Angles

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

(\frac{11\pi}{10}) lies in the third quadrant. The reference angle is (\frac{11\pi}{10}-\pi=\frac{\pi}{10}).

Correct answer: A. \(135^\circ,\ 495^\circ\)

Coterminal angles have the same initial and terminal arms, and their measures differ by a whole multiple of \(360^\circ\). Here, \(495^\circ-135^\circ=360^\circ\), so A is correct. In B, the difference is only \(90^\circ\). Exam tip: check the difference first.

Correct answer: A. (\frac{5\pi}{72})

(12^\circ30'=12.5^\circ=\frac{25}{2}^\circ). In radians it is (\frac{25}{2}\times\frac{\pi}{180}=\frac{5\pi}{72}).

Correct answer: C. \(\frac{7}{4}\)

One complete revolution measures \(2\pi\) radians. Therefore, the number of revolutions is \(\frac{7\pi}{2}\div 2\pi=\frac{7}{4}\). Hence, the correct answer is \(\frac{7}{4}\) revolutions. \(\frac{3}{2}\) revolutions equals only \(3\pi\) radians, so it is not correct. Exam tip: To convert radians into revolutions, divide the angle by \(2\pi\).

Correct answer: B. \(45^\circ\)

Coterminal angles have the same terminal side, so they differ by an integral multiple of \(360^\circ\). Thus, \(-315^\circ+360^\circ=45^\circ\), and \(45^\circ\) lies in the required interval \(0^\circ\le\theta<360^\circ\). Although \(30^\circ\) is nearby, it does not differ from \(-315^\circ\) by a whole multiple of \(360^\circ\). Exam tip: to express a negative angle in this interval, first add \(360^\circ\).

Correct answer: C. (229.2^\circ)

(4) radians (=4\times\frac{180^\circ}{\pi}\approx229.2^\circ). Use (\pi\approx3.14) for approximation.

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

For the same arc length (\theta) is inversely proportional to radius. When radius doubles the angle becomes half as (\frac{3\pi}{10}).

Correct answer: C. \(\frac{3\pi}{4}\)

One complete revolution measures \(2\pi\) radians. Therefore, \(\frac{3}{8}\) of a revolution is \(\frac{3}{8}\times 2\pi=\frac{3\pi}{4}\) radians. \(\frac{\pi}{2}\) represents only \(\frac{1}{4}\) of a revolution, so it is not correct. Exam tip: To convert a fractional revolution into radians, multiply the fraction by \(2\pi\).

Correct answer: B. \(50^\circ\)

Coterminal angles differ by an integral multiple of \(360^\circ\). Subtract the greatest suitable multiple of \(360^\circ\): \(1440^\circ=4\times360^\circ\). Thus, \(1490^\circ-1440^\circ=50^\circ\). Therefore, the least positive coterminal angle is \(50^\circ\). Although \(40^\circ\) is a close distractor, it does not differ from \(1490^\circ\) by a whole multiple of \(360^\circ\). Exam tip: divide the angle by \(360^\circ\) and take the positive remainder.

Correct answer: B. (\frac{8\pi}{7})

In the third quadrant the angle is (\pi+) reference angle. Hence (\pi+\frac{\pi}{7}=\frac{8\pi}{7}).

Correct answer: A. (45^\circ) and (405^\circ)

(405^\circ-45^\circ=360^\circ) so they are coterminal. The difference must be a multiple of (360^\circ).

Correct answer: B. \(14\pi\) m

The arc-length formula is \(s=r\theta\), where \(\theta\) must be in radians. \(40^\circ=40\times\frac{\pi}{180}=\frac{2\pi}{9}\) radians. Therefore, \(s=63\times\frac{2\pi}{9}=14\pi\) m. Hence, \(14\pi\) m is correct. A nearby distractor such as \(16\pi\) m can result from an incorrect degree-to-radian conversion or multiplication. Exam tip: always convert the angle to radians before applying the arc-length formula.