An angle has a measure of 16pi/15 radians. What is the measure of the angle, in degrees?

1. TRANSLATE the problem information

• Given information:
  • Angle measure: \(\frac{16\pi}{15}\) radians
  • Need to find: measure in degrees

2. TRANSLATE the conversion approach

• To convert radians to degrees, multiply by \(\frac{180°}{\pi}\) radians
• This conversion factor comes from the fact that \(180° = \pi\) radians

3. SIMPLIFY the multiplication

• Set up: \(\frac{16\pi}{15}\) radians \(\times\) \(\frac{180°}{\pi}\) radians
• Multiply numerators and denominators: \(\frac{16\pi \times 180°}{15 \times \pi}\)
• Cancel the π terms: \(\frac{16 \times 180°}{15}\)
• Calculate: \(\frac{2880°}{15} = 192°\)

Answer: 192° (or 192 degrees, or 192)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Using the wrong conversion factor (\(\frac{\pi}{180°}\) instead of \(\frac{180°}{\pi}\))

Students sometimes remember that π and 180 are related but get confused about which way the conversion goes. They might calculate \(\frac{16\pi}{15} \times \frac{\pi}{180°}\), which would give a much smaller result. This conceptual confusion about the direction of unit conversion leads to completely incorrect answers.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during the fraction multiplication

Students might correctly set up \(\frac{16\pi}{15} \times \frac{180}{\pi}\) but then make mistakes like:

• Not canceling the π terms properly
• Miscalculating \(16 \times 180 = 2880\)
• Incorrectly dividing 2880 by 15

These computational errors can lead to various incorrect numerical answers, causing confusion when selecting from answer choices.

The Bottom Line:

This problem tests both unit conversion knowledge and careful algebraic manipulation. Success requires knowing the correct conversion relationship and executing the arithmetic precisely.