Find the area of each triangle ABC.


A = 56.80°, b = 32.67 in., c = 52.89 in.

Verified step by step guidance

1. Identify the given values: angle \(A = 56.80^\circ\), side \(b = 32.67\) inches, and side \(c = 52.89\) inches.
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: \(\text{Area} = \frac{1}{2} \times 32.67 \times 52.89 \times \sin(56.80^\circ)\).
4. Calculate \(\sin(56.80^\circ)\) using a calculator or trigonometric table.
5. Multiply the values together 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 finding unknown angles or sides in non-right triangles, which is essential for calculating the area when given two sides and an 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 triangle's angles and sides are known, the area can also be determined using the formula A = (1/2) * a * b * sin(C), where a and b are two sides, and C is the included angle. This approach is particularly useful for triangles that are not right-angled.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. These functions are fundamental in trigonometry and are used to solve for unknown angles and sides in triangles. Understanding how to apply these functions is crucial for using the Law of Sines and calculating the area of triangles based on given angles and side lengths.