In Exercises 13–16, find the area of the triangle having the given measurements. Round to the nearest square unit.C = 42°, a = 4 feet, b = 6 feet

Verified step by step guidance

1. Identify the formula for the area of a triangle using two sides and the included angle: \( \text{Area} = \frac{1}{2}ab\sin(C) \).
2. Substitute the given values into the formula: \( a = 4 \) feet, \( b = 6 \) feet, and \( C = 42^\circ \).
3. Calculate \( \sin(42^\circ) \) using a calculator or trigonometric table.
4. Multiply the values: \( \frac{1}{2} \times 4 \times 6 \times \sin(42^\circ) \).
5. Round the result to the nearest square unit 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 given problem.

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 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.

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 crucial for calculating the area of the triangle using the angle C. Understanding how to apply these functions is fundamental for solving problems involving triangles in trigonometry.