The number of radians in a 720^circ angle can be written as api, where a is a constant. What is...

1. TRANSLATE the problem information

• Given information:
  • We have a \(720\)-degree angle
  • Need to express the radian measure as \(\mathrm{a}\pi\)
  • Need to find the value of constant \(\mathrm{a}\)

2. INFER the conversion approach

• We need the fundamental relationship between degrees and radians
• Key insight: \(180° = \pi\) radians
• Since \(720°\) is a multiple of \(180°\), we can use this relationship directly

3. INFER the multiplication relationship

• Notice that \(720° = 4 \times 180°\)
• Since \(180° = \pi\) radians, we have:
• \(720° = 4 \times 180° = 4 \times \pi\) radians \(= 4\pi\) radians

4. SIMPLIFY to find the constant

• We have \(720° = 4\pi\) radians
• The problem asks for the form \(\mathrm{a}\pi\), so \(\mathrm{a} = 4\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the fundamental conversion relationship between degrees and radians, or trying to use a more complex conversion formula unnecessarily.

Students might try to memorize a conversion formula like "multiply by \(\pi/180\)" but then make arithmetic errors, or they might not recall the basic relationship that \(180° = \pi\) radians. This leads to confusion about how to approach the conversion systematically.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students understand the fundamental relationship between degrees and radians (\(180° = \pi\) radians) and can apply proportional reasoning to find equivalent measures. The key insight is recognizing that \(720°\) is exactly 4 times the benchmark angle of \(180°\).