Question

find coterminal, reference, and quadrant (radians)
score: 2/5 penalty: none
question
for the rotation \\(\frac{47\pi}{9}\\), find the coterminal angle from \\(0 \leq \theta < 2\pi\\), the quadrant, and the reference angle.
answer attempt 1 out of 2
the coterminal angle is \\(\square\\), which lies in quadrant \\(\square\\), with a reference angle of \\(\square\\).
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Explanation:

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

First, convert $2\pi$ to ninths: $2\pi = \frac{18\pi}{9}$. We subtract multiples of this from $\frac{47\pi}{9}$ to get an angle in $0 \leq \theta < 2\pi$.
Number of full rotations: $\lfloor\frac{47}{18}
floor = 2$ (since $2 \times 18 = 36$)
$\frac{47\pi}{9} - 2\times\frac{18\pi}{9} = \frac{47\pi}{9} - \frac{36\pi}{9} = \frac{11\pi}{9}$

Step2: Identify the quadrant

Compare $\frac{11\pi}{9}$ to quadrant bounds:
$\pi < \frac{11\pi}{9} < \frac{3\pi}{2}$, so it is in Quadrant 3.

Step3: Calculate reference angle

For angles in Quadrant 3, reference angle = $\theta - \pi$
$\frac{11\pi}{9} - \pi = \frac{11\pi}{9} - \frac{9\pi}{9} = \frac{2\pi}{9}$

Answer:

The coterminal angle is $\frac{11\pi}{9}$, which lies in Quadrant 3, with a reference angle of $\frac{2\pi}{9}$.