Find the area of each triangle ABC.


A = 59.80°, b = 15.00 cm, C = 53.10°

Verified step by step guidance

1. Identify the given values: angle A = 59.80°, side b = 15.00 cm, and angle C = 53.10°.
2. Use the fact that the sum of angles in a triangle is 180° to find angle B: B = 180° - A - C.
3. Apply the Law of Sines to find side a: \( \frac{a}{\sin A} = \frac{b}{\sin B} \).
4. Calculate side a using the equation from the Law of Sines.
5. Use the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times b \times a \times \sin C \).

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 side lengths or angles in non-right triangles, which is essential for solving the given problem.

Area of a Triangle

The area of a triangle can be calculated using various formulas, one of which is A = 1/2 * a * b * sin(C), where 'a' and 'b' are two sides of the triangle and 'C' is the included angle. This formula is derived from the basic definition of area and is particularly useful when two sides and the included angle are known, as in the given problem.

Angle Sum Property

The Angle Sum Property states that the sum of the interior angles of a triangle is always 180 degrees. In this problem, knowing two angles allows us to find the third angle by subtracting the sum of the known angles from 180 degrees. This is crucial for applying the Law of Sines and calculating the area of triangle ABC.