Problem 67 - Algebra and Trigonometry with Analytic Geometry

Question

Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength \(\lambda\) and the distance between the two radio towers is equal to \(\frac{1}{2} \lambda\), then the intensity \(I\) of the signal in the direction \(\theta\) is given by $$ I=\frac{1}{2} I_{0}[1+\cos (\pi \sin \theta)] $$ where \(I_{0}\) is the maximum intensity. Approximate \(I\) in terms of \(I_{0}\) for each \(\theta\). (a) \(\theta=0\) (b) \(\theta=\pi / 3\) (c) \(\theta=\pi / 7\)

Step-by-Step Solution

Answer

(a) I = I_0, (b) I = 0.25 I_0, (c) I = 0.855 I_0.

Step 1: Understanding the Formula

The given formula for intensity is \( I = \frac{1}{2} I_0 [1 + \cos(\pi \sin \theta)] \). To approximate the intensity at different angles \( \theta \), we will substitute the specific value of \( \theta \) into this formula and solve for \( I \) in terms of \( I_0 \).

Step 2: Substituting θ = 0

For \( \theta = 0 \), the formula becomes \( I = \frac{1}{2} I_0 [1 + \cos(\pi \sin 0)] \). Since \( \sin 0 = 0 \), it follows that \( \cos(0) = 1 \). Therefore, the intensity is \( I = \frac{1}{2} I_0 [1 + 1] = I_0 \).

Step 3: Substituting θ = π/3

For \( \theta = \frac{\pi}{3} \), substitute into the formula: \( I = \frac{1}{2} I_0 [1 + \cos(\pi \sin \frac{\pi}{3})] \). Calculate \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \). Therefore, \( \pi \sin \theta = \pi \frac{\sqrt{3}}{2} \). The cosine of this expression is \( \cos(\pi \cdot \frac{\sqrt{3}}{2}) \). Evaluating the cosine, we obtain \( -0.5 \). The intensity simplifies to \( I = \frac{1}{2} I_0 [1 - 0.5] = \frac{1}{4} I_0 \).

Step 4: Substituting θ = π/7

For \( \theta = \frac{\pi}{7} \), substitute into the formula: \( I = \frac{1}{2} I_0 [1 + \cos(\pi \sin \frac{\pi}{7})] \). Calculate \( \sin \frac{\pi}{7} \approx 0.435 \). Therefore, we compute \( \pi \sin \theta = \pi \cdot 0.435 \). The cosine of this value is \( \cos(\pi \cdot 0.435) \). Evaluating numerically, we receive \( \cos \approx 0.71 \). Thus, the intensity becomes \( I = \frac{1}{2} I_0 [1 + 0.71] = \frac{1}{2} I_0 \times 1.71 = 0.855 I_0 \).

Key Concepts

Signal Intensity

Signal intensity refers to the strength or brightness of a signal that is received at a certain point. In the context of radio waves, a higher signal intensity means a clearer and stronger broadcast can be heard at that location. It is represented by the variable \(I\) in the given problem, which measures how powerful the radio signal is in the direction given by the angle \(\theta\).

Understanding signal intensity is essential when dealing with multiple broadcasting towers. Different stations need to finely control signal emissions to avoid interference. It's like making sure each radio volume is just right, so everyone hears their favorite station without unnecessary overlap. By altering signal strength in certain directions, radio stations can effectively manage their broadcasts, ensuring clarity and reducing interference with nearby stations. This precise control helps expand listenership ranges effectively. Always consider how the distance between the broadcasting towers affects this intensity.

Radio engineers use mathematical formulas, such as the given \(I = \frac{1}{2} I_0 [1 + \cos (\pi \sin \theta)]\), to determine and optimize signal intensity. This equation incorporates trigonometric principles to calculate the radio wave's reach and strength depending on \(\theta\), which can vary based on placement and other factors.

Cosine Function

The cosine function is a fundamental concept in mathematics, especially in trigonometry. It is one of the basic trigonometric functions used to describe the relationship between angles and the ratios of different sides in a right triangle. It's represented by \(\cos(\theta)\), and it usually describes the ratio of the adjacent side to the hypotenuse. In our exercise, the cosine function helps to determine how the signal intensity changes with different angles.

Cosine ranges between -1 and 1. This property is used in the intensity formula \(1 + \cos(\pi \sin(\theta))\), where \(\theta\) varies. When \(\theta = 0\), \(\cos(0) = 1\), leading to maximum signal intensity \(I = I_0\). For other angles, such as \(\theta = \pi/3\) or \(\theta = \pi/7\), the cosine value changes, affecting the signal intensity. Here, the formula creatively uses the cosine function to model how signals interfere and combine.

This interference manifests as changes in signal intensity at different angles. Thus, when understanding how radio signals spread, it's crucial to apply trigonometric functions like cosine to predict and control the behaviour of these waves effectively.

Radio Wave Interference

Radio wave interference occurs when radio signals from different sources cross paths and mix. This mixing can cause unintended effects, such as decreased signal clarity or noise, impacting the listener's experience. To mitigate this, radio stations carefully design their signal patterns to control intensity and direction.

The challenge with multiple towers is ensuring that the signals sent out do not interfere destructively where they meet. Destructive interference reduces signal strength, which occurs due to the overlapping of out-of-sync waves – imagine one wave's peak meeting another's trough. The formula \(I = \frac{1}{2} I_0 [1 + \cos (\pi \sin(\theta))]\) helps predict where such interferences might occur based on \(\theta\).

Through understanding interference and using mathematical equations, such as incorporating both signal intensity and trigonometric functions, engineers can precisely adjust angles and distances between towers. This engineering allows for a smoother broadcasting experience, where signals are sent out clearly without overlap and interference is minimized, maintaining a high-quality signal to all listeners within the effective range.