Problem Analysis #30

[THOUGHT] Use the formula for area with two sides and the included angle: $\text{Area}=\frac12 \cdot AB \cdot AC \cdot \sin(\angle BAC)$. Given $\cos(\angle BAC)=-\frac35$, we get $\sin(\angle BAC)=\sqrt{1-\cos^2}=\sqrt{1-\frac{9}{25}}=\frac45$ (positive since the angle is between $0^\circ$ and $180^\circ$). Then compute the area and verify with Python. [/THOUGHT] [PYTHON] import sympy as sp

AB = 5 AC = 6 cosA = -sp.Rational(3,5) sinA = sp.sqrt(1 - cosA**2) # positive sine for an angle in (0, pi) area = sp.Rational(1,2) * AB * AC * sinA print(sinA) print(area) [/PYTHON]

[THOUGHT] The computed sine is $4/5$, so the area is $\frac12\cdot 5\cdot 6\cdot \frac45=12$. $\boxed{12}$