Angles

(225^\circ=\frac{5\pi}{4}) and (r=\frac{s}{\theta}=\frac{25\pi}{5\pi/4}=20). Convert the angle to radians first.

(144^\circ=\frac{4\pi}{5}) and (40\pi=\frac{1}{2}r^2\cdot\frac{4\pi}{5}) gives (r=10). Keep the angle in radians in the area formula.

(2\theta=\frac{22\pi}{9}) and ( \frac{22\pi}{9}-2\pi=\frac{4\pi}{9} ). After multiplying, subtract (2\pi).

( \alpha+\beta=\frac{20\pi}{10}=2\pi ), which is coterminal with (0). In principal angle form, (2\pi) is taken as (0).

(875^\circ-720^\circ=155^\circ), and the reference angle is (180^\circ-155^\circ=25^\circ). Find the principal angle before the reference angle.

( -1190^\circ+1440^\circ=250^\circ ), and the reference angle is (250^\circ-180^\circ=70^\circ).

( -\theta=-\frac{5\pi}{4} ), and ( -\frac{5\pi}{4}+2\pi=\frac{3\pi}{4} ). Add (2\pi) to a negative angle.

To find a principal angle, subtract multiples of \(360^\circ\) from the given angle. Since \(635^\circ-360^\circ=275^\circ\), the principal angle of \(635^\circ\) is \(275^\circ\). In contrast, \(725^\circ-720^\circ=5^\circ\), so it is not correct. Exam tip: Coterminal angles differ by an integral multiple of \(360^\circ\).

Subtracting three complete revolutions, where \(2\pi=\frac{12\pi}{6}\), gives \(\frac{41\pi}{6}-\frac{36\pi}{6}=\frac{5\pi}{6}\). Now \(\frac{5\pi}{6}=150^\circ\), which lies between \(\frac{\pi}{2}\) and \(\pi\); hence its terminal side lies in the second quadrant. The fourth quadrant contains angles between \(\frac{3\pi}{2}\) and \(2\pi\). Exam tip: reduce a large angle by subtracting multiples of \(2\pi\) before identifying its quadrant.

( \theta-\pi=\frac{14\pi}{5}-\frac{5\pi}{5}=\frac{9\pi}{5} ), and it is in (0) to (2\pi). First subtract and then check the range.

(3\theta=\frac{15\pi}{6}=\frac{5\pi}{2}), and its principal angle is ( \frac{\pi}{2} ). Hence the terminal side lies on the positive (y)-axis.

( \frac{7\pi}{18}\times\frac{180^\circ}{\pi}=70^\circ ). Multiply by ( \frac{180^\circ}{\pi} ) to convert radians to degrees.

One complete revolution measures \(2\pi\) radians. Hence, adding or subtracting \(2n\pi\) from \(\theta\) leaves its terminal arm unchanged. Adding \(n\pi\) generally gives the opposite arm. Exam tip: use \(2\pi\) for coterminal angles.

Coterminal angles have the same terminal side because they differ by an integral multiple of \(2\pi\). Here, \(\frac{9\pi}{4}-2\pi=\frac{9\pi}{4}-\frac{8\pi}{4}=\frac{\pi}{4}\). Therefore, the terminal side is coterminal with \(\frac{\pi}{4}\). Although \(\frac{\pi}{2}\) may seem close, it represents \(90^\circ\), whereas \(\frac{\pi}{4}=45^\circ\). Exam tip: reduce an angle to the interval \([0,2\pi)\) by adding or subtracting \(2\pi\) as needed.

In the third quadrant the principal angle is (180^\circ+35^\circ=215^\circ). Add or subtract the reference angle according to the quadrant.

In the fourth quadrant ( \theta=2\pi-\frac{\pi}{9}=\frac{17\pi}{9} ). A full revolution in radians is (2\pi).

( \frac{7\pi}{9}=140^\circ ), which is greater than (90^\circ), so no positive complementary angle is possible. First make the units same.

( \frac{11\pi}{6}=330^\circ ), and the other angle is (330^\circ-275^\circ=55^\circ). First make the units of both angles the same.

Subtracting one full revolution, \(765^\circ-720^\circ=45^\circ\). Since \(45^\circ\) lies between \(0^\circ\) and \(90^\circ\), the terminal side is in the first quadrant. The positive \(y\)-axis would require an angle of \(90^\circ\), or a coterminal angle. Exam tip: reduce any angle by multiples of \(360^\circ\) before identifying its quadrant or axis.

The arc-length formula is \(s=r\theta\), where \(\theta\) must be in radians. Thus, \(r=\frac{s}{\theta}=\frac{12\pi}{3\pi/4}=12\pi\times\frac{4}{3\pi}=16\) cm. Therefore, option C is correct. An error such as choosing 12 cm can result from dividing incorrectly by \(\frac{3\pi}{4}\). Exam tip: In arc-length questions, first check that the angle is expressed in radians.