Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Surveying To find the distance between two points \(A\) and \(B\) that lie on opposite banks of a river, a surveyor lays off a line segment \(A C\) of length 240 yards along one bank and determines that the measures of \(\angle B A C\) and \(\angle A C B\) are \(63^{\circ} 20^{\prime}\) and \(54^{\circ} 10^{\prime}\), respectively (see the figure). Approximate the distance between \(A\) and \(B\).

Step-by-Step Solution

Answer

The approximate distance between points A and B is 220.32 yards.

Step 1: Understand the Problem

We need to find the distance between points \(A\) and \(B\) which are on opposite banks of a river. We have a triangle \(\triangle ABC\) with \(AC = 240\) yards, \(\angle BAC = 63^{\circ} 20^{\prime}\), and \(\angle ACB = 54^{\circ} 10^{\prime}\). We will use the law of sines to find \(AB\).

Step 2: Calculate the Missing Angle

First, we determine the third angle of the triangle, \(\angle ABC\), using the fact that the sum of angles in a triangle is \(180^{\circ}\). \(\angle ABC = 180^{\circ} - 63^{\circ} 20^{\prime} - 54^{\circ} 10^{\prime} = 62^{\circ} 30^{\prime}\).

Step 3: Apply the Law of Sines

The law of sines states that \(\frac{AC}{\sin \angle ABC} = \frac{AB}{\sin \angle ACB}\). Thus, \(AB = AC \times \frac{\sin \angle ACB}{\sin \angle ABC}\).

Step 4: Substitute the Values

Substitute values into the equation: \(AC = 240\) yards, \(\angle ACB = 54^{\circ} 10^{\prime}\), \(\angle ABC = 62^{\circ} 30^{\prime}\).\[ AB = 240 \times \frac{\sin(54^{\circ} 10^{\prime})}{\sin(62^{\circ} 30^{\prime})} \]

Step 5: Calculate Sines of Angles

Calculate \(\sin(54^{\circ} 10^{\prime})\) and \(\sin(62^{\circ} 30^{\prime})\) using a calculator: \(\sin(54^{\circ} 10^{\prime}) \approx 0.811\) and \(\sin(62^{\circ} 30^{\prime}) \approx 0.883\).

Step 6: Compute the Distance AB

Substitute the sine values back into the equation: \[ AB = 240 \times \frac{0.811}{0.883} \approx 240 \times 0.918 \approx 220.32 \text{ yards}\].

Key Concepts

Triangle Geometry

Understanding the basics of triangle geometry is crucial, especially when it comes to solving problems involving distances and angles. Triangles are simple, three-sided shapes, but there are a few important properties to keep in mind:

• The sum of the interior angles in any triangle is always 180 degrees. This rule helps us find any unknown angle if we know the other two.
• Triangles can have different forms such as isosceles, scalene, or equilateral, but in any type, angle properties remain sacred.

In the given problem, triangle geometry assists in calculating a missing angle, ensuring we leverage each angle's role effectively while navigating the triangle structure. Triangle edges, by contrast, are explored using known trigonometry rules derived from these angles, making triangle geometry the backbone of the solution.

Angle Calculation

Calculating angles is a crucial aspect of solving many geometry problems. It's especially vital in triangle-related exercises as knowing the angle measures can help unlock other unknown metrics. To effectively calculate angles:

• Remember that in a triangle, the sum of all angles equals 180 degrees. Use this to determine the unknown angle if the other two angles are provided.
• To find the missing angle in the given exercise, subtract the sum of the known angles from 180 degrees. This direct approach provides clarity and precision to your solution, ensuring accurate angle calculation is straightforward.

The ability to calculate angles with confidence and accuracy is a necessary skill for any geometry challenge, directly impacting the solution's correctness.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent play a key role in solving triangle problems. In our exercise, the law of sines serves as a bridge between known and unknown values:

• The law of sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where \(a, b, c\) are the sides opposite to angles \(A, B, C\) respectively.
• This formula allows for the calculation of a side of the triangle when two angles and one side are known, as utilized in our example.

Access to a calculator simplifies the process of finding the sine of angles, translating complex tasks into manageable steps. Mastery of trigonometric functions and their applications is essential for efficiently solving geometrical problems.