Angles

The difference between coterminal angles is an integral multiple of (360^\circ). Use this rule when the terminal side is the same.

Coterminal angles have the same initial and terminal sides, so their difference is an integral multiple of \(360^\circ\). Since \(30^\circ+360^\circ=390^\circ\), \(390^\circ\) is a positive coterminal angle of \(30^\circ\). The difference between \(300^\circ\) and \(30^\circ\) is \(270^\circ\), which is not a multiple of \(360^\circ\). Exam tip: use \(\theta+360^\circ n\), where \(n\) is an integer, to find coterminal angles.

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(45^\circ-360^\circ=-315^\circ\), \(-315^\circ\) is a negative coterminal angle of \(45^\circ\). The angle \(-45^\circ\) differs from \(45^\circ\) by \(90^\circ\), so it is not coterminal. Exam tip: subtract \(360^\circ\) from a given angle to obtain a negative coterminal angle.

(400^\circ-360^\circ=40^\circ). Subtract (360^\circ) from a large angle to get the principal angle.

Coterminal angles differ by an integral multiple of 360°. Here, \((-30^\circ+360^\circ)=330^\circ\), which lies between 0° and 360°. Although 300° is a nearby distractor, its difference from \(-30^\circ\) is 330°, not a multiple of 360°. Exam tip: To convert a negative angle into the 0° to 360° range, first add 360°.

(720^\circ \div 360^\circ=2) complete revolutions. Divide by (360^\circ) to count revolutions.

One half revolution is (180^\circ), and (540^\circ\div180^\circ=3). Divide by (180^\circ) for half revolutions.

In angular measurement, one degree is divided into 60 equal parts, and each part is called one minute. Therefore, \(1^\circ=60'\). Since \(30'\) equals half a degree, it is not correct. Exam tip: remember \(1^\circ=60'\) and \(1'=60''\) for degree-minute-second conversions.

\(315^\circ\) lies between \(270^\circ\) and \(360^\circ\), so its terminal side is in the fourth quadrant. \(225^\circ\) lies in the third quadrant. Exam tip: place an angle between \(0^\circ\) and \(360^\circ\) before identifying its quadrant.

Since (60'=1^\circ), (30'=\frac{1}{2}^\circ). Divide minutes by (60) to convert into degrees.

Since \(1'=60''\), convert \(90''\) into minutes by dividing by 60: \(90''\div 60=\frac{3}{2}'\). Therefore, \(\frac{3}{2}'\) is correct. Note that \(\frac{1}{2}'\) equals only \(30''\). Exam tip: divide by 60 when converting seconds to minutes.

(15^\circ \times \frac{\pi}{180^\circ}=\frac{\pi}{12}). Simplify (15) and (180) by (15).

Coterminal angles have the same initial and terminal sides, so their measures differ by a multiple of \(360^\circ\). Since \(405^\circ-45^\circ=360^\circ\), option C is correct. Exam tip: check the difference first.

(225^\circ \times \frac{\pi}{180^\circ}=\frac{5\pi}{4}). Remembering (45^\circ=\frac{\pi}{4}) also helps.

(\frac{5\pi}{6}\times\frac{180^\circ}{\pi}=150^\circ). Cancel (\pi) first and then multiply.

(\frac{7\pi}{6}\times\frac{180^\circ}{\pi}=210^\circ). Use (\frac{\pi}{6}=30^\circ).

(\frac{4\pi}{3}\times\frac{180^\circ}{\pi}=240^\circ). Since (\frac{\pi}{3}=60^\circ), four times it is (240^\circ).

(210^\circ) lies between (180^\circ) and (270^\circ), so it is in the third quadrant. Remember the boundary angles.

(315^\circ) lies between (270^\circ) and (360^\circ), so it is in the fourth quadrant. Decide the position using boundaries.

(100^\circ) lies between (90^\circ) and (180^\circ), so it is in the second quadrant. After (90^\circ) and before (180^\circ) is the second quadrant.