​A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 m and 21.3 m, and the angle between them is 42.2°. What is the area of the triangular lot?

Verified step by step guidance

1. Identify the formula for the area of a triangle when two sides and the included angle are known: \( \text{Area} = \frac{1}{2}ab\sin(C) \), where \( a \) and \( b \) are the sides, and \( C \) is the included angle.
2. Substitute the given values into the formula: \( a = 52.1 \text{ m} \), \( b = 21.3 \text{ m} \), and \( C = 42.2^\circ \).
3. Convert the angle from degrees to radians if necessary, using the conversion \( 1^\circ = \frac{\pi}{180} \text{ radians} \).
4. Calculate \( \sin(42.2^\circ) \) using a calculator or trigonometric table.
5. Substitute the values into the formula and simplify to find the area: \( \text{Area} = \frac{1}{2} \times 52.1 \times 21.3 \times \sin(42.2^\circ) \).

Key Concepts

Area of a Triangle

The area of a triangle can be calculated using the formula A = 1/2 * base * height. However, when two sides and the included angle are known, the area can also be found using the formula A = 1/2 * a * b * sin(C), where 'a' and 'b' are the lengths of the sides and 'C' is the included angle. This approach is particularly useful in trigonometry.

Sine Function

The sine function is a fundamental trigonometric function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the context of the area formula for triangles, the sine of the included angle is crucial as it helps determine the height of the triangle relative to the base formed by the two sides.

Trigonometric Ratios

Trigonometric ratios relate the angles and sides of triangles. In this problem, knowing the lengths of two sides and the included angle allows the use of trigonometric ratios to find the area. Understanding these ratios is essential for applying the correct formulas and solving problems involving triangles in trigonometry.