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

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 $left parenthesis 1 comma 0 right parenthesis$ 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 $StartFraction 565 pi Over 6 EndFraction$ is equal to $47 (2 π) + \frac{π}{6}$ , traveling a distance of $StartFraction 565 pi Over 6 EndFraction$ counterclockwise around a unit circle means traveling around the circle completely $47$ times and then another $StartFraction pi Over 6 EndFraction$ beyond that. That is, traveling $StartFraction 565 pi Over 6 EndFraction$ results in the same point as traveling $StartFraction pi Over 6 EndFraction$ . Traveling $StartFraction pi Over 6 EndFraction$ counterclockwise from the point $left parenthesis 1 comma 0 right parenthesis$ around a unit circle centered at the origin in the xy-plane results in the point $left parenthesis StartFraction StartRoot 3 EndRoot Over 2 EndFraction comma one half right parenthesis$ . Thus, the value of $cosine StartFraction 565 pi Over 6 EndFraction$ is the x-coordinate of the point $left parenthesis StartFraction StartRoot 3 EndRoot Over 2 EndFraction comma one half right parenthesis$ , which is $StartFraction StartRoot 3 EndRoot Over 2 EndFraction$ .

Choice A is incorrect. This is the value of $sine StartFraction 565 pi Over 6 EndFraction$ , not $cosine StartFraction 565 pi Over 6 EndFraction$ .

Choice B is incorrect. This is the value of the cosine of a multiple of $2 pi$ , not $StartFraction 565 pi Over 6 EndFraction$ .

Choice D is incorrect. This is the value of $StartStartFraction 1 OverOver tangent StartFraction 565 pi Over 6 EndFraction EndEndFraction$ , not $cosine StartFraction 565 pi Over 6 EndFraction$ .