Angles

Correct answer: B. (\frac{5\pi}{6})

Adding (6\pi) to (-\frac{31\pi}{6}) gives (\frac{5\pi}{6}). In exams, add multiples of (2\pi) to negative angles.

Correct answer: D. (165^\circ), second quadrant

(1245^\circ-1080^\circ=165^\circ), so the angle lies in the second quadrant. In exams, first subtract multiples of (360^\circ).

Correct answer: A. (\frac{5\pi}{48})

(18^\circ45'=\frac{75}{4}^\circ) and the radian measure is (\frac{75\pi}{720}=\frac{5\pi}{48}). In exams, first convert minutes into degrees.

Correct answer: C. (67.5^\circ)

From (s=r\theta), (\theta=\frac{3\pi}{8}), which is (67.5^\circ). In exams, keep the angle in radians in the arc formula.

Correct answer: D. (270^\circ)

(75\pi=\frac{1}{2}\times100\times\theta), so (\theta=\frac{3\pi}{2}=270^\circ). In exams, (\theta) is in radians in the area formula.

Correct answer: A. (260^\circ)

(-1540^\circ+1800^\circ=260^\circ), so the principal coterminal angle is (260^\circ). In exams, add multiples of (360^\circ) to negative angles.

Correct answer: C. (\frac{59\pi}{4})

(7) revolutions are (14\pi) and (135^\circ=\frac{3\pi}{4}), so the total is (\frac{59\pi}{4}). In exams, convert revolutions and extra angle separately.

Correct answer: B. (\frac{29\pi}{30})

The minute hand rotates (2\pi) in (60) minutes, so in (29) minutes it rotates (\frac{29\pi}{30}). In exams, use the time ratio.

Correct answer: D. (\frac{14\pi}{15})

(5) hours (36) minutes (=\frac{28}{5}) hours and the hour hand rate is (\frac{\pi}{6}) radians per hour. Hence the angle is (\frac{28}{5}\times\frac{\pi}{6}=\frac{14\pi}{15}).

Correct answer: A. (\frac{\pi}{7})

The principal angle of (\frac{41\pi}{7}) is (\frac{13\pi}{7}), which lies in the fourth quadrant. The reference angle is (2\pi-\frac{13\pi}{7}=\frac{\pi}{7}).

Correct answer: C. (\frac{13\pi}{18})

In the second quadrant, the principal angle is (\pi-\alpha). Hence (\pi-\frac{5\pi}{18}=\frac{13\pi}{18}).

Correct answer: D. (250^\circ)

The principal angle of (-\theta) is (110^\circ), so the principal angle of (\theta) is (-110^\circ+360^\circ=250^\circ). In exams, adjust with (360^\circ) after changing the sign.

Correct answer: B. (4)

One complete revolution is (360^\circ) and (\frac{1440^\circ}{360^\circ}=4). In exams, divide the difference by (360^\circ).

Correct answer: A. (73.34^\circ)

(20'=\frac{1}{3}^\circ) and (24''=\frac{1}{150}^\circ), so the total is (73.34^\circ). In exams, convert minutes and seconds separately into degrees.

Correct answer: C. Second quadrant

Subtracting (8\pi) from (\frac{107\pi}{12}) gives (\frac{11\pi}{12}). (\frac{11\pi}{12}) lies in the second quadrant.

Correct answer: D. (\frac{\pi}{6})

The smaller counterclockwise angle is (2\pi-\frac{11\pi}{6}=\frac{\pi}{6}). In exams, subtract the larger part from one full rotation.

Correct answer: A. (32) square cm

From (s=r\theta), (r=4), then (A=\frac{1}{2}r^2\theta=32). In exams, first find the radius and then apply the area formula.

Correct answer: C. (14)

(126^\circ=\frac{126\pi}{180}=\frac{7\pi}{10}=\frac{14\pi}{20}), so (m=14). In exams, make the denominator match the given form.

Correct answer: D. (15)

(300^\circ=\frac{5\pi}{3}=\frac{15\pi}{9}), so (p=15). In exams, convert degrees to radians and then match denominators.

Correct answer: B. (\frac{17\pi}{10})

(-\frac{83\pi}{10}+10\pi=\frac{17\pi}{10}). In exams, add a suitable large multiple of (2\pi) to bring the angle into the given interval.

Correct answer: A. (28\pi)

The wheel circumference is (70\pi=220) cm, so it makes (14) revolutions. The total angle is (14\times2\pi=28\pi) radians.

Correct answer: C. (90\pi) square cm

(144^\circ=\frac{4\pi}{5}) and (A=\frac{1}{2}\times225\times\frac{4\pi}{5}=90\pi). In exams, converting degrees into radians is essential.

Correct answer: D. (96^\circ)

The original angle is (\pi-\frac{7\pi}{15}=\frac{8\pi}{15}), which equals (96^\circ). In exams, keep the sum of supplementary angles as (180^\circ).

Correct answer: B. (52^\circ12')

The complementary angle is (90^\circ-37^\circ48'=52^\circ12'). In exams, remember (1^\circ=60') while borrowing.

Correct answer: C. Second quadrant

Since (\frac{\pi}{2}<\frac{9}{5}<\pi), the angle lies in the second quadrant. In exams, compare with (\frac{\pi}{2}\approx1.57) and (\pi\approx3.14).