Problem 50 - Algebra 2

Question

NAVIGATION Ships and airplanes measure distance in nautical miles. The formula 1 nautical mile \(=6077-31\) cos 2\(\theta\) feet, where \(\theta\) is the latitude in degrees, can be used to find the approximate length of a nautical mile at a cegrees, can be used to find the approximate length of a nautical mile at a certain latitude. Find the length of a nautical mile where the latitude is \(60^{\circ} .\)

Step-by-Step Solution

Answer

The length of a nautical mile at a latitude of \(60^{\circ}\) is 6092.5 feet.

Step 1: Identify the Given Information

We are given that the latitude \( \theta \) is \( 60^{\circ} \). The formula to calculate the length of a nautical mile is \( L = 6077 - 31 \cos(2\theta) \) feet.

Step 2: Substitute the Value of Latitude

Substitute \( \theta = 60^{\circ} \) into the formula. This gives us \( L = 6077 - 31 \cos(2 \times 60^{\circ}) \).

Step 3: Calculate the Double Angle

Calculate \( 2 \times \theta = 2 \times 60^{\circ} = 120^{\circ} \).

Step 4: Evaluate the Cosine Function

Find \( \cos(120^{\circ}) \). Since \( \cos(120^{\circ}) = \cos(180^{\circ} - 60^{\circ}) = -\cos(60^{\circ})\), and \( \cos(60^{\circ}) = 0.5 \), it follows that \( \cos(120^{\circ}) = -0.5 \).

Step 5: Substitute and Simplify

Substitute \( \cos(120^{\circ}) = -0.5 \) back into the formula: \( L = 6077 - 31(-0.5) \). Simplify to get \( L = 6077 + 15.5 \).

Step 6: Calculate the Result

Perform the addition: \( L = 6077 + 15.5 = 6092.5 \).

Key Concepts

Latitude

Latitude is a key geographical term used to describe a location on Earth's surface. It specifies how far north or south a place is from the Equator. The Equator is considered 0° latitude, while the poles are at 90° north and 90° south. Here are some important elements of latitude to remember:

• Latitude lines run horizontally around the Earth.
• They are measured in degrees, with each degree representing approximately 111 kilometers (or 69 miles) on the Earth's surface.
• Latitude affects the climate, weather, and often the daylight hours of a region.

For marine and air navigation, understanding latitude is crucial. It assists in calculating distances, like nautical miles, between various points. In this context, the formula for calculating nautical miles takes the latitude into account to adjust distances based on where you are on Earth.

Cosine Function

The cosine function is a fundamental concept in trigonometry. It relates the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. When dealing with angles expressed in degrees, the cosine function is used as part of various calculations, such as the nautical mile problem we are examining.

Key points about the cosine function include:

• It is periodic, with a cycle of 360°.
• Its range is between -1 and 1.
• At 0°, the cosine value is 1, while at 90°, it is 0.

In the nautical mile formula, the cosine function helps to adjust the basic length, accounting for the curvature of the Earth as it changes with latitude. Specifically, when the problem calls for the use of \( \cos(120^\circ) \), it connects with how different angles affect the distance calculations visually represented on Earth's surface.

Double Angle Formula

The double angle formula in trigonometry is used to simplify the calculation of functions when dealing with angles that are doubled. It involves the cosine of double angles, and in our formula, it appears as \( \cos(2\theta) \). The formula for the double angle of cosine is:

\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \]

By using this, calculations involving angles such as 120° (which is twice 60°) become easier to handle. Here's some additional insight on double angle formulas:

• The double angle formulas reduce complex expressions by using basic trigonometric identities.
• They are especially useful in trigonometry when dealing with power reduction and angle transformations.
• By simplifying expressions, you improve computational efficiency and accuracy.

Understanding this formula assists in the process of calculating the cosine of larger angles by breaking them down into simpler trigonometric components. This step is crucial for problems like the nautical mile calculation that involve doubling an angle to find the cosine value.