Problem 44 - 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 front property line makes angles of \(A\) and \(B\) with the two other boundaries, where \(A\) and \(B\) can be computed using the Law of Cosines and a scientific or graphing calculator to find the arccosine.

Step 1: Identifying the triangle

Here, the triangular parcel of land is represented by a triangle with sides of lengths 115 meters (frontage), and the two other boundaries are of lengths 76 meters and 92 meters.

Step 2: Applying the Law of Cosines to find the first angle

First, one must find the angle made by the frontage with one of the other boundaries. Let's designate this as angle A, and its opposite side as 'a'. 'b' and 'c' are the lengths of the other two sides. In this case, 'a'=115m, 'b'=76m, and 'c'=92m. The Law of Cosines states that \(cosA=\frac{b^{2}+c^{2}-a^{2}}{2bc}\). Substituting the given values into this formula, we get \(cosA=\frac{76^{2}+92^{2}-115^{2}}{2*76*92}\). This can be solved to find the cosine of angle A.

Step 3: Finding the actual angle

To find the actual angle, take the arccosine (or inverse cosine) of the number you calculated in Step 2. This can be done using a calculator set to give the answer in degrees.

Step 4: Applying the Law of Cosines to find the second angle

Next, the same steps can be repeated to find the angle made by the frontage line with the other boundary. In this case, 'a'=115m, 'b'=76m, and 'c'=92m, but with 'b' and 'c' interchanged from Step 2 as the sides adjacent to the angle have been swapped. The formula for angle B with \(cosB=\frac{b^{2}+c^{2}-a^{2}}{2bc}\) now looks like \(cosB=\frac{92^{2}+76^{2}-115^{2}}{2*92*76}\).

Step 5: Finding the actual second angle

As in Step 3, you find the actual angle B by taking the arccosine of the number computed in Step 4.

Key Concepts

Surveying

Surveying is a technique essential for determining the three-dimensional positions of points and the distances and angles between them. In the context of a triangular parcel of land, we use surveying to understand the layout by calculating the angles between the land's boundaries. This information helps in numerous practical applications, such as planning construction, delineating property boundaries, and determining the best use of the land.
The process often involves mathematics, especially geometry and trigonometry, to make precise computations in defining boundaries accurately. Surveyors frequently use tools and technology like theodolites or laser levels, but understanding the underlying math is key. For instance, when calculating angles, the Law of Cosines comes into play, which connects distances with angles in triangular settings, like in our exercise on the triangular parcel of land.

Triangle Geometry

The study of triangle geometry is fundamental in understanding various aspects of shapes and their properties. Triangles, being the simplest polygon with three edges and corners, are widely studied in mathematics. They can have various types, including equilateral, isosceles, and scalene, distinguished by their side lengths and angles.
In problems like our exercise with the land parcel, the triangle is scalene, meaning all sides are of different lengths. To solve such problems, we use relationships between the sides and angles, often through the Law of Cosines. This law is a crucial tool that extends the Pythagorean theorem to find an unknown angle when all three side lengths are known. The formula for any angle of a triangle could be written as \( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \), where \( a \), \( b \), and \( c \) are the side lengths.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for solving trigonometric equations, particularly when we need to determine angles from known values. In our triangle land parcel problem, once the cosine of an angle is determined using the Law of Cosines, the next step is to find the actual angle. This is where inverse trigonometric functions come in, specifically the arccosine function.
The arccosine function, written as \( \cos^{-1} \), is used to find an angle when its cosine value is known. For instance, if you calculate the cosine of an angle to be a certain value, applying the arccosine function on a calculator set to degrees gives you the angle in degrees. Such procedures are pivotal in land surveying and many engineering applications where knowing precise angles is necessary.