Chapter 15: Problem 24

Sketch on the same axes graphs of \(\sin t, \sin (t-\pi / 2)\), and \(\sin (t+\pi / 2)\), and observe which way the graph shifts.

Short Answer

Expert verified

\(\sin (t-\pi / 2)\) shifts right by \(\pi/2\) and \(\sin (t+\pi / 2)\) shifts left by \(\pi/2\).

Step by step solution

- Plot the graph of \(\sin t\)

Start by plotting the graph of the basic sine function \(\sin t\). This graph oscillates between -1 and 1, passing through the origin (0,0), reaching its maximum at \(\pi/2\), crossing zero again at \(\pi\), reaching its minimum at \(3\pi/2\), and completing the cycle at \(2\pi\).

- Plot the graph of \(\sin (t-\pi / 2)\)

The function \(\sin (t-\pi / 2)\) represents a horizontal shift of the basic sine function \(\sin t\) to the right by \(\pi/2\). This means that every point on the \(\sin t\) graph is moved \(\pi/2\) units to the right.

- Plot the graph of \(\sin (t+\pi / 2)\)

Next, plot the graph of \(\sin (t+\pi / 2)\). This function represents a horizontal shift of the basic sine function \(\sin t\) to the left by \(\pi/2\). Every point on the \(\sin t\) graph is moved \(\pi/2\) units to the left.

- Observe the Shifts

Comparing all three graphs, it can be observed that \(\sin (t-\pi / 2)\) is the graph of \(\sin t\) shifted right by \(\pi/2\) units, while \(\sin (t+\pi / 2)\) is the graph of \(\sin t\) shifted left by \(\pi/2\) units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sine function

The sine function, denoted as \( \sin t \), is a fundamental trigonometric function. It describes the relationship between an angle and the vertical coordinate of a point on the unit circle.
Its graph forms a wave-like curve that oscillates between -1 and 1. This wave is periodic and repeats every \( 2\pi \) units. The basic properties of the sine function include:

Amplitude: The maximum absolute value is 1.
Period: The function repeats every \( 2\pi \) units.
Zeroes: The function crosses the x-axis at multiples of \( \pi \).
Maximum Points: Peaks at \( \pi/2 \), \( 5\pi/2 \), etc.
Minimum Points: Troughs at \( 3\pi/2 \), \( 7\pi/2 \), etc.

This makes the sine function very useful for modeling wave behavior, sound, and other oscillatory processes.

horizontal shift

A horizontal shift is one type of trigonometric transformation applied to functions like sine. When a function \( f(t) \) becomes \( f(t-a) \), it is shifted horizontally.

If \( a \) is positive (e.g., \( \sin(t-\frac{\pi }{2}) \)), the graph shifts to the right by \( a \) units.
If \( a \) is negative (e.g., \( \sin(t+\frac{\pi }{2}) \)), the graph shifts to the left by \( \mid a \mid \) units.

This shift does not change the amplitude, period, or shape of the function; it only changes the starting point of the wave.

trigonometric transformations

Trigonometric transformations alter the appearance of trigonometric functions, making them more adaptable for various applications.
The main types include:

Amplitude Changes: Affects the height of the wave, given by \( A \) in \( A \sin(t) \).
Period Changes: Affects the length of one cycle, given by \( B \) in \( \sin(Bt) \), where the period becomes \( \frac{2\pi}{B} \).
Horizontal Shifts: As discussed, shifts the function left or right.
Vertical Shifts: Moves the function up or down, given by \( C \) in \( \sin(t) + C \).

These transformations can be combined to produce complex waveforms that can model real-world phenomena.

phase shift in trigonometry

Phase shift in trigonometry refers specifically to the horizontal shift in the graph of a trigonometric function. It essentially repositions the starting point of the wave.
In the sine function, \( \sin(t - \phi ) \), the phase shift is determined by \( \phi \). A few specifics are: