Problem 53 - Algebra and Trigonometry

Question

Find the area of a triangle with sides of length 10 and 22 and included angle \(10^{\circ} .\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 19.13 square units.

Step 1: Understand the Formula

To find the area of a triangle given two sides and the included angle, use the formula: \( \text{Area} = \frac{1}{2}ab \sin(C) \). Here, \( a = 10 \), \( b = 22 \), and \( C = 10^{\circ} \).

Step 2: Calculate the Sine of the Angle

Calculate the sine of the angle \( C = 10^{\circ} \). Use a calculator to find \( \sin(10^{\circ}) \).

Step 3: Substitute the Values into the Formula

Substitute the known values into the area formula: \( \text{Area} = \frac{1}{2} \times 10 \times 22 \times \sin(10^{\circ}) \).

Step 4: Perform the Calculation

Calculate \( \text{Area} = \frac{1}{2} \times 10 \times 22 \times \sin(10^{\circ}) \) to get the area. Use a calculator for the multiplication after finding \( \sin(10^{\circ}) \).

Step 5: Write the Final Answer

Compute the result to find the area of the triangle, ensuring the use of the calculator for precise values. The result provides the area in square units.

Key Concepts

Triangle Area Calculation

When it comes to finding the area of a triangle, especially when you don't have the height, there's a handy formula that uses the lengths of two sides and the sine of the included angle. This formula is useful because a triangle often doesn't come with clear height information in real-world applications. The equation is:\[ \text{Area} = \frac{1}{2}ab \sin(C) \]- Here, \(a\) and \(b\) are the lengths of the two sides.- \(C\) is the angle between those two sides.This method is particularly efficient for non-right triangles where traditional base-height calculations are cumbersome. In problems where only side lengths and an angle are given, this approach is both straightforward and incredibly effective.

Sine Function

The sine function is a key player in trigonometry and for triangle calculations. The sine of an angle in a triangle relates the angle to the ratio of the opposite side over the hypotenuse in a right triangle. However, for our current application, we aren't dealing with a right triangle, and yet, the sine function still plays a crucial role through trigonometric identities.- It is important to understand how to find the sine of an angle.- For angles not found on basic trigonometric tables, a calculator is typically used.For our example, finding \( \sin(10^{\circ}) \) requires the precision of a scientific calculator, as it gives us a necessary multiplier in the area formula. The sine function thus bridges the gap between angle measurements and length calculations.

Geometry

Geometry encompasses the study of shapes and their properties, and triangles are one of the building blocks in this field. Understanding the principles of geometry helps us efficiently solve problems related to area and angles. - Geometric formulas often derive from understanding shape properties and relationships. - With triangles, these include both standard calculations and trigonometric insights. In this scenario, we see geometry’s practical side by calculating the area with sides and angles. It's a reminder that geometry isn't just theoretical but also has tangible applications, like determining the needed material for a triangular section in construction or other fields.

Triangle Properties

Triangles have unique properties that make them both fascinating and highly applicable in various domains. The properties we often utilize include:- The sum of interior angles always equals \(180^{\circ}\).- Knowing two sides and the included angle can provide significant insights such as determining the area.These properties not only simplify multiple calculations but also provide avenues to appreciate the inherent symmetry and balancing of angles and lengths. By understanding these fundamentals, we are better equipped to dissect more complex problems, ensuring solutions are accurate and easy to comprehend.