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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 96
Capitolo 1, Problema 96

Identify the amplitude and period of the following functions.
f(π)=2sin2θf\(\left\)(\(\pi\)\(\right\))=2\(\sin\)2\(\theta\)

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1
Identify the general form of the sine function, which is \( f(\theta) = a \sin(b\theta) \), where \( a \) is the amplitude and \( \frac{2\pi}{b} \) is the period.
In the given function \( f(\theta) = 2\sin(2\theta) \), compare it with the general form to identify the values of \( a \) and \( b \). Here, \( a = 2 \) and \( b = 2 \).
The amplitude of a sine function is the absolute value of \( a \). Therefore, the amplitude is \( |2| = 2 \).
The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \). Substitute \( b = 2 \) into the formula to find the period.
Calculate the period: \( \frac{2\pi}{2} = \pi \). Thus, the period of the function is \( \pi \).

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Amplitude

Amplitude refers to the maximum value of a periodic function, particularly in trigonometric functions like sine and cosine. It indicates how far the function reaches above and below its midline. For the function f(θ) = 2sin(2θ), the amplitude is 2, meaning the function oscillates between -2 and 2.
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Period

The period of a function is the length of one complete cycle of the wave. For sine and cosine functions, the period can be determined from the coefficient of the variable inside the function. In f(θ) = 2sin(2θ), the period is calculated as 2π divided by the coefficient of θ, which is 2, resulting in a period of π.
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Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. They are periodic functions, meaning they repeat their values in regular intervals. Understanding their properties, including amplitude and period, is essential for analyzing wave-like behaviors in various applications.
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