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Question

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

Match each of the degree measures of angles in standard position on the left with its mathematically equivalent radian measure on the right. Which radian measure corresponds to each degree measure?

  • −135∘-135^\circ−3π4-\frac{3\pi}{4} radians
  • 480∘480^\circ8π3\frac{8\pi}{3} radians
  • −300∘-300^\circ−5π3-\frac{5\pi}{3} radians
  • 585∘585^\circ13π4\frac{13\pi}{4} radians

Answer

The correct pairings are: −135∘-135^\circ matches with −3π4-\frac{3\pi}{4} radians; 480∘480^\circ matches with 8π3\frac{8\pi}{3} radians; −300∘-300^\circ matches with −5π3-\frac{5\pi}{3} radians; and 585∘585^\circ matches with 13π4\frac{13\pi}{4} radians.
Each degree measure is multiplied by π180∘\frac{\pi}{180^\circ} and simplified to find its equivalent radian measure. This process yields the unique matching pairs: −135∘-135^\circ to −3π4-\frac{3\pi}{4} radians, 480∘480^\circ to 8π3\frac{8\pi}{3} radians, −300∘-300^\circ to −5π3-\frac{5\pi}{3} radians, and 585∘585^\circ to 13π4\frac{13\pi}{4} radians.

Step-by-Step Solution

1
Recall the formula to convert degrees to radians.
Radian measure = Degree measure ×π180∘\times \frac{\pi}{180^\circ}
Since a full circle is 360∘360^\circ or 2π2\pi radians, the conversion ratio simplifies to π\pi radians per 180∘180^\circ.
2
Convert the first degree measure, −135∘-135^\circ, to radians.
−135∘×π180∘=−135π180=−3π4-135^\circ \times \frac{\pi}{180^\circ} = -\frac{135\pi}{180} = -\frac{3\pi}{4} radians
Dividing the numerator and denominator by their greatest common divisor, 4545, simplifies the fraction to −34-\frac{3}{4}.
3
Convert the second degree measure, 480∘480^\circ, to radians.
480∘×π180∘=480π180=8π3480^\circ \times \frac{\pi}{180^\circ} = \frac{480\pi}{180} = \frac{8\pi}{3} radians
Dividing the numerator and denominator by their greatest common divisor, 6060, simplifies the fraction to 83\frac{8}{3}.
4
Convert the third degree measure, −300∘-300^\circ, to radians.
−300∘×π180∘=−300π180=−5π3-300^\circ \times \frac{\pi}{180^\circ} = -\frac{300\pi}{180} = -\frac{5\pi}{3} radians
Dividing the numerator and denominator by their greatest common divisor, 6060, simplifies the fraction to −53-\frac{5}{3}.
5
Convert the fourth degree measure, 585∘585^\circ, to radians.
585∘×π180∘=585π180=13π4585^\circ \times \frac{\pi}{180^\circ} = \frac{585\pi}{180} = \frac{13\pi}{4} radians
Dividing the numerator and denominator by their greatest common divisor, 4545, simplifies the fraction to 134\frac{13}{4}.

Key Concept

Converting degree measures to equivalent radian measures using the conversion factor π180∘\frac{\pi}{180^\circ}.

Alternative Method

Alternatively, you can recall key benchmark angles on the unit circle (such as 45∘=π445^\circ = \frac{\pi}{4} radians and 60∘=π360^\circ = \frac{\pi}{3} radians) and express each angle as an integer multiple of these benchmarks. For example, −135∘-135^\circ is −3×45∘-3 \times 45^\circ, which corresponds to −3×π4=−3π4-3 \times \frac{\pi}{4} = -\frac{3\pi}{4} radians.
Estimated Time:1m 30s
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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