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Chapter 4: Problem 3

State the amplitude and period of the function defined by each equation. $$y=\sin 2 x$$

Short Answer

Expert verified
The amplitude of the function \(y=\sin 2x\) is 1, and the period of the function is \(\pi\).

Step by step solution

01

Identify the Amplitude

The amplitude of a function in the form \(y=A\sin Bx\) or \(y=A\cos Bx\) is given by the absolute value of the coefficient of the sine or cosine function, in this case, A. In the function \(y=\sin 2x\), the coefficient of sine is 1. So, the absolute value of 1, which is 1, is the amplitude of this function.
02

Identify the Period

The period of a function in the form \(y=A\sin Bx\) or \(y=A\cos Bx\) is given by \(\frac{2\pi}{|B|}\). So, in the function \(y=\sin 2x\), B is 2. Therefore, the period of this function is \(\frac{2\pi}{2}\), which simplifies to \(\pi\).

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Key Concepts

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

Amplitude
When we talk about amplitude in the context of trigonometric functions, think of it as the "height" of the wave. In simpler terms, it tells us how far the wave stretches above and below its central axis.
For a function given in the form \( y = A \sin Bx \) or \( y = A \cos Bx \), the amplitude is represented by the absolute value of the coefficient \( A \). The amplitude is always a positive number.
  • In the function \( y = \sin 2x \), \( A \) equals 1. Therefore, the amplitude is \( |1| = 1 \).
  • This implies that the wave extends 1 unit above and 1 unit below the horizontal axis.
Understanding amplitude helps visualize how "tall" the wave of the sine function is at any given point.
Period
The period of a sine or cosine function indicates how long it takes for the function to complete one full cycle, or wave. A complete cycle goes from the starting point, reaches the maximum and minimum values, and then returns to the starting value again.
For the general sine function formula \( y = A \sin Bx \), the period is calculated using \( \frac{2\pi}{|B|} \). This formula shows how the coefficient \( B \) affects the stretch or compression of the wave.
  • In the function \( y = \sin 2x \), the value of \( B \) is 2.
  • Plugging this into our formula, the period becomes \( \frac{2\pi}{2} = \pi \).
This means the function repeats its pattern every \( \pi \) units along the x-axis, making it quicker to complete one cycle compared to the standard sine wave, whose period is \( 2\pi \).
Sine Function
The sine function is a fundamental concept in trigonometry. It represents a smooth, wave-like oscillation, which is useful in various areas such as physics, engineering, and signal processing. The standard sine function is written as \( y = \sin x \) and has properties that can be transformed using different coefficients.
  • When the coefficient \( A \) is applied, the amplitude of the wave changes, affecting how tall the wave becomes.
  • The coefficient \( B \) modifies the period, changing how quickly the wave repeats.
  • The sine function naturally ranges between -1 and 1 without transformations.
These transformations allow us to adapt the sine function to model real-world phenomena with accuracy and efficiency.
Mathematics Education
Trigonometric functions like sine are a key part of mathematics education because they build foundational skills for understanding waves and oscillations. Educators emphasize not only their application in pure math but also in real-world contexts.
A clear understanding of terms such as amplitude and period can significantly enhance a student's ability to solve problems involving waves and oscillations.
  • Concepts of amplitude and period help in predicting and analyzing periodic events like sound waves or tides.
  • Learning the sine function's properties prepares students for advanced math courses, including calculus and beyond.
By making these concepts relatable and easy to understand, education ensures students can apply these skills in various scientific and technical disciplines.

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Most popular questions from this chapter

Graph at least one full period of the function defined by each equation. $$y=\frac{1}{2} \sin \frac{\pi x}{3}$$

Use a graphing utility to graph each function. $$y=\frac{\sin x}{x}$$

Use a graphing utility to graph each function. $$y=\sin |x|$$

Graph at least one full period of the function defined by each equation. $$y=2 \sin \pi x$$

Graph at least one full period of the function defined by each equation. $$y=|-2 \cos 3 x|$$
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