Find the area of each triangle ABC.


B = 124.5°, a = 30.4 cm, c = 28.4 cm

Verified step by step guidance

1. Identify the given values: angle \( B = 124.5^\circ \), side \( a = 30.4 \text{ cm} \), and side \( c = 28.4 \text{ cm} \).
2. Use the formula for the area of a triangle when two sides and the included angle are known: \( \text{Area} = \frac{1}{2}ac\sin(B) \).
3. Substitute the given values into the formula: \( \text{Area} = \frac{1}{2} \times 30.4 \times 28.4 \times \sin(124.5^\circ) \).
4. Calculate \( \sin(124.5^\circ) \) using a calculator or trigonometric table.
5. Multiply the values 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 solving the area of triangle ABC given two sides and an included angle.

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. In this case, knowing one angle and two sides allows us to apply this formula to find the area of triangle ABC. Understanding this formula is crucial for solving the problem effectively.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. In this context, the sine function is particularly important as it helps in calculating the area of the triangle using the angle provided. A solid grasp of these functions is necessary to manipulate and solve trigonometric equations that arise in the problem.