Question

In triangle $ ABC $, if $ a = 5,\ b = 4,\ \cos(A - B) = \frac{31}{32} $, then $ c = ? $

• A. 8
• B. \( \sqrt{41} \)
• C. 6
• D. \( \sqrt{24} \)

Hint:

Use cosine rule expressions to substitute \( \cos A \) and \( \cos B \) into angle difference identity.

Answer 1

The Correct Option is C

Use identity: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] From cosine rule: \[ \cos A = \frac{b^2 + c^2 - a^2}{2bc},\quad \cos B = \frac{a^2 + c^2 - b^2}{2ac} \] Substitute \( a = 5,\ b = 4 \), and simplify to solve for \( c \) The result leads to \( c = 6 \)