Problem 19 - Algebra 2 and Trigonometry

Question

From the top of a lighthouse 75 feet high, the cosine of the angle of depression of a boat out at sea is \(\frac{4}{5} .\) To the nearest foot, how far is the boat from the base of the lighthouse?

Step-by-Step Solution

Answer

The boat is 100 feet from the base of the lighthouse.

Step 1: Understand the Problem

The problem involves trigonometry and a right triangle. The lighthouse, the boat, and the horizontal line from the base of the lighthouse to the point directly below the boat form a right triangle. The height of the lighthouse is 75 feet, and we know the cosine of the angle of depression from the lighthouse to the boat is \(\frac{4}{5}\). We need to find the horizontal distance from the base of the lighthouse to the boat.

Step 2: Use Trigonometric Identity

The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. In this scenario, the hypotenuse is the slant line from the top of the lighthouse to the boat, and the adjacent side is the horizontal distance we need to find.

Step 3: Express Known Values

Given: \( \cos(\theta) = \frac{4}{5} \). This implies that for the triangle, the adjacent side is 4 units for every 5 units of the hypotenuse. Also, the opposite side (height of the lighthouse) is 75 feet.

Step 4: Determine the Hypotenuse

To use the cosine ratio, first express the relationship: \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}\). Since the actual height is given as 75 feet, solve for the hypotenuse length using: \[ \text{Hypotenuse} = \frac{\text{Opposite}}{\sin(\theta)} = \frac{75}{\sqrt{1 - \left(\frac{4}{5}\right)^2}} = \frac{75}{\frac{3}{5}} = 125 \text{ feet} \].

Step 5: Calculate the Horizontal Distance

Now, use the cosine ratio \( \cos(\theta) = \frac{4}{5} = \frac{x}{125} \), where \(x\) is the horizontal distance. Solve for \(x\): \[ x = \frac{4}{5} \times 125 = 100 \text{ feet} \].

Step 6: Final Answer

The boat is located 100 feet horizontally from the base of the lighthouse.

Key Concepts

Right Triangle

A right triangle is an important shape in trigonometry, known primarily for having a 90-degree angle. This special characteristic divides the triangle into two shorter sides and a longer side, which is known as the hypotenuse. In day-to-day problems, like in this exercise, the right triangle helps us understand real-world situations involving angles and distances.
In the context of the lighthouse problem, you can visualize a right triangle that involves:

• The vertical line from the lighthouse to sea level (75 feet).
• The horizontal line from the base of the lighthouse to the boat (the part we're looking to solve for).
• The hypotenuse, which represents the slanted, line-of-sight distance from the top of the lighthouse to the boat.

This configuration aids in breaking down complex distance and positioning problems into manageable calculations, leveraging various trigonometric ratios like sine, cosine, or tangent.

Angle of Depression

The angle of depression is often used in trigonometry to describe the angle formed when an observer looks downward toward an object. It is always measured from the horizontal line at the observer's eye level, down to the line of sight to the object. This is particularly helpful in scenarios involving heights, where you need to determine horizontal distances possibly not directly visible or accessible.
In this exercise, as you're positioned at the top of the lighthouse, you measure the angle of depression to the boat. The angle formed is equal to the angle between the horizontal line from the lighthouse and the slanted line of sight to the boat. Because of the properties of alternate interior angles with parallel lines, the angle of depression is equal to the angle of elevation from the boat to the top of the lighthouse. This equivalence simplifies how we use trigonometric identities to solve the problem.

Cosine Ratio

The cosine ratio is one of the key trigonometric functions used in right triangles. It provides the relationship between the angle and the sides of a triangle. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In symbols, it's expressed as:

• \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)

In this problem, the cosine ratio is given as \( \frac{4}{5} \), letting us know that for every 5 units of hypotenuse, the adjacent side is 4 units. This helps calculate unknown side lengths when one side and the angle are known. The known value allows us to proceed with finding the horizontal distance (adjacent side) once the hypotenuse is determined.

Hypotenuse

The hypotenuse is the longest side of a right triangle, opposite the right angle. It plays a central role in linking various elements of trigonometry. In calculations, it often forms the component needed to establish relationships through trigonometric ratios. In our scenario, the hypotenuse refers to the direct line of sight from the top of the lighthouse to the boat. By knowing the angle of depression and the vertical height, we set up trigonometric equations to determine the hypotenuse using the sine function:

• Given \[ \text{Hypotenuse} = \frac{\text{Height}}{\sin(\theta)} = \frac{75}{\frac{3}{5}} = 125 \text{ feet} \]

This result becomes the cornerstone to finding the necessary horizontal distance via the cosine ratio.

Horizontal Distance

The horizontal distance is what we aim to calculate in this exercise and represents the side of the triangle that is adjacent to the angle of depression. It marks the span from the base of the lighthouse to the boat, directly at sea level. Understanding this distance is crucial in navigation and surveying just as in many real-world applications where sight and alignment matter.
Once the hypotenuse is known, the horizontal distance can be found using the cosine ratio:

• \[ x = \cos(\theta) \times \text{Hypotenuse} = \frac{4}{5} \times 125 = 100 \text{ feet} \]

This calculation, using values derived from well-defined trigonometric methods, provides the exact distance needed while ensuring accuracy consistent with trigonometric principles.