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

An angle in standard position measures −468∘-468^\circ. What is the radian measure of the coterminal angle that lies in the interval [0,2π)[0, 2\pi)?

  1. A
    5π7\frac{5\pi}{7}
  2. 7π5\frac{7\pi}{5}Answer
  3. C
    π6\frac{\pi}{6}
  4. D
    3π5\frac{3\pi}{5}
  5. E
    9π5\frac{9\pi}{5}

Answer

The correct radian measure of the coterminal angle is 7π5\frac{7\pi}{5}.
To find the correct radian measure, first determine the positive coterminal angle in degrees by adding multiples of 360∘360^\circ until the angle lies in the interval [0∘,360∘)[0^\circ, 360^\circ). Adding 720∘720^\circ to −468∘-468^\circ results in 252∘252^\circ. Convert this angle to radians by multiplying by π180∘\frac{\pi}{180^\circ}, which yields 252π180\frac{252\pi}{180}. Dividing the numerator and denominator by their greatest common divisor, 3636, simplifies the expression to 7π5\frac{7\pi}{5} radians.

Step-by-Step Solution

1
Find a positive coterminal angle in degrees by adding multiples of 360∘360^\circ to the initial angle.
−468∘+360∘=−108∘-468^\circ + 360^\circ = -108^\circ, and −108∘+360∘=252∘-108^\circ + 360^\circ = 252^\circ.
Adding 720∘720^\circ (two full rotations) to −468∘-468^\circ shifts the angle into the standard positive range [0∘,360∘)[0^\circ, 360^\circ) while maintaining the same terminal ray.
2
Convert the coterminal angle from degrees to radians by multiplying by the conversion factor π180∘\frac{\pi}{180^\circ}.
252∘×π180∘=252π180252^\circ \times \frac{\pi}{180^\circ} = \frac{252\pi}{180} radians.
Since π\pi radians corresponds to 180∘180^\circ, multiplying by π180∘\frac{\pi}{180^\circ} changes the unit of measure from degrees to radians.
3
Simplify the fraction 252π180\frac{252\pi}{180} by dividing both the numerator and the denominator by their greatest common divisor.
Dividing 252252 and 180180 by their greatest common divisor of 3636 yields 7π5\frac{7\pi}{5} radians.
Simplifying the fraction expresses the final radian measure in its standard, reduced form.

Key Concept

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

Alternative Method

Convert the initial angle of −468∘-468^\circ directly to radians first by multiplying by π180∘\frac{\pi}{180^\circ}, yielding −13π5-\frac{13\pi}{5} radians. To find the positive coterminal angle in the interval [0,2π)[0, 2\pi), add multiples of 2π2\pi radians (which is 10π5\frac{10\pi}{5}): −13π5+10π5=−3π5-\frac{13\pi}{5} + \frac{10\pi}{5} = -\frac{3\pi}{5}, and then −3π5+10π5=7π5-\frac{3\pi}{5} + \frac{10\pi}{5} = \frac{7\pi}{5} radians.
Estimated Time:2m 0s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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