Angles

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

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.

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

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

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

Sector perimeter is (2r+r\theta), so (14+7\theta=22). This gives (\theta=\frac{8}{7}).

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.

(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.

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

(-\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.

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

(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.

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).

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

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).

In the third quadrant, the angle is (\pi+\alpha). Hence (\pi+\frac{2\pi}{11}=\frac{13\pi}{11}).

(-725^\circ+1080^\circ=355^\circ). In exams, keep adding multiples of (360^\circ) to a negative angle.

From (A=\frac{1}{2}rs), (r=8), then (\theta=\frac{s}{r}=\frac{25}{8}). In exams, find the radius first from arc length and area.

(15^\circ) per second gives (900^\circ) per minute, and (900^\circ=5\pi) radians. In exams, convert the time unit first.