Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the area of the triangle having the indicated angle and sides. \(C=110^{\circ}, \quad a=6, \quad b=10\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 28.68 square units.

Step 1: Identify the Given Parameters

The angle \( C = 110^{\circ} \) and two sides of the triangle are given, side a = 6 units and side b = 10 units.

Step 2: Plug Values Into Formula

The area of a triangle with sides a and b and an included angle C can be calculated using the formula, \( Area = \frac{1}{2}ab \sin C \). Thus, substitute the given values into the formula to get \( Area = \frac{1}{2} * 6 * 10 * \sin(110^{\circ}) \).

Step 3: Calculate the Area

Perform the multiplication and utilise the sine function on your calculator to find the area. Make sure your calculator is set to degrees.

Key Concepts

Triangle Area

Finding the area of a triangle is a key application of trigonometry and geometry. Specifically, when you know two sides of the triangle and the angle between them, you can use a special formula. This formula is: \[Area = \frac{1}{2}ab \sin C\]Where:

• \( a \) and \( b \) are the lengths of the known sides
• \( C \) is the angle between these two sides

This formula is perfect when you don't have the height of the triangle straight away but have these other measurements.
In the context of the triangle given in the problem, the formula simplifies the process by using the sine of the angle instead of calculating an actual height. Calculators often have a sine function button used to find \( \sin C \) for obtaining precise results. Just ensure your calculator is set to the correct unit, in this case, degrees. This formula is an efficient way to solve triangle area problems easily and swiftly.

Sine Rule

The sine rule is a vital tool in trigonometry. Although it was not directly used in this problem, understanding it provides fundamental knowledge for solving various triangle problems. The sine rule states:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This rule is used primarily in situations where you have:

• Two angles and one side
• Two sides and an angle that isn't between them

Although the original problem used the formula for the area of a triangle, understanding how the sine rule works can help solve related problems.
It helps find unknown sides or angles, extending its usefulness beyond just finding areas. It connects the angles and sides, forming a harmonious relationship throughout the triangle, showing trigonometry's elegance.
Whenever you're tackling triangles, having the sine rule in your toolset makes for a more comprehensive understanding of how triangles work.

Geometry

Geometry is the study of shapes and their properties. Triangles, as a fundamental shape in geometry, represent a core area of study. In problems like the one given, knowledge of geometry helps you understand how shapes interrelate and how you can calculate different properties like angles and areas.
Triangles come in various types, such as equilateral, isosceles, and scalene, each with unique properties but common foundational principles. The problem involves a scalene triangle with all sides and angles different. These principles remain constant regardless of the type.
When you know a triangle's certain sides and angles, tools like the Pythagorean theorem, trigonometric ratios, and properties specific to triangle types come into play. Applying these geometrical principles ensures you solve problems accurately while understanding the deeper relationship between various elements in triangles.
For students, grasping how geometry operates in problems, like calculating areas using trigonometry, adds depth and confidence to their mathematical journey.