Question:An angle has a measure of 225^circ. What is the measure of this angle in radians?Express your answer as a...

1. TRANSLATE the problem information

• Given information:
  • Angle measure: \(225^\circ\)
  • Need to find: equivalent measure in radians
  • Express as: fraction in terms of \(\pi\) in lowest terms

2. INFER the approach needed

• This is a unit conversion problem requiring the degree-to-radian formula
• Since we need the answer in "lowest terms," we'll need to simplify the resulting fraction

3. Apply the conversion formula

• Use: \(\mathrm{radians} = \mathrm{degrees} \times (\pi/180^\circ)\)
• Substitute: \(\mathrm{radians} = 225 \times (\pi/180)\)
• \(\mathrm{radians} = 225\pi/180\)

4. SIMPLIFY the fraction to lowest terms

• Find the GCD of 225 and 180 using prime factorization:
  • \(225 = 3^2 \times 5^2 = 9 \times 25\)
  • \(180 = 2^2 \times 3^2 \times 5 = 4 \times 45\)
  • \(\mathrm{GCD}(225, 180) = 3^2 \times 5 = 45\)
• Reduce: \(225\pi/180 = (225 \div 45)\pi/(180 \div 45) = 5\pi/4\)

Answer: \(5\pi/4\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't remember or incorrectly recall the degree-to-radian conversion formula.

Some students might try to use \(180/\pi\) instead of \(\pi/180\), or might forget the conversion factor entirely. Without the correct formula, they get stuck at the setup stage and resort to guessing.

This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY skill: Students apply the conversion formula correctly to get \(225\pi/180\) but fail to reduce it to lowest terms.

They might not recognize that the fraction needs simplification, or they might make arithmetic errors when finding the GCD of 225 and 180. This results in answers like \(225\pi/180\) or incorrectly simplified versions.

This may lead them to select an answer that isn't in simplest form or contains calculation errors.

The Bottom Line:

This problem tests both formula recall and fraction simplification skills. Students need to remember the specific conversion factor and then execute multi-step arithmetic accurately to reach the simplified form.