Angles

( \frac{11\pi}{15}\times\frac{180^\circ}{\pi}=132^\circ). Calculate (180^\circ\div15=12^\circ) and multiply.

(-\frac{9\pi}{8}\times\frac{180^\circ}{\pi}=-202.5^\circ). The negative sign in radians remains in degrees.

( \frac{37\pi}{12}-\frac{24\pi}{12}=\frac{13\pi}{12} ). Subtract multiples of (2\pi) in radians.

(-\frac{41\pi}{10}+\frac{60\pi}{10}=\frac{19\pi}{10}). Add enough multiples of (2\pi) to a negative angle.

( \frac{52\pi}{9}-\frac{36\pi}{9}=\frac{16\pi}{9} ). Use (2\pi=\frac{18\pi}{9}) while subtracting.

(-\frac{29\pi}{5}+\frac{30\pi}{5}=\frac{\pi}{5}). Keeping the same denominator makes radian calculation easy.

To obtain the principal coterminal angle, subtract a multiple of \(360^\circ\) so that the result lies between \(0^\circ\) and \(360^\circ\). \(2210^\circ-6\times360^\circ=2210^\circ-2160^\circ=50^\circ\). Hence, \(50^\circ\) is correct. \(40^\circ\) is not coterminal with \(2210^\circ\), since their difference is not an integral multiple of \(360^\circ\). Exam tip: divide the angle by \(360^\circ\) and use the remainder.

(-1855^\circ+2160^\circ=305^\circ). Add a suitable multiple of (360^\circ) to a large negative angle.

(1485^\circ-45^\circ=1440^\circ=4\times360^\circ), so they are coterminal. If the difference is a multiple of (360^\circ), consider them coterminal.

(-640^\circ+720^\circ=80^\circ), and (80^\circ) lies in the first quadrant. First convert a negative angle into a positive coterminal angle.

( \frac{3\pi}{2}<\frac{23\pi}{14}<2\pi ), so it lies in the fourth quadrant. Compare radian limits with a common denominator.

( \frac{\pi}{2}<\frac{6\pi}{11}<\pi ), so the terminal side lies in the second quadrant. The interval between ( \frac{\pi}{2} ) and ( \pi ) is the second quadrant.

( \pi<\frac{15\pi}{11}<\frac{3\pi}{2} ), so it lies in the third quadrant. After ( \pi ) and before ( \frac{3\pi}{2} ) is the third quadrant.

( -\frac{5\pi}{8}+2\pi=\frac{11\pi}{8} ), and it lies in the third quadrant. Add (2\pi) to a negative radian angle to check its position.

(30'=\frac{30}{60}^\circ=0.5^\circ), so the value is (72.5^\circ). Do not write minutes directly as decimals.

(12'=\frac{12}{60}^\circ=0.2^\circ), so (58^\circ12'=58.2^\circ). Divide minutes by (60).

(0.4^\circ\times60'=24'), so (84.4^\circ=84^\circ24'). Multiply the decimal part by (60).

(0.125^\circ\times60'=7.5'), and (0.5'\times60''=30''). Convert the decimal part step by step into minutes and seconds.

(27^\circ=97200''), and (18'=1080''), so the total is (98316''). Remember (1^\circ=3600'') and (1'=60'').

Since \(1^\circ=3600''\), \(7395''=2\times3600''+195''\). Now, \(195''=3\times60''+15''=3'15''\). Therefore, \(7395''=2^\circ3'15''\), so option B is correct. Option C has an incorrect conversion of the remaining seconds into minutes and seconds. Exam tip: first divide total seconds by 3600 for degrees, then divide the remainder by 60 for minutes.