What is the coterminal angle of ( \frac{29\pi}{4} ) between (0) and (2\pi)?

Correct answer: ( \frac{5\pi}{4} )

The direct answer is option A, \(\frac{5\pi}{4}\). Coterminal angles differ by complete turns of \(2\pi\). We subtract enough complete turns to place the angle between 0 and \(2\pi\): \(\frac{29\pi}{4}-2\pi=\frac{21\pi}{4}\), and subtract \(2\pi\) again to get \(\frac{13\pi}{4}\), and once more to get \(\frac{5\pi}{4}\). Equivalently, \(2\pi=\frac{8\pi}{4}\), so \(\frac{29\pi}{4}-3\cdot\frac{8\pi}{4}=\frac{5\pi}{4}\), which lies in the required interval. Option A is correct. Option B, \(\frac{3\pi}{4}\), would result from subtracting too much or using an incorrect number of quarter-turns. Option C, \(\frac{7\pi}{4}\), is another possible-looking angle but is not coterminal with the given angle. Option D, \(\frac{\pi}{4}\), is also not obtained by subtracting a whole number of full turns. Remember: reduce by multiples of \(2\pi\), not by arbitrary fractions of a turn.