I was doing my homework and was stuck on a problem for a really long time, and eventually checked the answer key to try and see if I could understand the solution. This is the first part but I still don't understand what it is saying.

For this question, we're going to use our filled in unit circle:

By definition, answers the question "What is the angle between \(-\pi/2\) and \(\pi/2\) with a sine of \(\frac{-\sqrt{3}}{2}\)?" Since the sine is the vertical coordinate of the corresponding terminal point, we see that the only such angle in the interval is \(-\pi/3\) which means that \(\arcsin\left(-\frac{\sqrt{3}}{2}\right)= -\frac{\pi}{3}\)

Can someone explain to me how this works? Because aren't there two intervals where the angle is? Sorry if this doesn't make sense. All help is greatly appreciated.

roomtempiq Jan 4, 2024