Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Area of a Triangular Lot A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 meters and 21.3 meters, and the angle between them is \(42.2^{\circ} .\) What is the area of the lot?

Step-by-Step Solution

Answer

The area of the triangular lot is approximately 373.19 square meters.

Step 1: Identify the Formula

To find the area of a triangle when two sides and the included angle are known, use the formula: \[ \text{Area} = \frac{1}{2}ab\sin(C) \] where \( a \) and \( b \) are the sides of the triangle, and \( C \) is the angle between them.

Step 2: Substitute the Known Values

Substitute the values from the problem into the formula: \( a = 52.1 \) meters, \( b = 21.3 \) meters, and \( C = 42.2^{\circ} \). The equation becomes: \[ \text{Area} = \frac{1}{2} \times 52.1 \times 21.3 \times \sin(42.2^{\circ}) \]

Step 3: Calculate the Sine of the Angle

Find \( \sin(42.2^{\circ}) \) using a calculator. \[ \sin(42.2^{\circ}) \approx 0.6718 \]

Step 4: Perform the Multiplication

Now substitute the sine value back into the equation and perform the multiplications: \[ \text{Area} = \frac{1}{2} \times 52.1 \times 21.3 \times 0.6718 \] First, calculate \( 52.1 \times 21.3 \), then multiply it by \( 0.6718 \), and finally divide by 2.

Step 5: Final Calculation

Calculate the result: \[ \text{Area} \approx \frac{1}{2} \times 1110.63 \times 0.6718 \approx 373.19 \] So, the area of the triangular lot is approximately 373.19 square meters.

Key Concepts

Area of a Triangle

When calculating the area of a triangle, it is important to know the relationship between the sides and angles. There are different methods for different kinds of triangles. For a triangle with two known sides and the included angle, the area can be found using a specific formula.
To find the area of a triangle in such cases, the formula to use is:

• \( \text{Area} = \frac{1}{2}ab\sin(C) \)

This formula is derived from dividing the triangle into two right-angled triangles. Here, \( a \) and \( b \) are the lengths of the known sides, and \( C \) is the angle between these sides.
By applying the sine function, we determine the height of the reflected triangle. Multiplying the base and height provides the area calculation. The factor \( \frac{1}{2} \) comes from the standard area formula for triangles.

Sine Function

The sine function is an essential part of trigonometry that helps in understanding the relationship between the angles and sides of triangles. It is particularly useful in right-angle triangles, but extends to all triangles when applied correctly.
The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Symbolically, it's written as:

• \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)

In the context of the current exercise, we use the angle between two sides of the triangle. By knowing the sine of this angle, you're able to calculate other measurements like the area.
Calculators and trigonometric tables are commonly used tools to find and use the sine values of angles in problem-solving scenarios.

Law of Sines

The Law of Sines is a powerful principle in trigonometry that allows the solving of unknown sides and angles in any triangle, not just right-angled ones. The law states that:

• \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

This means each side length of a triangle divided by the sine of its opposite angle is equivalent to the other side ratios. This concept is useful when trying to find unknown lengths or angles of non-right triangles where certain other measures are known.
Although not directly utilized in the initial exercise, the understanding of the Law of Sines is crucial for more complex triangular problems one might encounter in trigonometry. Coupled with the Law of Cosines, it forms the backbone of solving many trigonometric problems in geometry.