Problem Analysis #30

To solve for the area of triangle $ABC$ given that:

• $AB = 5$
• $AC = 6$
• $\cos(\angle BAC) = -\dfrac{3}{5}$

we can use the standard formula for the area of a triangle when two sides and the included angle are known:

Step 1: Use the Pythagorean identity to find $\sin(\angle BAC)$

We are given:

Using the identity:

Since $\theta = \angle BAC$ is between $0^\circ$ and $11.5^\circ$ (because cosine is negative), it lies in the second quadrant where sine is positive, so we take the positive root.

Step 2: Plug into the area formula

Wait — this seems off. Let's recompute carefully:

Yes, correct.

Final Answer

[THOUGHT] [PYTHON] import math

ab = 5 ac = 6 cos_theta = -3/5 sin_theta = math.sqrt(1 - cos_theta**2) area = 0.5 * ab * ac * sin_theta area [/PYTHON]

[THOUGHT] [PYTHON] import math

ab = 5 ac = 6 cos_theta = -3/5 sin_theta = math.sqrt(1 - cos_theta**2) area = 0.5 * ab * ac * sin_theta area [/PYTHON]

\boxed{12}

[THOUGHT] [PYTHON] import math

ab = 5 ac = 6 cos_theta = -3/5 sin_theta = math.sqrt(1 - cos_theta**2) area = 0.5 * ab * ac * sin_theta area [/PYTHON]

\boxed{12}

[THOUGHT] \boxed{12}