In triangle JKL, \(\mathrm{cos(K) = \frac{24}{51}}\) and angle J is a right angle. What is the value of \(\mathrm{cos(L)}\)?

1. TRANSLATE the problem information

• Given information:
  • Triangle JKL with right angle at J
  • \(\mathrm{cos(K) = \frac{24}{51}}\)
• This tells us that KL is the hypotenuse (opposite the right angle)

2. INFER what cos(K) = 24/51 means for the triangle

• Since \(\mathrm{cosine = \frac{adjacent}{hypotenuse}}\), we can set up:
  • Side adjacent to angle K = \(\mathrm{24n}\) (for some scale factor n)
  • Hypotenuse \(\mathrm{KL = 51n}\)
• The adjacent side to angle K is JK, so \(\mathrm{JK = 24n}\)

3. INFER the strategy to find cos(L)

• We need \(\mathrm{cos(L) = \frac{adjacent\;to\;L}{hypotenuse} = \frac{JL}{KL}}\)
• We know \(\mathrm{KL = 51n}\), but need to find JL
• Use Pythagorean theorem to find the missing side

4. SIMPLIFY using Pythagorean theorem

• \(\mathrm{(KL)^2 = (JK)^2 + (JL)^2}\)

\(\mathrm{(51n)^2 = (24n)^2 + (JL)^2}\)

\(\mathrm{2601n^2 = 576n^2 + (JL)^2}\)

\(\mathrm{(JL)^2 = 2601n^2 - 576n^2 = 2025n^2}\)

\(\mathrm{JL = 45n}\)

5. SIMPLIFY to find the final answer

• \(\mathrm{cos(L) = \frac{JL}{KL} = \frac{45n}{51n} = \frac{45}{51}}\)
• Reduce the fraction: \(\mathrm{\frac{45}{51} = \frac{15}{17}}\) (use calculator: \(\mathrm{15 \div 17 \approx 0.8824}\))

Answer: 15/17 or 0.882 or .8823 or .8824

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not correctly identify which sides correspond to "adjacent" for each angle. They might think the adjacent side to K is JL instead of JK, leading to incorrect setup of the cosine ratio.

This confusion about which side is adjacent to which angle can cause them to set up the wrong equation and arrive at an incorrect answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the Pythagorean theorem but make arithmetic errors when calculating \(\mathrm{(51)^2 - (24)^2}\) or when simplifying the final fraction \(\mathrm{\frac{45}{51}}\) to \(\mathrm{\frac{15}{17}}\).

This may lead them to select an incorrect numerical answer or get stuck on complex fractions.

The Bottom Line:

Success on this problem requires clearly visualizing the right triangle and understanding that cosine relationships for the two acute angles use different adjacent sides. The key insight is that once you know one acute angle's cosine, you can find all three sides and therefore the other angle's cosine.