A rotating ray, starting from a certain position, rotates two full turns in the counterclockwise direction (opposite to the clock hands) and then produces an additional angle of 30°. What are the sexagesimal (degree) and circular (radian) measures of this angle?

Q.A rotating ray, starting from a certain position, rotates two full turns in the counterclockwise direction (opposite to the clock hands) and then produces an additional angle of 30°. What are the sexagesimal (degree) and circular (radian) measures of this angle?

Since the ray is rotating in the counterclockwise direction (opposite to the clock hands), \[ \therefore \text{The angle will be positive.} \] A single complete rotation of the ray produces an angle of \(360^\circ\). \[ \therefore \text{Two complete rotations will produce } 2 \times 360^\circ = 720^\circ \] After completing two full rotations, the ray rotates an additional \(30^\circ\). So, in sexagesimal (degree) measure, the total angle is: \[ 720^\circ + 30^\circ = 750^\circ \] Now, since \(180^\circ = \pi\) radians, \[ 750^\circ = \frac{750\pi}{180} = \frac{25\pi}{6} = 4\frac{1}{6}\pi \text{ radians} \]