Problem Analysis #30

[THOUGHT] We know two sides AB = 5, AC = 6 and the cosine of the included angle A. The area formula is: Area = (1/2) * AB * AC * sin(A). We have cos(A) = -3/5, so we can use sin^2(A) = 1 - cos^2(A) to find sin(A), then compute the area. I’ll verify numerically in Python. [/THOUGHT] [PYTHON] import math

AB = 5 AC = 6 cosA = -3/5

sinA = math.sqrt(1 - cosA**2) # positive sine for angle between 0 and pi area = 0.5 * AB * AC * sinA print(area) [/PYTHON]

[THOUGHT] Using the formula for area with two sides and included angle, the Python computation gives area = 12. I will report 12 as the final answer. [ \text{Area} = \frac12 \cdot 5 \cdot 6 \cdot \sin(\angle BAC) = 12 ] [/THOUGHT] \boxed{12}