Point 10 — Unit circles with complicated angles

What do we see?

Complicated angles on a unit circle.

What do we understand?

x = rcosθ and y = rsinθ. Reduce the angle modulo 2π before evaluating.

Question 1 of 2

Point N lies on a unit circle in the xy-plane and has coordinates (1, 0). Point O is the center of the circle and has coordinates (0, 0). Point M also lies on the unit circle, and the measure of angle NOM is $\frac{562 π}{3}$ radians. If the coordinates of point M are (a, b), where a and b are constants, what is the value of a?