Question:In a right triangle, the sine of one of the two acute angles is 2/3. What is the cosine of...

1. TRANSLATE the problem information

• Given information:
  • We have a right triangle
  • One acute angle has sine = \(\frac{2}{3}\)
  • We need the cosine of the other acute angle

2. INFER the key relationship

• In a right triangle, the two acute angles are complementary (they add up to \(90°\))
• If one angle is \(\alpha\), then the other angle is \((90° - \alpha)\)
• This means we need \(\cos(90° - \alpha)\) when we know \(\sin(\alpha) = \frac{2}{3}\)

3. INFER the solution strategy

• Use the complementary angle identity: \(\cos(90° - \theta) = \sin(\theta)\)
• Therefore: \(\cos(90° - \alpha) = \sin(\alpha) = \frac{2}{3}\)

Answer: D) \(\frac{2}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the complementary angle relationship and instead try to construct the triangle using Pythagorean theorem.

They think: "If \(\sin(\alpha) = \frac{2}{3}\), then \(\text{opposite} = 2\) and \(\text{hypotenuse} = 3\). So \(\text{adjacent} = \sqrt{3^2 - 2^2} = \sqrt{5}\). Therefore \(\cos(\alpha) = \frac{\sqrt{5}}{3}\)."

But they're finding \(\cos(\alpha)\), not cos of the OTHER angle. This leads them to select Choice C (\(\frac{\sqrt{5}}{3}\)).

Second Most Common Error:

Conceptual confusion about trigonometric signs: Students might think that since one trig function is positive, the complementary one should be negative, or they make sign errors in their calculations.

This may lead them to select Choice A (\(-\frac{2}{3}\)) or Choice B (\(-\frac{\sqrt{5}}{3}\)).

The Bottom Line:

The key insight is recognizing that this is fundamentally about complementary angle relationships, not about constructing and analyzing the triangle from scratch. The complementary angle identity gives you the answer directly.