Angles

Arc (s=r\theta=8\times\frac{3\pi}{4}=6\pi), so perimeter is (2r+s=16+6\pi). In exams, do not forget to add two radii in sector perimeter.

(410^\circ-(-310^\circ)=720^\circ), which is a multiple of (360^\circ). In exams, coterminal angles differ by (360^\circ k).

(\frac{23\pi}{5}-4\pi=\frac{3\pi}{5}), which lies in the second quadrant. So the correct quadrant is the second quadrant.

The hour hand rotates (2\pi) in (12) hours, and (2) hours (40) minutes (=\frac{8}{3}) hours. The angle is (\frac{8}{3}\times\frac{\pi}{6}=\frac{4\pi}{9}).

(\frac{37\pi}{6}-6\pi=\frac{\pi}{6}), which lies in the first quadrant. In the first quadrant, the reference angle is the angle itself.

One revolution is (2\pi) radians, so (x=\frac{15\pi}{2}\div2\pi=\frac{15}{4}). In exams, divide radians by (2\pi).

(s=r\theta), so (r=\frac{10\pi}{\frac{2\pi}{3}}=15). In exams, multiply by the reciprocal while dividing by a fraction.

(72^\circ=\frac{72\pi}{180}=\frac{2\pi}{5}), so (k=2). In exams, always simplify the fraction.

(\frac{11\pi}{9}\times\frac{180^\circ}{\pi}=220^\circ), which lies in the third quadrant. In exams, converting to degrees can make quadrant identification easier.

(1560^\circ-1440^\circ=120^\circ), which lies in the second quadrant. The reference angle is (180^\circ-120^\circ=60^\circ).

(-\frac{25\pi}{6}+ \frac{36\pi}{6}=\frac{11\pi}{6}), which lies in the fourth quadrant. In exams, keep adding (2\pi) to a negative radian angle.

The difference is (\frac{14\pi}{7}=2\pi), so the angles are coterminal. In exams, match the difference with a multiple of (2\pi).

(245^\circ=\frac{245\pi}{180}=\frac{49\pi}{36}), so (m=49). In exams, make the denominator match the given form.

(\frac{17\pi}{12}) is in the third quadrant, so the reference angle is (\frac{17\pi}{12}-\pi=\frac{5\pi}{12}). In exams, use (\theta-\pi) in the third quadrant.

(\frac{4}{3}\times\frac{180^\circ}{\pi}\approx76.4^\circ), so the nearest value is (76^\circ). In exams, use (\pi\approx3.14) for approximation.

From (s=r\theta), (r=9), then (A=\frac{1}{2}r^2\theta=81). In exams, find the radius first.

(999^\circ-720^\circ=279^\circ), which lies in the fourth quadrant. In exams, decide the quadrant only after finding the principal angle.

In the second quadrant, the angle is \(\pi-\alpha\). Hence \(\pi-\frac{\pi}{9}=\frac{8\pi}{9}\).

(540^\circ-360^\circ=180^\circ). In exams, choose the principal angle according to the given interval.

(40^\circ=\frac{2\pi}{9}) and (s=21\times\frac{2\pi}{9}=\frac{14\pi}{3}). In exams, convert degrees into radians and apply (s=r\theta).