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Correct, we can use law of sines to fill the missing angles.

Law of cosines could as well give the missing angles, but it is longer, so we logically go for the shorter law of sines.

so once i get all the sides through the law of cosine, can i just go back to using the law of sines to find the rest?

Hi there! The shaded part represents the interval for which the inverse sine and inverse cosine exists.

Inverse cos exists in the interval $\left\lbrack0,\pi\right\rbrack$.

Inverse sin exists in the interval $\left\lbrack-\frac{\pi}{2},\frac{\pi}{2}\right\rbrack$.

This intervals were agreed on by mathematicians. We need to note the intervals so that we return an acceptable angle.