Problem 42 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Sketch an appropriate figure, unless the figure is given. A communications satellite is in orbit \(35,300 \mathrm{km}\) directly above the earth's equator. What is the greatest latitude from which a signal can travel from the earth's surface to the satellite in a straight line? The radius of the earth is \(6400 \mathrm{km}\)

Step-by-Step Solution

Answer

The greatest latitude from which a signal can travel to the satellite is approximately 8.682°.

Step 1: Understanding the Problem

We need to find the greatest latitude on Earth from which a signal can be sent directly to a satellite located 35,300 km above the equator. The Earth's radius is given as 6400 km. The task is to find the angle at which the signal, traveling in a straight line, just grazes the Earth, known as the latitude angle.

Step 2: Setting Up the Geometry

Visualize the Earth as a circle and the satellite as a point directly above the Earth's center. Let the Earth's center be O, the point on the Earth's surface be A (maximum latitude point), and the satellite be S. The distance OS (from Earth's center to the satellite) equals the Earth's radius plus the satellite's altitude: \[ OS = 6400 ext{ km} + 35300 ext{ km} = 41700 ext{ km} \] Here, line OA is the Earth's radius (6400 km), and AS is the line of sight where the signal is sent.

Step 3: Applying Trigonometry

Use the law of cosines in triangle OAS to find the angle \( \theta \) at O, which represents the angle from the center of the Earth to the point of contact on its surface. The cosine law is: \[ OS^2 = OA^2 + AS^2 - 2 \cdot OA \cdot AS \cdot \cos(\theta) \] Since AS is the tangent line, its length isn't needed directly. Instead, set AS to a point at the horizon (meaning it's perpendicular to OA), and the triangle OAS becomes a right triangle with OS as the hypotenuse.

Step 4: Using Right Triangle Trigonometry

For the right triangle scenario where \( AS \) is tangent, recognize the trigonometric relation from the Earth's radius (OA) and the distance from the center to the satellite (OS). The cosine of angle \( \theta \) is: \[ \cos(\theta) = \frac{OA}{OS} = \frac{6400}{41700} \] Solve for \( \theta \): \[ \theta = \cos^{-1}(\frac{6400}{41700}) \approx 81.318^{\circ} \]

Step 5: Calculating Latitude

Latitude on Earth's surface ranges from 0° at the equator to 90° at the poles. The relation from the center \( \theta \) to latitude is: \[ ext{Latitude} = 90^{\circ} - \theta \] Thus, the maximum latitude from which the signal can be sent is: \[ 90^{\circ} - 81.318^{\circ} \approx 8.682^{\circ} \]

Key Concepts

geometry in mathematics

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. It is essential for many fields including engineering, physics, and architecture. In the context of Earth and satellite problems, geometry helps us understand the relationships between different points and lines in space.
For example, envisioning Earth as a circle, and satellites as points in orbit, is a geometric approach. This allows us to calculate distances and angles, crucial for determining how signals can travel between the Earth and these satellites.
Using geometric figures, such as circles and triangles, helps visualize these concepts and solve complex problems involving orbits and latitudes. Thus, a strong grasp of geometric principles is fundamental when working through problems related to space and satellite communications.

law of cosines

The law of cosines is a powerful tool in trigonometry, extending the Pythagorean theorem to any type of triangle. It relates the lengths of a triangle's sides to the cosine of one of its angles. For a triangle with sides \(a\), \(b\), and \(c\), and opposite angles \(A\), \(B\), and \(C\), the law states:

• \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\)

In the context of satellite communication, the law of cosines becomes indispensable. By treating the Earth and satellite as parts of a triangle, the law helps us calculate angles and distances. For instance, finding the angle of elevation necessary for a signal to travel to a satellite or calculating how far a signal might reach.
This law is indispensable for determining the position of Earth's objects in relation to satellites, aiding in the prediction of signal coverage and communication capabilities.

right triangle trigonometry

Right triangle trigonometry involves the study of triangles where one angle is exactly 90 degrees. This type of trigonometry is centered on the relationships between the angles and sides of right triangles. The primary functions—sine, cosine, and tangent—help describe these relationships. In a right triangle:

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

In Earth and satellite problems, right triangle trigonometry is crucial. By considering a scenario where the line of sight is tangent to Earth's surface, we can form right triangles. For example, if the line AS is tangent at the horizon, we apply right triangle trigonometry to find the angle of elevation needed.
Understanding these concepts allows for practical applications, such as determining how high satellites must be to cover certain latitudes or what viewing angles are possible from Earth's surface.

angle calculation

Calculating angles is a fundamental part of solving trigonometry problems and is widely used in various fields. In our satellite problem, we need to calculate the angle formed by a point on Earth's surface and the satellite overhead. This involves using trigonometric identities and algorithms, primarily involving cosine, due to the oblique configurations often encountered in these scenarios.
Subsequently, determining the latitude requires recognizing that the degrees between zenith (directly overhead) and the tangent line to Earth's surface accounts for how far signals can travel. Thus, it's essential to:

• Understand angle relationships, especially how they interact with circle geometry.
• Apply inverse trigonometric functions like \( \cos^{-1} \) to retrieve angle measures needed for precise calculations.

These calculations are vital for satellites, allowing us to define the range of effective signal transmission and optimize communication systems.

earth and satellite problems

Earth and satellite problems involve understanding the interaction between bodies in space, requiring grasp of multiple disciplines, including geometry and trigonometry. These problems often involve resolving complex orbital paths, angles of communication signals, and distances.
For example, knowing the maximum latitude a signal can be transmitted from involves calculating various geometric and trigonometric elements to ensure effective communication. With satellites positioned above the equator, determining how these signals behave relative to various points on Earth is critical.
Successfully addressing Earth and satellite problems enhances communication technologies, built on accurately predicting how signals travel through space. It underpins technologies like GPS and satellite-based internet, supporting global connectivity and navigation.