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Tips and answer to finding principal value
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X =\[\sin^{-1} (\sqrt{2}/2)\]
look at a cheat sheet if you do not remember which angle (number) has sine of \[\frac{\sqrt{2}}{2}\] you will see that it is \[\frac{\pi}{4}\] and although there are infinitely many others, this is the only one in the interval \[[-\frac{\pi}{2},\frac{\pi}{2}]\]
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