Question

find coterminal, reference, and quadrant (radians)
score: 3/5 penalty: none
question
for the rotation (-\frac{21pi}{4}), find the coterminal angle from (0 leq \theta < 2pi), 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: Add multiples of $2\pi$

To find a coterminal angle in $0 \leq \theta < 2\pi$, add $3\times2\pi = 6\pi = \frac{24\pi}{4}$ to $-\frac{21\pi}{4}$:
$$-\frac{21\pi}{4} + \frac{24\pi}{4} = \frac{3\pi}{4}$$

Step2: Identify the quadrant

$\frac{\pi}{2} < \frac{3\pi}{4} < \pi$, so it lies in Quadrant II.

Step3: Calculate reference angle

For angles in Quadrant II, reference angle = $\pi - \theta$:
$$\pi - \frac{3\pi}{4} = \frac{4\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{4}$$

Answer:

The coterminal angle is $\frac{3\pi}{4}$, which lies in Quadrant II, with a reference angle of $\frac{\pi}{4}$.