Problem 35 - Algebra 2

Question

A radio transmitter sends out two signals, one for voice communication and another for data. Suppose the equation of the voice wave is \(v=10 \sin \left(2 t-30^{\circ}\right)\) and the equation of the data wave is \(d=10 \cos \left(2 t+60^{\circ}\right) .\) Draw a graph of the waves when they are combined.

Step-by-Step Solution

Answer

Graph the waves as individual plots and combine them to view the resultant wave.

Step 1: Understand the given equations

We have two wave equations: the voice wave as \( v = 10 \sin(2t - 30^\circ) \) and the data wave as \( d = 10 \cos(2t + 60^\circ) \). These are sinusoidal equations representing waveforms.

Step 2: Convert angles to radians

Since trigonometric functions often work with radians, let's convert the angles from degrees to radians: \(-30^\circ = -\frac{\pi}{6}\) and \(+60^\circ = \frac{\pi}{3}\). Thus, the equations become \( v = 10 \sin(2t - \frac{\pi}{6}) \) and \( d = 10 \cos(2t + \frac{\pi}{3}) \).

Step 3: Find the resultant wave equation

The two waveforms can be represented by a single function as they are transmitted together. This can be calculated by superimposing the two: \( r = v + d = 10 \sin(2t - \frac{\pi}{6}) + 10 \cos(2t + \frac{\pi}{3}) \).

Step 4: Simplify the resultant equation if possible

Check if it is possible to combine the sine and cosine terms into a single sinusoidal equation. Using the sum-to-product identities or numeric evaluations might aid in the simplification for specific plots. If simplification isn't straightforward, plotting without simplifying is a valid solution.

Step 5: Sketch the resultant wave

Set up a graph with time \( t \) on the x-axis and amplitude on the y-axis. Plot the functions \(10 \sin(2t - \frac{\pi}{6})\), \(10 \cos(2t + \frac{\pi}{3})\), and their resultant \( r(t) = 10 \sin(2t - \frac{\pi}{6}) + 10 \cos(2t + \frac{\pi}{3})\). Use a graphing tool if necessary to ensure accurate representation.

Key Concepts

Wave Equations

Wave equations are mathematical models that describe the behavior of waves, such as sound, light, and radio waves. In this scenario, they represent the two signals transmitted by the radio transmitter. Each equation follows the form of a sinusoidal function, which is typical for wave equations. The equations are:

• Voice Wave: \( v = 10 \sin(2t - 30^\circ) \)
• Data Wave: \( d = 10 \cos(2t + 60^\circ) \)

These equations tell us about the amplitude, frequency, and phase shift of the waves. Understanding wave equations is crucial for analyzing how different waves interact with one another. The amplitude here is 10, indicating the height of the waves. The term \( 2t \) reflects the frequency, dictating how fast the wave oscillates. Lastly, the phase shift components \( -30^\circ \) and \(+60^\circ \) adjust the starting point of the wave along its cycle.

Angle Conversion

Angles in wave equations are initially given in degrees. However, trigonometric functions, especially in calculus and advanced mathematics, commonly use radians. This requires converting angles from degrees to radians. This conversion is essential, as calculations involving trigonometric functions often become simpler in radians. To convert from degrees to radians, use the formula:\[ \text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right) \]

• \(-30^\circ\) becomes \(-\frac{\pi}{6}\)
• \(+60^\circ\) becomes \(\frac{\pi}{3}\)

This conversion matters when you calculate adjustments in phase or perform mathematical simplifications. It's particularly useful in higher-level math and physics, where standardized units help streamline processes and ensure consistency in calculations.

Superposition of Waves

Superposition of waves is a fundamental concept in physics and engineering where two or more waves overlap and combine. The resulting wave is the sum of the individual wave equations. In our example, the combined signal from two different equations is found by adding them together:\[ r = v + d = 10 \sin(2t - \frac{\pi}{6}) + 10 \cos(2t + \frac{\pi}{3}) \]This principle explains phenomena such as interference patterns, constructive and destructive interference, and is widely used in both theoretical and applied physics. Wave superposition results in a new wave pattern that may have different amplitude and phase characteristics than the original waves. Finding the resultant wave helps in understanding how individual signals contribute to a final output.

Graphing Sinusoidal Functions

Graphing sinusoidal functions can help visualize the behavior and combination of waves. In this case, we have three important graphs: the graph of the voice wave, the data wave, and their combination or resultant wave. Plotting these can show us how the signals vary over time.Steps to graph these functions:

• Set up a graph with time \( t \) on the x-axis and amplitude on the y-axis.
• Plot the voice wave function \( 10 \sin(2t - \frac{\pi}{6}) \).
• Plot the data wave function \( 10 \cos(2t + \frac{\pi}{3}) \).
• Combine them to graph the resultant wave \( r = 10 \sin(2t - \frac{\pi}{6}) + 10 \cos(2t + \frac{\pi}{3}) \).

Using graphing tools can enhance accuracy and make the process easier. Visualizing sinusoidal functions is key to understanding their cycles, peaks, and interactions, giving insights into how they might behave in real-world applications, such as in communication systems like radio transmitters.