Question

calculating the area of a triangle using trigonometric area formula
which triangles area would be calculated using the trigonometric area formula?

Explanation:

The trigonometric area formula for a triangle is $\frac{1}{2}ab\sin C$, which requires two side lengths and the included angle (the angle between those two sides). We check each triangle:

1. First triangle: Has sides $PN=6$, $QN=5$, and included $\angle N=40^\circ$ (fits the formula's requirements).
2. Second triangle: The given angle $\angle Y=110^\circ$ is not between the labeled sides $XY$ (unlabeled) and $YZ=4$, so we lack two sides with their included angle.
3. Third triangle: The given angle $\angle C=22^\circ$ is not between sides $AB=4$ and $BC=6$, so it does not match the formula's needs.
4. Fourth triangle: Only has all three side lengths, no included angle provided, so it does not use the trigonometric area formula (Heron's formula would be used instead).

Answer:

The first triangle (with vertices P, Q, N, sides 6, 5, and included 40° angle at N)