Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Radio direction finders are at points \(A\) and \(B\), which are 3.46 miles apart on an east-west line, with \(A\) west of \(B\). From \(A,\) the bearing of a certain radio transmitter is \(47.7^{\circ}\); from \(B,\) the bearing is \(302.5^{\circ} .\) Find the distance of the transmitter from \(A\).

Step-by-Step Solution

Answer

The transmitter is approximately 1.93 miles away from point A.

Step 1: Understand the Bearings

The problem gives us bearings from two points, \(A\) and \(B\), to a transmitter. A bearing is an angle measured clockwise from the north direction.\- From \(A\), the bearing to the transmitter is \(47.7^{\circ}\).\- From \(B\), the bearing to the transmitter is \(302.5^{\circ}\).

Step 2: Identify Triangle and Angles

The points form a triangle \( \triangle ABT \), where \(T\) is the position of the transmitter. We need to find the angles \( \angle ATB \) formed by the intersection of the lines from \(A\) and \(B\) towards \(T\). \( \angle ATB = 180^{\circ} - (90^{\circ} - 47.7^{\circ}) - (302.5^{\circ} - 270^{\circ}) \). Calculating gives \( \angle ATB = 180^{\circ} - 42.3^{\circ} - 32.5^{\circ} = 105.2^{\circ}\).

Step 3: Use Law of Sines

The Law of Sines states that \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). In \(\triangle ABT\), \(a = 3.46\), \(\angle ATB = 105.2^{\circ}\), \( \angle TAB = 32.5^{\circ}\), and \( \angle ABT = 42.3^{\circ}\).Using the Law of Sines: \(\frac{BT}{\sin 32.5^{\circ}} = \frac{3.46}{\sin 105.2^{\circ}}\).

Step 4: Calculate Transmitter Distance

We want to find the distance \(BT\), so using the equation from Step 3 and solving for \(BT\):\[BT = \frac{3.46 \cdot \sin 32.5^{\circ}}{\sin 105.2^{\circ}}\]Using a calculator, we find \(BT \approx 1.93\) miles.

Key Concepts

Bearings

Bearings are an essential concept in navigation and trigonometry. They express directions as angles, measuring clockwise from the north. This makes them crucial in mapping paths and positions. For example, a bearing of \( 47.7^{\circ} \) means you would turn \( 47.7^{\circ} \) clockwise from true north to find your direction. From the problem, we have bearings from two points, \( A \) and \( B \). These bearings help define the direction from these points to a common transmitter. Understanding bearings involves: - Recognizing "0 degrees" as north, "90 degrees" as east. - "180 degrees" means south, and "270 degrees" means west.The bearings guide us in forming the triangle needed to solve the problem further. When bearing from \( B \) is \( 302.5^{\circ} \), it indicates a direction slightly clockwise from due west.

Law of Sines

The Law of Sines is a powerful tool used in trigonometry to solve triangles, specifically non-right triangles. The law states that in any triangle, the ratio of each side length to the sine of its opposite angle is constant. This can be expressed with the formula: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This formula is instrumental in finding unknown sides or angles of a triangle when certain other angles and sides are known. For the triangle in our exercise, \( \triangle ABT \), we knew:- The side \( AB \) is 3.46 miles.- The bearing angles allow us to calculate interior angles \( \angle ATB \), \( \angle TAB \), and \( \angle ABT \).Using these angles, we apply the Law of Sines to find the unknown distance \( BT \), providing the complete distance solution from one point to a transmitter.

Triangle Geometry

Understanding triangle geometry is central to solving this type of problem, where the distance needs to be calculated indirectly. A triangle is defined by its three sides and three angles, and by knowing some of these, you can find others using geometrical theorems.In the context of bearings and directions:- The bearings help locate the angles.- These angles are essential to determine the internal angles of the triangle, which further aids in finding unknown lengths.In our problem:- \( \triangle ABT \) could be analyzed because two of its bearings were known.- Using subtraction and supplemental angle properties, we could find \( \angle ATB \), an interior angle critical for using the Law of Sines.These processes highlight how bearings integrate with triangle geometry, forming a cohesive method for resolving unknown distances in navigation and surveying tasks. Thus, a solid grasp of triangle properties and laws makes these computations possible and quite straightforward.