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Difficulty: MediumUnit Circle and Angle Measures (Radians and Degrees)

A point on the unit circle starts at (1,0)(1, 0) and undergoes a counterclockwise rotation of 570∘570^\circ about the origin. What is the radian measure of the angle in the interval [0,2π)[0, 2\pi) that corresponds to the point's final position?

  1. A
    6π7\frac{6\pi}{7}
  2. B
    17π6\frac{17\pi}{6}
  3. 7π6\frac{7\pi}{6}Answer
  4. D
    76π\frac{7}{6\pi}
  5. E
    3π4\frac{3\pi}{4}

Answer

The correct answer is the option representing 7π6\frac{7\pi}{6} radians.
The correct answer represents 7π6\frac{7\pi}{6} radians. To find the position on the unit circle after a counterclockwise rotation of 570∘570^\circ, we first determine the coterminal angle within one full rotation (360∘360^\circ). Subtracting 360∘360^\circ from 570∘570^\circ gives 210∘210^\circ. We then convert this angle to radians by multiplying by π180∘\frac{\pi}{180^\circ}, which simplifies to 7π6\frac{7\pi}{6} radians.

Step-by-Step Solution

1
Find a coterminal angle for 570∘570^\circ within the standard [0∘,360∘)[0^\circ, 360^\circ) range.
570∘−360∘=210∘570^\circ - 360^\circ = 210^\circ
Since a full rotation is 360∘360^\circ, subtracting 360∘360^\circ yields an angle in the same position on the unit circle but within one full rotation.
2
Convert the coterminal angle from degrees to radians by multiplying by π180∘\frac{\pi}{180^\circ}.
210∘×π180∘=210π180=7π6210^\circ \times \frac{\pi}{180^\circ} = \frac{210\pi}{180} = \frac{7\pi}{6} radians
To convert degrees to radians, multiply by the conversion factor π180∘\frac{\pi}{180^\circ}.
3
Verify that the resulting angle is in the interval [0,2π)[0, 2\pi).
Since 0≤7π6<2π0 \le \frac{7\pi}{6} < 2\pi, the angle is in the correct interval.
The question requires the final angle to be in the interval [0,2π)[0, 2\pi).

Key Concept

Finding coterminal angles and converting degree measures to radian measures on the unit circle.

Alternative Method

Convert the rotation to radians first: 570∘×π180∘=19π6570^\circ \times \frac{\pi}{180^\circ} = \frac{19\pi}{6} radians. Then, subtract 2π2\pi to find the coterminal angle: 19π6−2π=7π6\frac{19\pi}{6} - 2\pi = \frac{7\pi}{6} radians.
Estimated Time:1m 15s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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