What is the least positive integer (n) such that \(\frac{5\pi}{12}+\frac{n\pi}{6}\) and \(\frac{17\pi}{12}\) are coterminal?

Correct answer: 6

Coterminal angles differ by \(2k\pi\), where \(k\) is an integer. Hence, \(\frac{5\pi}{12}+\frac{n\pi}{6}=\frac{17\pi}{12}+2k\pi\). Subtracting \(\frac{5\pi}{12}\) gives \(\frac{n\pi}{6}=\pi+2k\pi\), so \(n=6+12k\). The least positive value is obtained for \(k=0\), giving \(n=6\). With \(n=5\), the difference is \(\frac{5\pi}{6}\), not an integral multiple of \(2\pi\). Exam tip: test coterminality by checking whether the angle difference is an integral multiple of \(2\pi\).