Problem 25 - Algebra 2

Question

TRAVEL The cruise ship Reno sailed due west 24 miles before turning south. When the Reno became disabled and radioed for help, the rescue boat found that the fastest route to her covered a distance of 48 miles. The cosine of the angle at which the rescue boat should sail is \(0.5 .\) Find the angle \(\theta\) , to the nearest tenth of a degree, at which the rescue boat should travel to aid the Reno.

Step-by-Step Solution

Answer

\(\theta = 60.0^\circ\)

Step 1: Understand the Problem

The cruise ship Reno sailed 24 miles due west and then turned south before becoming disabled. A rescue boat sails at an angle \(\theta\) from its initial direction. The cosine of this angle is given as 0.5, and the distance from the rescue boat to the disabled Reno is 48 miles. We need to find the angle \(\theta\).

Step 2: Use the Cosine Function

We have the cosine of angle \(\theta\) given as 0.5. The cosine function relates the lengths of the adjacent side and the hypotenuse of a right triangle to the angle: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\). In this context, the adjacent side is along the west direction (24 miles), and the hypotenuse is the route to the Reno (48 miles).

Step 3: Calculate the Angle \(\theta\)

We have \(\cos(\theta) = 0.5\). Using the inverse cosine function, we can find \(\theta\). Therefore, \(\theta = \cos^{-1}(0.5)\). This yields \(\theta = 60^\circ\).

Step 4: Round the answer

The problem asks for the angle to the nearest tenth of a degree. Since \(60\) is already a whole number, it can be represented as \(60.0\) to denote the tenths' place.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry and is often used to determine angles or side lengths in right triangles. It is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. In a formula, this is expressed as:

• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

The cosine function is particularly useful when you know two sides of a right triangle and want to find an angle, as demonstrated in this exercise involving the cruise ship Reno. With the adjacent side measuring 24 miles and the hypotenuse at 48 miles, the cosine function allows us to determine the angle needed for the rescue boat's path. This exemplary use of the cosine function not only finds the angle but ties together geometric and navigational concepts in trigonometry.

Inverse Trigonometric Functions

Inverse trigonometric functions are the tools we use to determine an angle when we have the value of a trigonometric function. For cosine, this is known as the 'arc cosine' or more commonly, the inverse cosine function. It is represented as \( \cos^{-1} \). When given \( \cos(\theta) = 0.5 \), the problem sets us up to find \( \theta \). To resolve \( \theta \), you take the inverse cosine of 0.5:

• \( \theta = \cos^{-1}(0.5) \)

This calculation yields an angle of \( 60^\circ \). This simplicity highlights the power of inverse trigonometric functions in solving real-world navigation problems, such as determining the correct bearing for a rescue mission.

Right Triangle Properties

Right triangles are a cornerstone of trigonometry; they come with a rich set of properties that make solving geometric problems straightforward. These triangles have one right angle (\( 90^\circ \)) and two other angles that sum to \( 90^\circ \).In our problem, the triangle formed by the path of the cruise ship and rescue boat is a right triangle:

• Westward travel (24 miles) forms one leg.
• The north-south leg turns out to make your unknown angle \( \theta \), joining the westward route to complete this right angle setup.
• The hypotenuse (48 miles) represents the path the rescue boat must take to reach the Reno.

Understanding these properties helps in visualizing how the missing parts relate and how trigonometric rules like the cosine function apply. By employing these properties and functions, you can easily navigate complex spatial problems in precise, logical steps.