Angles

The difference (\frac{5\pi}{2}-2\theta) must be a multiple of (2\pi). In the given interval (\theta=\frac{\pi}{4}) is suitable.

For arc length, \(s=r\theta\), where \(\theta\) must be measured in radians. Here \(s=550\) and \(\theta=\frac{5\pi}{6}\). Hence, \(r=\frac{s}{\theta}=\frac{550}{5\pi/6}=\frac{550\times6}{5\pi}=\frac{660}{\pi}\) m. Therefore, option D is correct. The value \(\frac{630}{\pi}\) may result from an error while multiplying or simplifying the fractions. Exam tip: Apply \(s=r\theta\) directly only when the angle is in radians.

Degree measure is (x\cdot\frac{180}{\pi}). (x\cdot\frac{180}{\pi}=60x) is possible only for (x=0).

(220^\circ) is in the third quadrant so the reference angle is (40^\circ). (220^\circ=\frac{11\pi}{9}) radians.

Coterminal angles differ by an integral multiple of \(2\pi\). Add \(2\pi\) to the given negative angle: \(-\frac{7\pi}{18}+2\pi=-\frac{7\pi}{18}+\frac{36\pi}{18}=\frac{29\pi}{18}\). This value lies between \(0\) and \(2\pi\), so it is the principal positive angle. The difference between \(\frac{25\pi}{18}\) and \(-\frac{7\pi}{18}\) is \(\frac{16\pi}{9}\), which is not an integral multiple of \(2\pi\). Exam tip: to convert a negative angle into its principal positive form, add \(2\pi\) as needed.

The interior angle of a regular polygon is (\frac{(n-2)\pi}{n}). From (\frac{(n-2)\pi}{n}=\frac{5\pi}{6}) we get (n=12).

The arc difference is ((13-5)\theta=8\theta). From (8\theta=6\pi) we get (\theta=\frac{3\pi}{4}).

The net angle is (\frac{17\pi}{6}-\frac{5\pi}{3}=\frac{7\pi}{6}). Take counterclockwise as positive and clockwise as negative.

The clockwise angle is (-\frac{14\pi}{3}). Adding (6\pi) gives the principal angle (\frac{4\pi}{3}).

For equal radii sector area is proportional to angle. The larger angle is (75^\circ\times\frac{8}{5}=120^\circ=\frac{2\pi}{3}).

(112^\circ 30') means (112.5^\circ), so the value is (5\pi/8). In exams, first convert minutes into degrees.

Multiplying radians by (180^\circ/\pi) gives (105^\circ). In exams, cancel (\pi) directly.

Adding (360^\circ) three times to (-725^\circ) gives (355^\circ). In exams, keep the answer between (0^\circ) and (360^\circ).

One complete revolution is (2\pi) radians, so the measure is (\frac{11}{6}\times2\pi=\frac{11\pi}{3}). In exams, take a full turn as (2\pi).

Multiplying (-\frac{17\pi}{18}) by (180^\circ/\pi) gives (-170^\circ). The negative sign shows direction.

The coterminal angle of (1234^\circ) is (154^\circ), which lies in the second quadrant. In exams, first find the remainder after division by (360^\circ).

Adding (2\pi) five times to (-\frac{29\pi}{6}) gives (\frac{\pi}{6}). Always keep the principal angle in the given interval.

Using (s=r\theta), (\theta=\frac{35}{14}=\frac{5}{2}) radians. In exams, divide arc length by radius.

(135^\circ=\frac{3\pi}{4}) and (s=8\times\frac{3\pi}{4}=6\pi). In exams, first convert degrees into radians.

Coterminal angles must differ by an integral multiple of (2\pi). The difference for (-\frac{7\pi}{4}) does not satisfy this.