Question

find an angle \\( \theta \\) coterminal to \\( \frac{25\pi}{6} \\), where \\( 0 \leq \theta < 2\pi \\).

Explanation:

Step1: Subtract full rotations ($2\pi$)

Since $2\pi = \frac{12\pi}{6}$, subtract it from $\frac{25\pi}{6}$:
$\frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6}$

Step2: Subtract another full rotation

$\frac{13\pi}{6}$ is still greater than $2\pi$, so subtract $\frac{12\pi}{6}$ again:
$\frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}$

Step3: Verify range

Check that $0 \leq \frac{\pi}{6} < 2\pi$, which is true.

Answer:

$\frac{\pi}{6}$