt{\frac{49}{625}}}{2}=\frac{9}{25}, \frac{16}{25}.$$
Since $\sin^2(x)=\frac{9}{25}, \frac{16}{25}, \sin(x)=\pm \frac 35, \pm \frac 45$
Check these values with the function $f(S)=\sin(2\cdot\sin^{-1}(S))$, (where $S$ is the solutions i.e. $\pm\frac 35, \pm \frac 45$) to see the ones which provide the correct solution, and note the values which work are $\frac 35, \frac 45$.
We can see them graphed here:
