The measure of angle R is 4pi/3 radians. If the measure of angle T is 3/4 times the measure of...

1. TRANSLATE the problem information

• Given information:
  • Angle R = \(\frac{4\pi}{3}\) radians
  • Angle T = \(\frac{3}{4}\) times angle R
  • Need to find angle T in degrees
• What this tells us: We need to find \(\mathrm{T} = \frac{3}{4} \times \mathrm{R}\), then convert to degrees

2. INFER the most efficient approach

• Since we need the final answer in degrees, we have two options:
  • Convert R to degrees first, then multiply by 3/4
  • Find T in radians first, then convert to degrees
• Either approach works, but finding T in radians first often involves cleaner arithmetic

3. SIMPLIFY to find T in radians

• \(\mathrm{T} = \frac{3}{4} \times \frac{4\pi}{3}\)
• \(\mathrm{T} = \frac{3 \times 4\pi}{4 \times 3} = \frac{12\pi}{12} = \pi\) radians

4. SIMPLIFY the conversion to degrees

• Using the conversion formula: \(\mathrm{degrees} = \mathrm{radians} \times \frac{180°}{\pi}\)
• \(\mathrm{T} = \pi \times \frac{180°}{\pi} = 180°\)

Answer: B. 180

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when multiplying fractions, getting \(\frac{3}{4} \times \frac{4\pi}{3} = \frac{12\pi}{7}\) or similar incorrect results.

This leads to wrong radian values, which then convert to incorrect degree measures, causing them to select wrong answer choices or abandon the problem entirely.

Second Most Common Error:

Missing conceptual knowledge: Students forget the radian-to-degree conversion formula or confuse it with the degree-to-radian conversion (using \(\frac{\pi}{180°}\) instead of \(\frac{180°}{\pi}\)).

This may lead them to calculate \(\mathrm{T} = \pi \times \frac{\pi}{180°} = \frac{\pi^2}{180°}\), yielding a completely wrong numerical result and causing confusion about which answer choice to select.

The Bottom Line:

This problem tests both fraction arithmetic fluency and unit conversion knowledge. Success requires careful fraction manipulation and correct application of the radian-degree conversion formula.