Problem 35 - Precalculus Mathematics for Calculus

Question

Distance Across a River To find the distance across a river, a surveyor chooses points \(A\) and \(B\), which are \(200 \mathrm{ft}\) apart on one side of the river (see the figure). She then chooses a reference point \(C\) on the opposite side of the river and finds that \(\angle B A C \approx 82^{\circ}\) and \(\angle A B C \approx 52^{\circ} .\) Approximate the distance from \(A\) to \(C\). GRAPH CANT COPY

Step-by-Step Solution

Answer

The distance from A to C is approximately 219 feet.

Step 1: Analyze Triangle ABC

The surveyor forms a triangle \( \triangle ABC \) with the known distance \( AB = 200 \text{ feet} \) and angles \( \angle BAC = 82^{\circ} \) and \( \angle ABC = 52^{\circ} \). We need to find \( AC \), the distance from \( A \) to \( C \).

Step 2: Find the Third Angle in Triangle ABC

To use the Law of Sines, we need all angles in \( \triangle ABC \). The sum of angles in a triangle is \( 180^{\circ} \), so find the missing angle \( \angle ACB \) as follows: \( \angle ACB = 180^{\circ} - 82^{\circ} - 52^{\circ} = 46^{\circ} \).

Step 3: Apply the Law of Sines

Use the Law of Sines which states \( \frac{AC}{\sin(\angle ABC)} = \frac{AB}{\sin(\angle ACB)} \). Plug in the known values: \( \frac{AC}{\sin(52^{\circ})} = \frac{200}{\sin(46^{\circ})} \).

Step 4: Solve for AC

Rearrange the equation to solve for \( AC \): \( AC = \frac{200 \cdot \sin(52^{\circ})}{\sin(46^{\circ})} \).

Step 5: Calculate AC Using Trigonometric Values

Calculate the sine values using a calculator: \( \sin(52^{\circ}) \approx 0.7880 \) and \( \sin(46^{\circ}) \approx 0.7193 \). Substitute and calculate \( AC \): \( AC = \frac{200 \times 0.7880}{0.7193} \approx 219.01 \text{ feet} \).

Key Concepts

Law of Sines

When dealing with non-right triangles in trigonometry, the **Law of Sines** is a powerful tool that helps find unknown side lengths and angles. This law is based on the relationship between the sides and angles of a triangle.
To understand the Law of Sines, consider a triangle with sides labeled as \( a, b, \) and \( c \), and their opposite angles labeled as \( A, B, \) and \( C \), respectively. The formula is:

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

This equation means that the ratio of the length of a triangle side over the sine of its opposite angle is the same for all three sides of the triangle.
In the context of solving real-world problems, like finding distances in surveying, the Law of Sines can be applied once we know at least one angle-side opposite pair completely, which allows us to solve for the missing elements of the triangle using given data and simple trigonometric calculations.
Using this law simplifies problems involving non-right triangles, making them accessible through basic trigonometric operations.

Triangle Angle Sum

One of the fundamental properties of triangles is the **Triangle Angle Sum**, which states that the sum of the internal angles in any triangle is always \( 180^{\circ} \). This property forms the basis for several triangle calculations.
In practical applications, such as surveying a land area or calculating distances, knowing two angles of a triangle allows us to find the third.
If two angles of a triangle are known, the third angle can be calculated using:

• \( \angle C = 180^{\circ} - \angle A - \angle B \)

This principle helps complete the set of data needed to apply the Law of Sines or Cosines.
Understanding that the angle sum is always \( 180^{\circ} \) helps avoid errors in calculations, ensuring accuracy when solving problems that involve other trigonometric principles and ensuring a solution can be consistently applied.

Surveying Applications

In the field of **Surveying**, trigonometry plays a crucial role, especially when it comes to measuring hard-to-access distances, like across rivers or rugged terrains. Surveyors frequently use concepts like the Law of Sines and Triangle Angle Sum to compute accurate measurements without needing to physically traverse the difficult terrain.
In our scenario, the surveyor can determine the distance across a river by forming a triangle with well-positioned and accessible points along both sides of the river. They measure two angles and the length of one side directly.
Using these measurements:

• The surveyor employs the Triangle Angle Sum to find any missing angles, completing the triangle's set of information.
• The Law of Sines then computes the unknown distances, allowing the surveyor to apply simple and reliable mathematical technique to solve what otherwise might be a tricky measurement problem.

Surveying with trigonometric principles ensures practical, non-invasive methods to achieve high precision, essential for government projects, construction, and environmental study areas, thereby making these mathematical principles invaluable.