Triangle ABC is isosceles with AB = AC. If \(\sin(\angle B) = \frac{119}{169}\), what is the value of \(\sin(\angle C)\)?

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC is isosceles with \(\mathrm{AB = AC}\)
  • \(\sin(\mathrm{angle\ B}) = \frac{119}{169}\)
  • Need to find \(\sin(\mathrm{angle\ C})\)

2. INFER the key relationship

• Since \(\mathrm{AB = AC}\), we have an isosceles triangle
• In any isosceles triangle, the base angles are equal
• The base angles are the angles opposite the equal sides
• Angle B is opposite side AC, angle C is opposite side AB
• Since \(\mathrm{AB = AC}\), we get \(\mathrm{angle\ B = angle\ C}\)

3. Apply the equal angles relationship

• If \(\mathrm{angle\ B = angle\ C}\), then their sine values must be equal
• Since \(\sin(\mathrm{angle\ B}) = \frac{119}{169}\)
• Therefore \(\sin(\mathrm{angle\ C}) = \frac{119}{169}\)

Answer: B \(\left(\frac{119}{169}\right)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize which angles are the base angles in an isosceles triangle. They might think that since the triangle is isosceles "somehow the angles are related" but don't make the precise connection that the angles opposite the equal sides are equal.

This confusion leads them to guess or try to use trigonometric identities they remember, potentially selecting Choice A \(\left(\frac{120}{169}\right)\) thinking there's some calculation needed.

Second Most Common Error:

Conceptual confusion about isosceles triangles: Students remember that "angles are equal in isosceles triangles" but confuse which angles. They might think the vertex angle (angle A) equals the base angles, leading to more complex trigonometric calculations that don't match any answer choice.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

The key insight is recognizing that isosceles triangle properties give us the answer immediately - no complex trigonometry needed, just the fundamental property that base angles are equal.