1. **Problem statement:** We have a circle with center at point $O(0,0)$ and a point $N(1,0)$ on the circle. Point $M(a,b)$ is also on the circle. The angle $\angle NOM$ measures $\frac{218\pi}{3}$ radians. Given that $b$ is constant, find the value of $a$. 2. **Key information:** - Center $O$ at $(0,0)$ - Point $N$ on circle at $(1,0)$ - Radius $r$ is distance $ON = 1$ - Angle $\angle NOM = \frac{218\pi}{3}$ radians - Point $M(a,b)$ lies on the circle with radius $1$ - $b$ is constant 3. **Formula and approach:** The coordinates of a point on a circle of radius $r$ centered at the origin with angle $\theta$ from the positive x-axis are: $$ x = r \cos \theta, \quad y = r \sin \theta $$ Here, $r=1$, so: $$ a = \cos \theta, \quad b = \sin \theta $$ 4. **Simplify the angle:** Since angles on a circle are periodic with period $2\pi$, reduce $\frac{218\pi}{3}$ modulo $2\pi$: $$ \frac{218\pi}{3} = 72\pi + \frac{2\pi}{3} $$ Because $72\pi$ is a multiple of $2\pi$ (since $72\pi = 36 \times 2\pi$), the effective angle is: $$ \theta = \frac{2\pi}{3} $$ 5. **Calculate $a$ and $b$:** $$ a = \cos \frac{2\pi}{3} = -\frac{1}{2} = -0.5 $$ $$ b = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \approx 0.866 $$ 6. **Since $b$ is constant, $b = \frac{\sqrt{3}}{2}$, so $a = -0.5$.** **Final answer:** $a = -0.5$ **Correct choice:** c. -0.5