Question

calculating the area of a triangle using trigonometric area formula
which triangles area would 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. We need to find a triangle where two - side lengths and the included angle are given.

Step2: Analyze first triangle

In the first triangle, side lengths $AB = 4$, $BC = 5$ and the included angle $\angle B=25^{\circ}$. We can use the formula $A=\frac{1}{2}(AB)(BC)\sin B=\frac{1}{2}\times4\times5\times\sin25^{\circ}$.

Step3: Analyze second triangle

In the second triangle, side lengths $PR = 6$, $QR = 5$ and the non - included angle $\angle R = 40^{\circ}$. We cannot use the trigonometric area formula directly as the angle is not the included angle between the two given sides.

Step4: Analyze third triangle

In the third triangle, we are given three side lengths $DE = 12$, $EF = 10$, $DF = 7$ and no angles. We cannot use the trigonometric area formula.

Answer:

The first triangle (with side lengths 4 and 5 and included angle $25^{\circ}$)