How to use double angle identities to find $\sin x$ and $\cos x$ from $\sin 2x $? If $\sin 2x =\frac{5}{13}$ and $0^\circ < x < 45^\circ$, find $\sin x$ and $\cos x$.
The answers should be $\frac{\sqrt{26}}{26}$ and $\frac{5\sqrt{26}}{26}$
Ideas
The idea is to  use double angle identities. One such identity is $\sin 2x=2\sin x\cos x$.
It's easy to use it to find $\sin 2x$ from known $\sin x$ and $\cos x$. But here it's the other way around.

Suppose we had a right triangle with an angle $2x$, and $\sin2x=\frac{5}{13}$. Further suppose that the hypotenuse of the triangle was 13. We can deduce that the side oppoites $2x$ must be 5. Applying the Pythagorean theorem to find the other side we have
$$13^2=5^2+(\text{adjacent side})^2$$
$$169-25=(\text{adjacent side})^2$$
$$144=(\text{adjacent side})^2$$
implying that the side opposite angle $2x$ is $12$. This allows us to state that
$$\cos 2x=\frac{12}{13}$$
Which is easier to work with because
$$\cos2x=\cos^2x-\sin^2x=2\cos^2x-1$$
Substituting we have
$$\frac{12}{13}=2\cos^2x-1$$
$$\cos^2x=\frac{25}{26}$$
$$\cos x=\frac{5}{\sqrt{26}}=\frac{5\sqrt{26}}{26}$$
Applying the Pythagorean identity we have
$$\cos^2x+\sin^2x=1$$
$$\frac{25}{26}+\sin^2x=1$$
$$\sin^2x=\frac{1}{26}$$
$$\sin x=\frac{1}{\sqrt{26}}=\frac{\sqrt{26}}{26}$$
