Angles

Sector area is ( \frac{1}{2}rs ), so (64=\frac{1}{2}\times r\times16) gives (r=8). When arc length is given, ( \frac{1}{2}rs ) is very useful.

( \theta=\frac{s}{r}=\frac{6\pi}{18}=\frac{\pi}{3}=60^\circ ). First find the radian angle and then convert it into degrees.

( \frac{5\pi}{6} ) lies in the second quadrant and the reference angle is ( \pi-\frac{5\pi}{6}=\frac{\pi}{6} ). In the second quadrant use ( \pi-\theta ).

( \frac{7\pi}{6} ) lies in the third quadrant and ( \frac{7\pi}{6}-\pi=\frac{\pi}{6} ). In the third quadrant use ( \theta-\pi ).

( \frac{11\pi}{6} ) lies in the fourth quadrant and (2\pi-\frac{11\pi}{6}=\frac{\pi}{6}). In the fourth quadrant use (2\pi-\theta).

(210^\circ) lies in the third quadrant and (210^\circ-180^\circ=30^\circ). In the third quadrant subtract (180^\circ).

(330^\circ) lies in the fourth quadrant and (360^\circ-330^\circ=30^\circ). In the fourth quadrant use (360^\circ-\theta).

(135^\circ-(-225^\circ)=360^\circ), so they are coterminal angles. If the difference is a multiple of (360^\circ), the angles are coterminal.

The terminal side lies on the positive (y)-axis at (90^\circ). Axis angles are not placed in quadrants.

(30'=\frac{1}{2}^\circ) so (7^\circ 30'=7.5^\circ) and (7.5^\circ\times\frac{\pi}{180}=\frac{\pi}{24}). Convert minutes into degrees first.

(18^\circ=\frac{18\pi}{180}=\frac{\pi}{10}). Multiply by ( \frac{\pi}{180} ) to convert degrees to radians.

(54^\circ=\frac{54\pi}{180}=\frac{3\pi}{10}). It is important to write the fraction in simplest form.

(126^\circ=\frac{126\pi}{180}=\frac{7\pi}{10}). Dividing by (18) makes simplification faster.

(198^\circ=\frac{198\pi}{180}=\frac{11\pi}{10}). An angle slightly greater than (180^\circ) gives a radian value slightly greater than (\pi).

(252^\circ=\frac{252\pi}{180}=\frac{7\pi}{5}). Divide the degree measure by (180) and attach (\pi).

(-315^\circ=\frac{-315\pi}{180}=-\frac{7\pi}{4}). Keep the negative sign until the final answer.

(33^\circ45'=33.75^\circ=\frac{135^\circ}{4}) and the radian value is ( \frac{3\pi}{16} ). Convert minutes into degrees first.

(2^\circ15'=2.25^\circ) and (2.25^\circ\times \frac{\pi}{180}=\frac{\pi}{80}). Convert small degree measures carefully into decimals.

\( \frac{\pi}{20}\times\frac{180^\circ}{\pi}=9^\circ\). Divide \(180^\circ\) by the denominator.

( \frac{13\pi}{18}\times\frac{180^\circ}{\pi}=130^\circ). If the denominator is (18), take (180^\circ\div18=10^\circ).