Question

The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point \(P\) in the diagram equals the degree measure of \(\operatorname{arc} P E\). At what latitude \(\theta\) is the circumference of the circle of latitude at \(P\) half the distance around the equator?

Short answer

\(\theta = \cos^{-1}( \frac{1}{2} )\) or approximately 60 degrees. This means the latitude where the circle's circumference is half that of the equator's is around 60 degrees.

Step-by-step solution

Define the given

The given point is at a latitude \(\theta\) and \(P\) is the point at which this latitude intersects the Earth. The Earth's equator is a great circle, and therefore its radius \(r\) is the radius of the Earth \(R\). The radius of the circle of latitude at \(P\) is \(R \cos(\theta)\), since that forms the adjacent side of the right triangle within the spherical cross section of the Earth.

Setting up the equation

We're looking for the latitude where the circle's circumference is half that of the equator's. The equator's circumference is \(2 \pi R\) and the circumference of the circle of latitude at \(P\) is \(2 \pi R \cos(\theta)\). Setting these two equal to each other, we get: \[2 \pi R \cos(\theta) = \frac{1}{2} \times 2 \pi R\]

Solving the equation

Solving the above equation for \(\theta\), we divide both sides by \(2 \pi R\) to get \[\cos(\theta) = \frac{1}{2}\]. Taking the inverse cosine of both sides gives \(\theta = \cos^{-1}( \frac{1}{2})\].

Key concepts

Latitude

Latitude is an important concept in geography and is used to describe the location of a point on the Earth's surface. It is measured in degrees from the equator, which is set at 0°, to the poles, which are at 90° North and South. Latitude lines, or parallels, run horizontally around the globe and indicate how far a location is north or south of the equator.
A unique aspect of latitude is that each degree can be subdivided into 60 minutes, and each minute into 60 seconds, which allows for precise positioning.

• Locations with higher latitudes are closer to the poles.
• Lower latitudes are closer to the equator.

Understanding latitude is crucial for navigation, climate studies, and understanding the Earth's geometry.

Circumference

The circumference of a circle is the distance around it. For the Earth, this includes significant circles like the equator and circles of latitude. The length of the circumference gives valuable information about distances on the Earth's surface. The formula used to determine the circumference is based on the radius of the circle.
For a circle of latitude at a point, the circumference is given by \(2 \pi R \cos(\theta)\), where \(R\) is the Earth's radius and \(\theta\) is the latitude.

• This formula adjusts because the actual radius at any latitude (other than 0°, the equator) is reduced by the \(\cos(\theta)\).

Understanding this calculation helps in visualizing how the actual distances vary with latitude.

Cosine Function

The cosine function is a fundamental concept in trigonometry, often used to relate the angles and sides of a right triangle. In the context of circles of latitude, the cosine function helps calculate the radius and the circumference of these circles.
The cosine of an angle in a right triangle is the length of the adjacent side divided by the hypotenuse. In our scenario, the cosine function adjusts the Earth's radius to find the radius of the circle of latitude: \(R \cos(\theta)\).

• It provides a simple yet effective way to factor in the Earth's curvature as we move away from the equator.
• By solving equations involving the cosine function, we can find crucial angles, such as the latitude at which certain distances are observed, like when a circle's circumference is half the equator's.

Right Triangle

Right triangles are everywhere in trigonometry and geography. A right triangle is defined by one of its angles being precisely 90 degrees. In the context of circles of latitude, a right triangle is formed between the latitude circle, the Earth's radius, and the equator line.
This triangle allows us to apply trigonometric ratios to solve for unknowns such as the radius of the latitude circle. Since the circle of latitude at any given point forms the base, we can calculate its length using trigonometric functions like cosine:

• The hypotenuse is the radius \(R\) of the Earth.
• The adjacent side is \(R \cos(\theta)\), representing the radius of the circle of latitude.

Using right triangle principles allows for straightforward calculations and visual understanding of Earth's geometrical properties.