Angles

To convert degrees to radians multiply by ( \frac{\pi}{180} ). The same rule applies to angles like (75^\circ).

In angular measurement, 1 degree is divided into 60 minutes; that is, \(1^\circ=60'\). Therefore, the correct answer is 60'. Note that 90' equals \(1.5^\circ\), so it is not correct. Exam tip: Remember \(1^\circ=60'\) and \(1'=60''\).

In angular measurement, 1 minute is divided into 60 seconds. Therefore, \(1' = 60''\), so 60'' is correct. 30'' represents half a minute, not one full minute. Exam tip: Remember that \(1^\circ = 60'\) and \(1' = 60''\).

One degree contains \(60'\) (minutes). Therefore, \(2^\circ=2\times60'=120'\), so option C is correct. \(60'\) equals only \(1^\circ\), making it a close but incorrect option. Exam tip: To convert degrees to minutes, multiply the number of degrees by 60.

In angular measurement, \(1^\circ=60'\). Therefore, converting \(180'\) into degrees gives \(180\div60=3\), so \(180'=3^\circ\). The option \(4^\circ\) is incorrect because it equals \(240'\). Exam tip: divide minutes by 60 to convert them into degrees.

( \frac{\pi}{12}\times \frac{180^\circ}{\pi}=15^\circ). Dividing (180^\circ) by the denominator is an easy method.

(225^\circ=\frac{225\pi}{180}=\frac{5\pi}{4}). Multiply by ( \frac{\pi}{180} ) to convert degrees to radians.

(135^\circ) lies between (90^\circ) and (180^\circ) so it is in the second quadrant. Remember the interval to decide the position.

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(420^\circ-360^\circ=60^\circ\), the coterminal angle between \(0^\circ\) and \(360^\circ\) is \(60^\circ\). \(90^\circ\) is a different angle and cannot be obtained from \(420^\circ\) by adding or subtracting a whole multiple of \(360^\circ\). Exam tip: add or subtract \(360^\circ\) until the angle lies in the required interval.

One degree contains 60 minutes, i.e. \(1^\circ=60'\). Therefore, \(3^\circ=3\times60'=180'\). The nearby option 150' equals \(2.5^\circ\), not 3°. Exam tip: multiply degrees by 60 to convert them into minutes.

(75^\circ=\frac{75\pi}{180}=\frac{5\pi}{12}). Multiply by ( \frac{\pi}{180} ) to convert degrees to radians.

(105^\circ=\frac{105\pi}{180}=\frac{7\pi}{12}). Simplify the fraction before choosing the answer.

(165^\circ=\frac{165\pi}{180}=\frac{11\pi}{12}). Multiples of (15^\circ) can lead to denominator (12).

(195^\circ=\frac{195\pi}{180}=\frac{13\pi}{12}). Divide the degree measure by (180) and attach ( \pi ).

(255^\circ=\frac{255\pi}{180}=\frac{17\pi}{12}). Dividing by (15) is useful for simplification.

(285^\circ=\frac{285\pi}{180}=\frac{19\pi}{12}). Counting in ( \frac{\pi}{12} ) steps helps for such angles.

(345^\circ=\frac{345\pi}{180}=\frac{23\pi}{12}). Be careful with angles near (360^\circ).

To convert degrees to radians, multiply the angle by \(\frac{\pi}{180}\): \((-135^\circ)\times\frac{\pi}{180}=-\frac{135\pi}{180}=-\frac{3\pi}{4}\). Hence, the correct answer is \(-\frac{3\pi}{4}\). Note that \(-\frac{\pi}{4}\) corresponds to \(-45^\circ\), not \(-135^\circ\). Exam tip: Always retain the negative sign when converting a negative angle to radians.

( \frac{5\pi}{12}\times \frac{180^\circ}{\pi}=75^\circ). Cancel ( \pi ) and divide (180^\circ) by the denominator.

( \frac{7\pi}{12}=\frac{7\times 180^\circ}{12}=105^\circ). Use ( \frac{180^\circ}{\pi} ) for radians to degrees.