Find two negative and three positive angles, expressed in radians, for which the point on the unit circle that corresponds to each angle is $\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$.
Answer
The three positive angles are \(\boxed{\frac{10\pi}{3}, \frac{16\pi}{3}, \frac{22\pi}{3}}\)
Step 1 ：The point \(\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\) on the unit circle corresponds to an angle of 240 degrees or \(\frac{4\pi}{3}\) radians in standard position.
Step 2 ：To find two negative angles, we can subtract multiples of \(2\pi\) (the period of the unit circle) from \(\frac{4\pi}{3}\).
Step 3 ：To find three positive angles, we can add multiples of \(2\pi\) to \(\frac{4\pi}{3}\).
Step 4 ：The two negative angles are \(\boxed{-\frac{2\pi}{3}, -\frac{8\pi}{3}}\)
Step 5 ：The three positive angles are \(\boxed{\frac{10\pi}{3}, \frac{16\pi}{3}, \frac{22\pi}{3}}\)
