A painter is going to apply paint to a triangular metal plate on a new building. Two sides measure 16.1 m and 15.2 m, and the angle between the sides is 125°. What is the area of the surface to be painted?

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: \( a = 16.1 \), \( b = 15.2 \), and \( C = 125^\circ \).
3. Calculate the sine of the angle: \( \sin(125^\circ) \).
4. Multiply the values: \( \frac{1}{2} \times 16.1 \times 15.2 \times \sin(125^\circ) \).
5. The result from the multiplication will give you the area of the triangular metal plate in square meters.

Key Concepts

Area of a Triangle

The area of a triangle can be calculated using the formula A = 1/2 * a * b * sin(C), where 'a' and 'b' are the lengths of two sides, and 'C' is the included angle between those sides. This formula is particularly useful when the angle is known, allowing for the direct calculation of the area without needing to find the height.

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, it helps to determine the height of the triangle relative to the base formed by the two sides, thus facilitating the area calculation.

Degrees and Radians

Angles can be measured in degrees or radians, with 180 degrees equivalent to π radians. In trigonometry, it is essential to ensure that the angle used in calculations is in the correct unit, as this affects the output of trigonometric functions like sine. For this problem, the angle of 125° must be used directly in the sine function to find the area.