Which triangle's area can be calculated using the trigonometric area formula $A=\frac{1}{2}absinC$?

• A. A triangle where all three angles are known ($AAA$ case)
• B. A triangle where only one side and one angle are known ($SA$ case)
• C. A triangle where two sides and the included angle are known ($SAS$ case)
• D. A triangle where all three sides are known ($SSS$ case)

Answer: (C)

Verified step by step guidance

1. Understand the trigonometric area formula for a triangle: \(A = \frac{1}{2}ab\sin C\), where \(a\) and \(b\) are two sides of the triangle, and \(C\) is the included angle between those sides.
2. Recognize that to use this formula, you need to know two sides and the angle between them, which corresponds to the SAS (Side-Angle-Side) case.
3. Note that if you only know all three angles (AAA), you cannot determine the area because the size of the triangle is not fixed without any side length.
4. If you know only one side and one angle (SA), you do not have enough information to apply this formula directly because you need two sides and the included angle.
5. If you know all three sides (SSS), you would use Heron's formula instead of the trigonometric area formula.