Problem 38 - Trigonometry

Question

Distance to a Rocket Tom and Fred are \(3.5\) miles apart watching a rocket being launched from Vandenberg Air Force Base. Tom estimates the bearing of the rocket from his position to be \(S 75^{\circ} \mathrm{W}\), while Fred estimates that the bearing of the rocket from his position is N \(65^{\circ} \mathrm{W}\). If Fred is due south of Tom, how far is each of them from the rocket?

Step-by-Step Solution

Answer

Tom is approximately 22.86 miles from the rocket and Fred is approximately 19.46 miles away.

Step 1: Understand the Geometrical Setup

Identify the positions of Tom, Fred, and the rocket on a coordinate setup given their bearings and distances. Tom is at point A, Fred at point B, and the rocket at point C. Tom and Fred are 3.5 miles apart, with Fred directly south of Tom, meaning they form a vertical line segment, AB, with Tom north of Fred.

Step 2: Analyze Bearings

Translate the bearings into angle measurements. Tom's bearing to the rocket is \(S 75^{\circ} W\), indicating a 75-degree angle from the south line westward, forming a 105-degree counterclockwise angle from the south line. Fred's bearing is \(N 65^{\circ} W\), forming a 65-degree angle westward from the north line, equivalent to a 245-degree angle counterclockwise.

Step 3: Convert Bearings to Triangle Angles

Calculate the interior angles of triangle \(ABC\). Angle \(CAB\) is \(105^{\circ}\) at Tom's position. The direction from Fred, angle \(CBA\), is \(65^{\circ}\). Since the points form a triangle, use the formula for the sum of angles in a triangle: \(180^{\circ}\). Therefore, angle \(ACB = 180^{\circ} - 105^{\circ} - 65^{\circ}\).

Step 4: Use Law of Sines

Apply the Law of Sines to find distances AC and BC. The law states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a,b,c\) are sides and \(A,B,C\) are the opposite angles. Calculate \(AC\) using \(\frac{AC}{\sin CBA} = \frac{3.5}{\sin ACB}\). Solve for \(BC\) similarly using \(\frac{BC}{\sin CAB} = \frac{3.5}{\sin ACB}\).

Step 5: Calculate Lengths AC and BC

First, determine angle \(ACB\) as \(10^{\circ}\). Substitute the known values: \(\frac{AC}{\sin 65^{\circ}} = \frac{3.5}{\sin 10^{\circ}}\). Solve for \(AC\), \(AC \approx \frac{3.5 \times \sin 65^{\circ}}{\sin 10^{\circ}}\). Similarly solve for \(BC\), \(BC \approx \frac{3.5 \times \sin 105^{\circ}}{\sin 10^{\circ}}\).

Step 6: Compute Final Distance Results

Calculate the approximate length of \(AC\) and \(BC\). \(AC \approx 22.86\) miles and \(BC \approx 19.46\) miles. Therefore, the distance of Tom from the rocket is about 22.86 miles and Fred is about 19.46 miles from the rocket.

Key Concepts

Law of Sines

The Law of Sines is a fundamental relationship in trigonometry that helps solve for unknown sides and angles in a triangle. This law states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]where \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. In the context of our example involving Tom, Fred, and the rocket, this law is the key to finding how far each of them is from the rocket.When we are given one side of the triangle and some angles, we can use the Law of Sines to find the unknown distances. For instance, if you know one angle and its opposite side, you can use this ratio to determine other unknown sides, provided you know at least one other angle or side. This translates the problem of bearings and angles into solvable equations with real-world applications. It’s especially powerful in navigation, surveying, and astronomy because it handles non-right triangles that are frequently encountered in these fields.

Bearings

Bearings are methods used to describe the direction or angle in navigation. They provide a precise way to indicate directions without the ambiguity of regular compass directions. A bearing is typically measured in degrees from the North, going clockwise. However, in certain cases, they can be given as a combination of cardinal directions (e.g., "S 75° W"). This indicates a 75-degree angle westward from the south line. Understanding bearings requires being comfortable with these circular measurements and often involves converting them into usable angles for trigonometric calculations.

Using Bearings to Determine Triangle Angles

When solving problems involving bearings, such as locating the position of a rocket in flight, it's crucial to convert these directional bearings into angles within a triangle. A bearing like "S 75° W" can determine the interior angles needed to apply the Law of Sines, turning the descriptions into numerical angles that can be used in calculations.

Triangle Angles

Triangle angles play a vital role in solving trigonometric problems, as they form the basis for applying various trigonometric laws, including the Law of Sines. A fundamental property of triangles is that the sum of their interior angles is always 180 degrees. This fact is indispensable when determining unknown angles in any triangle setup.In the scenario with Tom, Fred, and the rocket, the angles were essential to solve the problem. Firstly, by determining the bearings, we translated them into two known angles of the triangle \(ABC\): 105° for Tom's position and 65° for Fred's position. With these values, the remaining angle \(ACB\) could be calculated using:\[ ACB = 180° - CAB - CBA \]Knowing all three angles of the triangle allows us to effectively use trigonometric principles to find unknown sides. Understanding this interplay of angles is critical in navigation and geographical problem-solving scenarios. It allows predictions and calculations about distances based on angles delineated by real-world observations.