An angle measures 63 degrees. The complement of this angle is converted to radians. What is the measure of the...

1. TRANSLATE the problem information

• Given information:
  • An angle measures \(63°\)
  • We need the complement of this angle in radians

• What this tells us: We need to find what angle, when added to \(63°\), gives \(90°\), then convert that result to radians.

2. INFER the solution approach

• First find the complement in degrees, then convert to radians
• Complement means: \(\mathrm{complement + original\ angle = 90°}\)
• So \(\mathrm{complement = 90° - original\ angle}\)

3. Calculate the complement

• \(\mathrm{Complement = 90° - 63° = 27°}\)

4. INFER the conversion strategy

• Now we need to convert \(27°\) to radians
• Use the conversion formula: \(\mathrm{radians = degrees × (π/180°)}\)

5. Apply the conversion formula

• \(\mathrm{Radians = 27° × (π/180°)}\)
  \(\mathrm{= 27π/180}\)

6. SIMPLIFY the fraction

• \(\mathrm{27π/180}\) needs to be reduced to lowest terms
• Find GCD of 27 and 180: \(\mathrm{GCD(27, 180) = 9}\)
• \(\mathrm{27π/180}\)
  \(\mathrm{= (27÷9)π/(180÷9)}\)
  \(\mathrm{= 3π/20}\)

Answer: B) \(\mathrm{3π/20}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may find the supplement (90° difference) instead of the complement, or incorrectly calculate \(\mathrm{63° - 90°}\) instead of \(\mathrm{90° - 63°}\).

Some students confuse complementary angles (sum to \(90°\)) with supplementary angles (sum to \(180°\)), leading them to calculate \(\mathrm{180° - 63° = 117°}\), then convert this to radians. This gives \(\mathrm{117π/180 = 13π/20}\), which doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the complement as \(27°\) and correctly set up \(\mathrm{27π/180}\), but fail to simplify the fraction properly.

They might leave the answer as \(\mathrm{27π/180}\) and look for this among the choices, or incorrectly simplify it. Since \(\mathrm{27π/180}\) isn't simplified and doesn't appear in the answer choices, this leads them to select Choice C (\(\mathrm{27π/10}\)) thinking the denominator should be 10 instead of 180.

The Bottom Line:

This problem tests whether students can distinguish between complementary and supplementary angles, properly apply the degree-to-radian conversion formula, and execute fraction simplification accurately. The combination of these skills in sequence makes it easy to make errors at any step.