Question

Plot each sine wave. $$y=375 \sin \left(55 t+\frac{\pi}{4}\right)$$

Short answer

The sine wave starts at the phase shift \(\frac{\pi}{4}\) on the x-axis and oscillates between -375 and 375 on the y-axis with a period of \(\frac{2\pi}{55}\). Draw the curve through several periods to demonstrate the sine wave.

Step-by-step solution

Understanding the Sine Function

The general form of a sine function is \(y = A \sin(Bt + C)\) where \(A\) is the amplitude, \(B\) is the frequency, \(C\) is the phase shift, and \(t\) is the variable for time. In \(y=375 \sin(55t + \frac{\pi}{4})\), the amplitude is 375, the frequency is 55, the phase shift is \(\frac{\pi}{4}\), and \(t\) will be the variable on the x-axis when plotting the graph.

Plotting the Graph

To plot \(y=375 \sin(55t + \frac{\pi}{4})\), start by labeling the y-axis from -375 to 375 to denote the amplitude. The x-axis will represent time \(t\), where you'll show the phase shift of \(\frac{\pi}{4}\) and the period of the sine function. You can find the period by using the formula \(Period = \frac{2\pi}{B}\). Once this is noted, draw the sine curve starting from the phase shift and moving through one period, ensuring that the curve hits the maximum and minimum amplitudes correctly. Continue the wave for several periods to show its repetitive nature.

Key concepts

Amplitude of a Sine Function

Understanding the amplitude of a sine function is crucial when it comes to graphically representing the function. In mathematical terms, the amplitude refers to the height of the wave, which is measured from the central axis (often the x-axis) to the peak (or trough) of the wave. For the sine function given by

\( y = 375 \sin(55t + \frac{\pi}{4}) \),

the amplitude is represented by the coefficient in front of the sine, which is 375 in this case. This number tells us the maximum displacement from the central axis our sine wave will have. In other words, the graph will oscillate with peaks reaching up to 375 and troughs dipping down to -375 along the y-axis. The importance of amplitude is not only in its representation of peak to peak wave height but also in the interpretation of the function's intensity or strength in various application contexts, such as in physics with wave phenomena. To graph the sine wave accurately, the y-axis should be scaled to accommodate the full range of the wave from -375 to 375, ensuring the peaks and troughs are plotted correctly at these values.

Sine Function Period

Moving on from the maximum heights of the wave, let's explore the concept of the period of a sine function. The period represents the length of one complete cycle of the wave, after which the pattern repeats itself. It is the distance along the x-axis between two successive peaks (or any two corresponding points on the wave). For our sine wave described by

\( y = 375 \sin(55t + \frac{\pi}{4}) \),

the period can be calculated using the formula

\( Period = \frac{2\pi}{B} \),

where B is the frequency of the sine function. In this exercise, B is 55. After calculating, we get the period of the sine wave which is crucial for plotting the function correctly over the x-axis. Ensuring that one period is accurately represented on the graph allows the repetitive nature of the sine wave to be clearly visible. Subsequent periods should be evenly spaced to maintain the integrity of the wave's cyclical pattern. The knowledge of the period is also essential in engineering and science for understanding wave-like phenomena such as sound waves, light waves, or alternating current in electrical engineering.

Sine Function Phase Shift

Finally, the concept of phase shift in a sine function gives us information about the horizontal displacement of the wave. A phase shift either moves the wave to the left or right along the x-axis from the standard position of the sine function which starts at the origin. In layman's terms, the phase shift is the starting point of the sine wave.

Our function

\( y = 375 \sin(55t + \frac{\pi}{4}) \),

has a phase shift of \( \frac{\pi}{4} \) radians. This tells us that the starting point of the wave is shifted to the right by \( \frac{\pi}{4} \) radians. Graphically, this means that the point where the sine wave crosses the x-axis initially will be moved from the origin to this new position. When plotting the wave, you start drawing your curve at this phase shift, ensuring that each cycle of the wave is shifted accordingly. Understanding the phase shift is particularly useful when comparing two waves to see if they are in sync, or when trying to align waves as seen in signal processing.