<s>[INST] 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:
$\sin(2x) = 2 \sin(x) \cos(x)$
 [/INST] Since $\cos 2x=\pm\frac{7}{25}$, $\tan x=\frac{\sin 2x}{1+\cos 2x}\in\left\{\frac{24}{32},\,\frac{24}{18}\right\}=\left\{\frac{3}{4},\,\frac{4}{3}\right\}$ so $\sin x=\pm\frac{\tan x}{\sqrt{1+\tan^2 x}}\in\pm\left\{\frac{3}{5},\,\frac{4}{5}\right\}=\left\{-\frac{4}{5},\,-\frac{3}{5},\,\frac{3}{5},\,\frac{4}{5}\right\}$. 
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