Problem 58 - Algebra and Trigonometry

Question

Describe an applied problem that can be solved using the Law of Cosines but not the Law of sines.

Step-by-Step Solution

Answer

The straight-line return distance (d) and the angle that the boat should take to return directly can be found using the Law of Cosines. The navigational problem of a boat that leaves port and changes directions at a certain point before aiming to return, cannot be solved using the Law of Sines, making the Law of Cosines more applicable.

Step 1: Identifying the problem

The problem is a navigational one. Start with a boat leaving port, moving a certain distance in one direction, changing course, and then moving a certain distance before trying to return to port. You need to find the straight-line distance to return to port as well as the angle of return.

Step 2: Making the diagram

Create a diagram of the scenario. Let the port be the point A, first direction of the boat be AB, and the second direction be BC. Therefore, the triangle will be ABC, and we are looking for AC, the straight-line distance back to port, and the angle at which the boat has to return, \( \angle BAC \). Assume AB and BC to be known distances and \( \angle ABC \) to be the known angle.

Step 3: Applying Law of Cosines

Apply the Law of Cosines, which states in a triangle with sides of length a, b, and c and with the angle C opposite the side of length c, \( c^{2} = a^{2} + b^{2} - 2ab \cos C \). Here, AC^2 = AB^2 + BC^2 - 2*AB*BC*\(\cos \angle ABC\). Calculate AC by taking the square root of the result obtained.

Step 4: Finding the return angle

To find the angle \( \angle BAC \), also use the Law of Cosines but rearranged now for the angle calculation: \( \cos C = \frac{a^{2} + b^{2} - c^{2}}{2ab} \). In this case, \( \angle BAC = \cos^{-1} (\frac{AB^{2} + AC^{2} - BC^{2}}{2 * AB * AC}) \). Calculate this angle using an appropriate tool.

Key Concepts

Navigational Problems

Navigational problems are practical challenges often faced in marine and aerial settings. These scenarios involve finding the best route or direction to take when traveling between various points. In many cases, these problems are solved using trigonometric principles to ensure accuracy in navigation. In our exercise, the challenge involved a boat that changed its course after initially setting out from a port. To navigate accurately, understanding the exact distance back to port and the bearing or angle of return is crucial. This kind of problem highlights the real-world application of trigonometry, particularly the Law of Cosines, to deal with distances and angles where straightforward measurements are not possible. A navigational problem requires a detailed approach, employing both diagrams and mathematical techniques to achieve the most efficient and safe path for travel.

Triangle Properties

Triangles are one of the foundational shapes in mathematics, having special properties making them both unique and useful in solving various problems. Every triangle consists of three sides and three angles, with specific relationships between them. A key property is the sum of the internal angles always equals 180 degrees.

• In navigational problems like ours, we deal with a scenario forming a triangle for analysis—a vital step in determining distances and bearings accurately.
• Identifying sides and angles correctly is crucial to applying formulas like the Law of Cosines.

Understanding whether a triangle is scalene, isosceles, or equilateral can affect what calculations and formulas might be most effective. In our problem, because we knew two sides and the included angle of a triangle, the use of the Law of Cosines became a necessity rather than choosing the Law of Sines. Triangle properties not only simplify many problems, but they are also the base for diving into more complex trigonometric solutions.

Applied Trigonometry

Applied trigonometry is the utilization of trigonometric principles to solve practical real-world problems. In many fields, such as navigation, surveying, and architecture, trigonometry helps solve challenges involving distances and angles. The Law of Cosines, an essential trigonometric tool, is particularly useful in situations where you know either:

• Two sides and the included angle of a triangle
• Three sides of a triangle

This formula is effective where the Law of Sines falls short, such as when the known elements do not include an angle opposite a known side. In the exercise, applied trigonometry allowed us to calculate both the direct return distance for the boat and the angle for the returning direction—illustrating how mathematical concepts translate to solving concrete navigational issues. This application exemplifies how trigonometry bridges abstract mathematics with tangible situations, ensuring precision and efficiency in operations that require navigation or any form of structured spatial reasoning.