SAT Question f2495de4: Circles answer and explanation with Desmos solution

Choice C is correct. The cosine of an angle is equal to the cosine of $n(2\pi )$ radians more than the angle, where $n$ is an integer constant. Since $\frac{565\pi }{6}$ is equivalent to $47(2\pi )+\frac{\pi }{6}$, $\cos (\frac{565\pi }{6})$ can be rewritten as $\cos (47(2\pi )+\frac{\pi }{6})$, which is equal to $\cos (\frac{\pi }{6})$. Therefore, the value of $\cos (\frac{565\pi }{6})$ is equal to the value of $\cos (\frac{\pi }{6})$, which is $\frac{\sqrt{3}}{2}$.

Alternate approach: A trigonometric ratio can be found using the unit circle, that is, a circle with radius $1$ unit. The cosine of a number $t$ is the x-coordinate of the point resulting from traveling a distance of $t$ counterclockwise from the point $(1,0)$ around a unit circle centered at the origin in the xy-plane. A unit circle has a circumference of $2\pi$. It follows that since $\frac{565\pi }{6}$ is equal to $47(2\pi )+\frac{\pi }{6}$, traveling a distance of $\frac{565\pi }{6}$ counterclockwise around a unit circle means traveling around the circle completely $47$ times and then another $\frac{\pi }{6}$ beyond that. That is, traveling $\frac{565\pi }{6}$ results in the same point as traveling $\frac{\pi }{6}$. Traveling $\frac{\pi }{6}$ counterclockwise from the point $(1,0)$ around a unit circle centered at the origin in the xy-plane results in the point $(\frac{\sqrt{3}}{2},\frac{1}{2})$. Thus, the value of $\cos \frac{565\pi }{6}$ is the x-coordinate of the point $(\frac{\sqrt{3}}{2},\frac{1}{2})$, which is $\frac{\sqrt{3}}{2}$.