Problem 6 - Algebra and Trigonometry

Question

True or False The area of a triangle equals one-half the product of the lengths of two of its sides times the sine of their included angle

Step-by-Step Solution

Answer

True

Step 1 - Identify Triangle Area Formula

The area of a triangle can be calculated using the formula \( \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \)where \( a \) and \( b \) are the lengths of two sides of the triangle, and \( C \) is the included angle between these two sides.

Step 2 - Compare Given Statement

Compare the given statement to the formula identified in Step 1. The statement claims the area is equal to one-half the product of the lengths of two sides times the sine of their included angle.

Step 3 - Validate the Statement

Since the given statement accurately describes the formula for the area of a triangle, we conclude that the statement is true.

Key Concepts

trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is essential for understanding various geometric shapes and solving problems involving triangles. One of the key functions in trigonometry is the sine function, which helps find the area of triangles in specific scenarios. Trigonometry makes it easier to solve complex problems involving angles and lengths, whether in right or non-right triangles. It's used not only in academics but also in fields like engineering, physics, and architecture.

sine function

The sine function (sin) is one of the primary functions in trigonometry. It is particularly useful for finding the components of angles in triangles. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. In formula form: \( \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). Using the sine function in the context of triangle areas, as seen in the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \), demonstrates the versatile application of this function. Here, the angle C is not necessarily part of a right triangle, showcasing how sine helps in non-right triangle calculations as well.

triangle properties

Understanding triangle properties is crucial for using trigonometric formulas effectively. Triangles have several properties, including:

• Three sides and three angles
• Sum of all interior angles always equals 180 degrees
• If two sides and the included angle are known, we can determine the area using the formula \( \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \).

These properties help in solving problems related to triangles and are foundational to more advanced geometric and trigonometric concepts. The ability to apply these properties correctly is important in many practical applications, such as navigation and construction.