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Chapter 8: Problem 21

Find the area of the triangle. $$B=135.2^{\circ}, a=46.12 \mathrm{ft}, c=36.74 \mathrm{ft}$$

Short Answer

Expert verified
The area of the triangle is approximately 598.3 ft虏.

Step by step solution

01

Understand the Problem

We are given a triangle with an angle B and two sides a and c opposite to each other. We need to find the area of the triangle.
02

Apply the Sine Area Formula

The area of a triangle can be calculated using the formula: \[\text{Area} = \frac{1}{2} \times a \times c \times \text{sin}(B)\] We need angle B in degrees to use the sine function.
03

Substitute the Given Values

Substitute the given values into the formula: \[\text{Area} = \frac{1}{2} \times 46.12 \times 36.74 \times \text{sin}(135.2^{\text{\circ}})\]
04

Calculate the Sine of the Angle

Calculate \(\text{sin}(135.2^{\text{\circ}})\). Using a calculator, we find that \[\text{sin}(135.2^{\text{\circ}}) \, \approx \, 0.7071\]
05

Compute the Area

Now, substitute \(\text{sin}(135.2^{\text{\circ}})\) into the formula: \[\text{Area} = \frac{1}{2} \times 46.12 \times 36.74 \times 0.7071\] Calculate the product: \[\text{Area} \, \approx \, \frac{1}{2} \times 46.12 \times 36.74 \times 0.7071 = \frac{1}{2} \times 1196.6 = 598.3 \, \text{ft}^{2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Area Formula
The Sine Area Formula is a significant method in trigonometry to find the area of any triangle, particularly when we have two sides and the included angle. This formula is given by:

\[ \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \]
The formula works because it leverages the properties of the sine function associated with the given angle between the two sides. It simplifies the calculation otherwise required by the geometry formulas involving triangle height.
Trigonometry
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. Essential functions in trigonometry鈥攕ine, cosine, and tangent鈥攔elate the angles of a triangle to the lengths of its sides.

In our triangle area problem, the sine function is most crucial because it relates the angle to the proportion of the opposite side and hypotenuse in any right triangle:
\[ \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Understanding how to use these functions enables us to find unknown sides and angles and solve real-world problems efficiently.
Triangle Properties
Triangles are one of the most fundamental shapes in geometry, characterized by three sides, three angles, and three vertices. Key properties essential for understanding triangle area include:

  • Sum of Angles: The sum of the interior angles in any triangle is always 180 degrees.
  • Types of Triangles: Triangles can be classified by their sides (equilateral, isosceles, scalene) or their angles (acute, right, obtuse).
  • Area Formulas: Besides the Sine Area Formula, the area can also be found using the base-height method or Heron鈥檚 formula.

These properties are critical as they lay the foundation for more complex problem-solving techniques in trigonometry and geometry.

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