Problem 54 - Precalculus with Limits

Question

Surveying A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries?

Step-by-Step Solution

Answer

The angles that the frontage of 115 meters make with the other two boundaries can be found by using the law of cosines. Compute the first angle \(\theta_{1}\) using the sides of lengths 115, 76, and 92 meters. Next, compute the second angle \(\theta_{2}\) using the sides of lengths 115, 92, and 76 meters. Lastly, find the third angle \(\theta_{3}\) by subtracting the sum of \(\theta_{1}\) and \(\theta_{2}\) from 180 degrees.

Step 1: Calculate the first angle using law of cosines

We can begin by calculating one of the angles, say the angle opposite to the side of length 115m (\(\theta_{1}\)). By the law of cosines, the formula to calculate this angle is: \[ \theta_{1} = \arccos \left(\frac{76^2 + 92^2 - 115^2}{2 * 76 * 92}\right) \] After calculating using this formula, we should get the value of the first angle.

Step 2: Calculate the second angle using law of cosines

After calculating the first angle, we can calculate the next angle which is opposite to the side of length 76m (let's denote it as \(\theta_{2}\)). The formula to calculate \(\theta_{2}\) is: \[ \theta_{2} = \arccos \left(\frac{115^2 + 92^2 - 76^2}{2 * 115 * 92}\right) \] After computations, we will get the value of the second angle.

Step 3: Derive the third angle

As we know, the sum of the angles in a triangle equals 180 degrees. So, the third angle \(\theta_{3}\) (opposite to the side of length 92m) can be derived by subtracting the sum of the first two angles from 180 degrees: \[ \theta_{3} = 180 - (\theta_{1} + \theta_{2}) \] This will get us the value of the third angle.

Key Concepts

Triangle Measurement

When working with triangles, especially in surveying or geometry problems, measuring the sides accurately is essential. Each side of a triangle plays a pivotal role in determining the angles within it. In our specific problem, we have a triangle with three given sides: 115 meters, 76 meters, and 92 meters.

It's important to remember:

• The longest side of the triangle is often associated with the largest angle opposite it. In this case, 115 meters is the longest side.
• The measurements of these sides form the basis for applying trigonometric laws, such as the Law of Cosines, which helps in calculating the triangle's angles.

Before moving to angles, ensure the side lengths are consistent with your calculations and measurements.

Angle Calculation

Calculating angles in a triangle when the lengths of all sides are known is made straightforward with the Law of Cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. For our exercise, the Law of Cosines lets us find one angle using the formula:
\[ \theta_1 = \arccos \left(\frac{b^2 + c^2 - a^2}{2bc}\right) \]
In our case, to calculate the angle opposite to the longest side (115 meters), we'll use:
\[ \theta_1 = \arccos \left(\frac{76^2 + 92^2 - 115^2}{2 \times 76 \times 92}\right) \]
After calculating this angle, we move to the next angle. Using the same method:
\[ \theta_2 = \arccos \left(\frac{115^2 + 92^2 - 76^2}{2 \times 115 \times 92}\right) \]
The remaining angle is found by remembering the sum of angles in a triangle is 180 degrees:

• The third angle will be the remaining measure after summing the two calculated angles and subtracting from 180 degrees.

This method ensures precise and consistent angle measurements in any triangular setup.

Surveying Techniques

Surveying often involves measuring land areas that are not perfect geometries. Triangles are a common shape surveyed, requiring precise methods to determine unknown angles and sides. In surveying, the use of trigonometry, particularly the Law of Cosines, is standard practice for these calculations.

• Ensure accurate measurement of sides - Any errors in measuring sides lead to incorrect angle computation.
• Use reliable tools to compute angles - Calculators capable of trigonometric functions are essential.
• Cross-verify - After calculating angles, ensure their total equals 180 degrees for validation.

Surveying requires precision, and using mathematical principles like the Law of Cosines helps accurately map and understand land boundaries.