Question

which triangles area can be calculated using the trigonometric area formula?

Explanation:

Step1: Recall trigonometric area formula

The trigonometric area formula for a triangle is $A=\frac{1}{2}ab\sin C$, where $a$ and $b$ are two - side lengths of the triangle and $C$ is the included - angle between them.

Step2: Analyze first triangle

The first triangle (triangle $ABC$) only has side - lengths given ($AB = 16$, $BC = 24$, $AC = 28$) and no angle measures, so we cannot use the trigonometric area formula.

Step3: Analyze second triangle

In triangle $DEF$, we know two side - lengths ($DE$, $EF$) and the included angle $\angle E=108^{\circ}$. We can use the formula $A=\frac{1}{2}DE\times EF\times\sin E$.

Step4: Analyze third triangle

The third triangle only has one side - length ($17$) and one non - included angle ($47^{\circ}$), so we cannot use the trigonometric area formula.

Answer:

The triangle with side - lengths and an included angle (the second triangle, triangle $DEF$) can have its area calculated using the trigonometric area formula.