Which formula is used to calculate the area of a triangle using trigonometric functions?

The formula that is used to calculate the area of a triangle using trigonometric functions is ( \frac{1}{2} \times a \times b \times \sin(C) ). In this formula, ( a ) and ( b ) represent the lengths of two sides of a triangle, and ( C ) is the angle between those two sides.

This approach is derived from the properties of trigonometry, specifically the sine function, which relates to the area of a triangle when two sides and the included angle are known. The sine of an angle provides the ratio of the length of the opposite side to the hypotenuse in a right triangle, which in this case helps to calculate the height corresponding to the base formed by the two sides. Multiplying the two sides by the sine of the angle gives the effective height relative to the chosen base, which ultimately leads to the area formula for triangles.

The other options do not capture the relationship between the sides and angles of a triangle in the way that trigonometric functions do. For example, the first option describes the area using base and height, which is valid but does not use trigonometry. The third option gives a formula related to the semi-perimeter