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Multiple Choice
Identify the quadrant that the given angle is located in. 47π radians
A
Quadrant I
B
Quadrant II
C
Quadrant III
D
Quadrant IV
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Verified step by step guidance
1
Convert the given angle \( \frac{7\pi}{4} \) radians into degrees if necessary, but since the problem is in radians, we will work directly with radians.
Recall that a full circle in radians is \( 2\pi \), and each quadrant represents a quarter of the circle: \( \frac{\pi}{2} \) radians for Quadrant I, \( \pi \) radians for Quadrant II, \( \frac{3\pi}{2} \) radians for Quadrant III, and \( 2\pi \) radians for Quadrant IV.
Determine where \( \frac{7\pi}{4} \) lies by comparing it to the quadrant boundaries: \( \frac{7\pi}{4} \) is greater than \( \frac{3\pi}{2} \) (Quadrant III boundary) but less than \( 2\pi \) (Quadrant IV boundary).
Conclude that \( \frac{7\pi}{4} \) radians is located in Quadrant IV based on its position within the interval \( \left( \frac{3\pi}{2}, 2\pi \right) \).
Verify the result by visualizing the angle on the unit circle or by subtracting \( 2\pi \) to find the equivalent angle within one full rotation, confirming it lies in Quadrant IV.