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Ch. 4 - Graphs of the Circular Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 4 - Graphs of the Circular FunctionsProblema 45
Capitolo 5, Problema 45

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.
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Identify the key characteristics of the given graph, such as amplitude, period, phase shift, and vertical shift. These will help determine the values of \(a\), \(b\), and whether the function is sine or cosine.
Determine the amplitude \(a\) by finding the distance from the midline of the graph to its maximum or minimum value. This is the absolute value of \(a\) in the equation \(y = a \sin(bx)\) or \(y = a \cos(bx)\).
Calculate the period of the function by measuring the length of one complete cycle on the x-axis. Use the formula for period: \(\text{Period} = \frac{2\pi}{b}\). Solve for \(b\) as \(b = \frac{2\pi}{\text{Period}}\).
Decide whether the function is sine or cosine by examining the starting point of the graph at \(x=0\). If the graph starts at a maximum or minimum, it is likely a cosine function; if it starts at the midline going upward or downward, it is likely a sine function.
Write the equation in the form \(y = a \sin(bx)\) or \(y = a \cos(bx)\) using the values of \(a\) and \(b\) found, and include any phase shifts or vertical shifts if present.

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General Form of Sine and Cosine Functions

The equations y = a sin bx and y = a cos bx represent sinusoidal functions where 'a' controls the amplitude (height) and 'b' affects the period (frequency). Understanding these forms helps in modeling periodic behavior seen in graphs.
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Graph of Sine and Cosine Function

Amplitude and Period of Trigonometric Functions

Amplitude is the maximum value of the function, given by |a|, while the period is the length of one complete cycle, calculated as 2π/b. Identifying these from the graph is essential to determine the parameters 'a' and 'b' in the equation.
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Period of Sine and Cosine Functions

Phase Shift and Choosing Between Sine and Cosine

Sine and cosine functions differ by a horizontal shift (phase shift). By analyzing the graph's starting point and shape, one can decide whether to use sine or cosine to best fit the data without additional phase shifts.
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Phase Shifts
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