An angle has a measure of 108^circ. The measure of this angle in radians can be expressed as kpi, where...

1. TRANSLATE the problem information

• Given information:
  • Angle measure: \(108^\circ\)
  • Need to express in radians as \(k\pi\) where k is a constant
  • Find the value of k

• What this tells us: We need to convert degrees to radians and identify the coefficient of \(\pi\)

2. INFER the conversion approach

• To convert degrees to radians, we use the fundamental relationship: \(\pi \text{ radians} = 180^\circ\)
• This gives us the conversion factor: \(\frac{\pi \text{ radians}}{180^\circ}\)
• Once we get the radian measure in the form of (some number)\(\pi\), that number will be our k value

3. APPLY the degree-to-radian conversion

• Set up the conversion:
  \(108^\circ \times \frac{\pi \text{ radians}}{180^\circ} = \frac{108\pi}{180} \text{ radians}\)

• This can be written as: \(\frac{108}{180}\pi \text{ radians}\)

4. TRANSLATE this result into the required form

• The problem states the angle in radians equals \(k\pi\)
• So we have: \(k\pi = \frac{108}{180}\pi\)
• Therefore: \(k = \frac{108}{180}\)

5. SIMPLIFY the fraction to find k

• Find the greatest common divisor of 108 and 180
• \(108 = 36 \times 3\) and \(180 = 36 \times 5\)
• So: \(k = \frac{108}{180} = \frac{3}{5}\)

Answer: \(\frac{3}{5}\) (also acceptable as \(0.6\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(k = \frac{108}{180}\) but fail to fully simplify the fraction. They might reduce it partially (like getting 12/20 and stopping there) or make arithmetic errors in finding common factors.

This leads to selecting incorrect numerical values or getting confused about which form to submit.

Second Most Common Error:

Missing conceptual knowledge about degree-radian conversion: Students don't remember that \(\pi \text{ radians} = 180^\circ\), so they might guess at the conversion factor or use incorrect relationships like \(\pi \text{ radians} = 360^\circ\).

This leads to completely wrong setups and values that don't correspond to any reasonable answer.

The Bottom Line:

This problem tests both memorization of a key conversion formula and careful fraction arithmetic. Success requires knowing the fundamental degree-radian relationship and executing multi-step fraction simplification without computational errors.