Problem 26 - Algebra 2 and Trigonometry

Question

Two boats leave a dock to cross a river that is 80 meters wide. The first boat travels to a point that is 100 meters downstream from a point directly opposite the starting point, and the second boat travels to a point that is 200 meters downstream from a point directly opposite the starting point. a. Let \(x\) be the measure of the angle between the river's edge and the path of the first boat and \(y\) be the measure of the angle between the river's edge and the path of the second boat. Find \(\tan x\) and \(\tan y .\) b. Find the tangent of the measure of the angle between the paths of the boats.

Step-by-Step Solution

Answer

\(\tan x = 0.8\), \(\tan y = 0.4\), angle between paths: \(\approx 0.303\).

Step 1: Understand the problem

We need to find the tangent of angles for two boats as they cross a river. These angles are formed by the path of each boat and the perpendicular line from the starting point to their respective destinations on the other bank. The first boat lands 100 meters downstream and the second 200 meters, both starting from the same 80 meters width across the river.

Step 2: Identify the right triangles

For each boat, we form a right triangle. The width of the river is the opposite side (80 meters) for both triangles. The base, or adjacent side to the angle with the river's edge for the first boat is 100 meters, and for the second boat, it is 200 meters.

Step 3: Calculate \(\tan x\) for the first boat

Using the definition of tangent in a right triangle, \(\tan x = \frac{\text{opposite}}{\text{adjacent}}\). For the first boat, this is \(\tan x = \frac{80}{100} = 0.8\).

Step 4: Calculate \(\tan y\) for the second boat

Similarly, for the second boat, \(\tan y = \frac{\text{opposite}}{\text{adjacent}}= \frac{80}{200} = 0.4\).

Step 5: Calculate the tangent of the difference of angles \(x\) and \(y\)

The tangent of the angle between two paths is given by the formula \(\tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}\). Substitute the values \(\tan x = 0.8\) and \(\tan y = 0.4\): \[\tan(x-y) = \frac{0.8 - 0.4}{1 + 0.8 \times 0.4} = \frac{0.4}{1 + 0.32} = \frac{0.4}{1.32} \approx 0.303\].

Key Concepts

Right Triangle

In trigonometry, the right triangle is a fundamental shape. It consists of two legs that form a right angle (90 degrees) and a hypotenuse, which is the side opposite the right angle. As we explore problems involving right triangles, it's important to understand the roles of each side:

• The hypotenuse is the longest side of a right triangle.
• The opposite side is the side opposite the angle of interest.
• The adjacent side is the side next to the angle of interest, but not the hypotenuse.

In our exercise, each boat’s path across the river forms a right triangle. The river’s width (80 meters) represents the opposite side for both triangles, while the downstream distances (100 and 200 meters) represent the adjacent sides for the angles defined by each boat’s path.

Tangent Function

The tangent function is one of the primary functions in trigonometry, playing a crucial role in solving right triangle problems. It relates the measures of angles to the lengths of the opposite and adjacent sides of a right triangle. For an angle \( \theta \) in a right triangle:

• The tangent is the ratio of the length of the opposite side to the length of the adjacent side.
• Mathematically, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

In the context of the exercise, \( \tan x \) and \( \tan y \) are the tangents for the angles formed by each boat’s path with the riverbank. We calculated \( \tan x = 0.8 \) for the first boat and \( \tan y = 0.4 \) for the second, demonstrating how the tangent function helps us determine these values.

Angle Calculation

When faced with right triangle problems, calculating the angle often involves finding the tangent and using inverse trigonometric functions for precise angle measures. The exercise shows how we compute the tangents of angles \( x \) and \( y \):

• For the first boat, \( \tan x = \frac{80}{100} = 0.8 \).
• For the second boat, \( \tan y = \frac{80}{200} = 0.4 \).

Calculating the difference in angles involves understanding tangent differences. The formula \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \) enables us to find the tangent of the angle between the boat paths, yielding approximately \( 0.303 \). This approach highlights trigonometry’s power in solving geometric problems intuitively.

Math Problem Solving

Effective math problem solving in trigonometry requires clear understanding, breaking the problem into manageable steps, and applying mathematical principles correctly. Each step should:

• Clarify the problem, e.g., identifying right triangles as in the boat paths.
• Utilize known formulas, like the tangent function for angle calculations.
• Consider all given data, such as the river’s width and downstream distances.

In our exercise, by defining the problem and applying trigonometric functions, we systematically found the tangents of individual angles and their differences. Such step-by-step problem solving showcases the logical process in mathematics, making complex issues simpler and solvable.