Angles

Correct answer: C. (8\pi) cm

(72^\circ=\frac{2\pi}{5}) and (s=20\times \frac{2\pi}{5}=8\pi) cm. Convert degrees to radians before finding the arc.

Correct answer: D. 22 cm

When the angle is in radians, the arc-length formula is \(s=r\theta\). Thus, \(r=\frac{s}{\theta}=\frac{11\pi}{\pi/2}=11\pi\times\frac{2}{\pi}=22\) cm. The option \(22\pi\) cm is incorrect because the \(\pi\) terms cancel. Exam tip: Use this formula directly only when the angle is measured in radians.

Correct answer: A. (15\pi) square cm

Area is ( \frac{1}{2}r^2\theta=\frac{1}{2}\times100\times\frac{3\pi}{10}=15\pi ). Use this formula directly when the angle is in radians.

Correct answer: B. (18\pi) square cm

(45^\circ=\frac{\pi}{4}) and area is ( \frac{1}{2}\times144\times\frac{\pi}{4}=18\pi ). Convert the degree angle into radians first.

Correct answer: B. (2) radians

(25=\frac{1}{2}\times25\times\theta), so ( \theta=2 ) radians. Isolate the unknown angle in the area formula.

Correct answer: C. (8) cm

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.

Correct answer: B. (60^\circ)

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

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

( \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 ).

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

( \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 ).

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

( \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).

Correct answer: D. (30^\circ)

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

Correct answer: A. (30^\circ)

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

Correct answer: B. They are coterminal angles

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

Correct answer: D. (90^\circ)

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

Correct answer: C. ( \frac{\pi}{24} ) radians

(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.

Correct answer: A. ( \frac{\pi}{10} ) radians

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

Correct answer: B. ( \frac{3\pi}{10} ) radians

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

Correct answer: D. ( \frac{11\pi}{10} ) radians

(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).

Correct answer: A. ( \frac{7\pi}{5} ) radians

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

Correct answer: B. (-\frac{7\pi}{4}) radians

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

Correct answer: C. ( \frac{3\pi}{16} ) radians

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

Correct answer: D. ( \frac{\pi}{80} ) radians

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

Correct answer: B. (130^\circ)

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

Correct answer: C. (132^\circ)

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

Correct answer: D. (-202.5^\circ)

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