Find the area of each triangle ABC.
A = 42.5°, b = 13.6 m, c = 10.1 m

Verified step by step guidance

1. Identify the given values: angle \( A = 42.5^\circ \), side \( b = 13.6 \text{ m} \), and side \( c = 10.1 \text{ m} \).
2. Use the formula for the area of a triangle when two sides and the included angle are known: \( \text{Area} = \frac{1}{2}bc \sin A \).
3. Substitute the given values into the formula: \( \text{Area} = \frac{1}{2} \times 13.6 \times 10.1 \times \sin(42.5^\circ) \).
4. Calculate \( \sin(42.5^\circ) \) using a calculator or trigonometric table.
5. Multiply the values: \( \frac{1}{2} \times 13.6 \times 10.1 \times \sin(42.5^\circ) \) to find the area of the triangle.

Key Concepts

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. This law is particularly useful for solving triangles when given two angles and one side or two sides and a non-included angle.

Area of a Triangle

The area of a triangle can be calculated using various formulas, one of which is the formula A = 1/2 * base * height. However, when the height is not known, the area can also be determined using the formula A = (1/2) * a * b * sin(C), where a and b are two sides of the triangle and C is the included angle. This is particularly useful in non-right triangles.

Angle Measurement in Degrees

Angles in trigonometry are often measured in degrees, where a full circle is 360 degrees. Understanding how to convert between degrees and radians is essential, as many trigonometric functions use radians. In this problem, the angle A is given in degrees, which will be used in conjunction with the Law of Sines to find the remaining angles and ultimately the area of the triangle.