Problem 11 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle D E F, d=5.83, e=5.83, \mathrm{m} \angle F=48\)

Step-by-Step Solution

Answer

The area of \( \triangle DEF \) is approximately 12.6 square units.

Step 1: Identify the Known Elements

The problem provides two sides and the angle between them of triangle \( \triangle DEF \). The sides given are \( d = 5.83 \), \( e = 5.83 \), and the included angle \( \angle F = 48^\circ \).

Step 2: Use the Formula for the Area of a Triangle

For a triangle with two known sides and the included angle between them, the area \( A \) can be calculated using the formula: \[ A = \frac{1}{2} \cdot d \cdot e \cdot \sin(F) \] Where \( d \) and \( e \) are the sides of the triangle, and \( \angle F \) is the included angle.

Step 3: Calculate the Sine of the Angle

First, calculate \( \sin(48^\circ) \). Use a calculator to find that \[ \sin(48^\circ) \approx 0.7431 \].

Step 4: Substitute Values into the Area Formula

Substitute the known values into the formula for the area:\[ A = \frac{1}{2} \times 5.83 \times 5.83 \times 0.7431 \]

Step 5: Compute the Area

Calculate the area by performing the multiplication:\[ A = \frac{1}{2} \times 5.83 \times 5.83 \times 0.7431 \approx 12.6 \].

Step 6: Round to the Nearest Tenth

The calculated area \( \approx 12.6 \) is already in the nearest tenth.

Key Concepts

Trigonometry

Trigonometry is an exciting branch of mathematics that explores the relationships between the angles and sides of triangles. It's particularly useful in determining unknown angles or sides of a triangle when certain measurements are known. In our problem, trigonometry lets us find out the area of a triangle using its sides and included angle. With trigonometry, you’re not just limited to right-angled triangles; it applies to all triangles! This comes into play with geometric problems, physics applications, and even navigation. Trigonometry essentially lets us model and understand the world in various dimensions. It involves concepts like sine, cosine, and tangent, which help in these calculations.

Sine Function

The sine function is one of the most significant trigonometric functions used in various calculations involving angles and triangles. Understanding sine is crucial, especially when you're working with non-right triangles like in our example with triangle \( \triangle DEF \).

• The sine of an angle in a triangle is the ratio of the length of the opposite side to the length of the hypotenuse, specifically in a right-angled triangle. However, it can also be used for non-right triangles with the law of sines and area calculations.
• In our triangle, the sine function is used to determine the area using the formula: \[ A = \frac{1}{2} \cdot d \cdot e \cdot \sin(F) \]
• For angle \( F = 48^\circ \), we use calculators to find that \( \sin(48^\circ) \approx 0.7431 \).

The use of the sine function in this way allows for efficient calculations of areas, which is an excellent example of how powerful trigonometric functions can be.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. It’s at the heart of exploring the properties and measurements related to triangles. In this problem, geometry helps identify that we have an isosceles triangle since two sides are equal \(d = e = 5.83\). This information is vital for understanding the structure of the triangle and employing the correct formulas.Geometry also focuses on understanding and using specific properties of shapes, which helps in solving real-world problems, designing structures, and analyzing spaces. This mathematical discipline provides the tools to solve problems involving angles, lengths, areas, and volumes of various shapes, including triangles.

Triangle Properties

Triangles have unique properties that make them fundamental elements in geometry. Understanding these properties allows us to explore the relationships within the shape and with other components within a system or a problem.

• One main property of triangles is that the sum of their internal angles always equals \(180^\circ\). In the case of an isosceles triangle, we know two sides are equal, making two angles equal as well.
• The given triangle \( \triangle DEF \) has equal sides, which guides us directly towards using the cosine or sine rule for calculations.
• When finding the area of a triangle with two sides and the included angle, the formula \( A = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \) is ideal for these calculations.

Recognizing and using these properties of triangles efficiently aids in simplifying complex problems and calculations, such as finding areas and other measurements related to triangles in mathematics and real-world applications.