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Chapter 10: Problem 4

find the period of each function. $$y=4 \sin 2 x$$

Short Answer

Expert verified
The period of the function is \(\pi\).

Step by step solution

01

Identify the Function Type

The given function is a sine function. We have \(y = 4 \sin(2x)\). In general, a sine function is expressed as \(y = a \sin(bx + c) + d\). Here, \(a = 4\), \(b = 2\), \(c = 0\), and \(d = 0\).
02

Recall the Period Formula

For a sine function \(y = a \sin(bx + c) + d\), the period is calculated using the formula \( \frac{2\pi}{|b|} \). The period represents the length of one complete cycle of the sine wave before it repeats.
03

Substitute and Solve

In the formula \( \frac{2\pi}{|b|} \), substitute \(b = 2\). The period is then \( \frac{2\pi}{|2|} = \frac{2\pi}{2} = \pi\). Thus, the period of \(y = 4 \sin 2x\) is \(\pi\).

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Key Concepts

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

Sine Function
A sine function is a specific type of trigonometric function that models periodic phenomena. It describes how the sine curve or wave changes over time. A common usage of sine functions is to model repetitive cycles, like sound waves or tides.
The general form of a sine function is given by \(y = a \sin(bx + c) + d\). This function describes a wave with:
  • Amplitude \(a\), which determines the height from the midline to the peak or trough.
  • Frequency modifier \(b\), which affects the number of cycles displayed within a specific interval.
  • Horizontal shift \(c\), which translates the wave left or right.
  • Vertical shift \(d\), which moves the wave up or down.
Understanding each of these parameters helps you predict and analyze the behavior of trigonometric functions across different scenarios.
Period of a Function
The period of a sine function refers to the length of one complete wave cycle, after which the pattern starts repeating. It's a measure of how often a function returns to its initial state.
The formula to find the period in a function of the form \( y = a \sin(bx + c) + d \) is \( \frac{2\pi}{|b|} \). Here, \(b\) affects how stretched or compressed the wave appears:
  • If \(b > 1\), the wave compresses, meaning more cycles fit in the same space.
  • If \(0 < b < 1\), the wave stretches, resulting in fewer cycles.
  • If \(b < 0\), it only affects the mirror image direction but not the period itself.
The understanding of period is crucial for analyzing repeating events in real-life applications like signal processing and time series analysis.
Wave Cycle
A wave cycle in the context of a sine wave is the complete journey of the wave from the start to its repeat point. Imagine drawing a wave that begins at its midline, ascends to its peak, descends back to midline, drops to its trough, and rises back again to the starting point at midline.
Each cycle is characterized by:
  • A crest, representing the highest point.
  • A trough, representing the lowest point.
  • Two crossings of the central axis or midline, one ascending and one descending.
In mathematical terms, understanding this cycle enables you to predict the behavior of the wave across intervals, allowing calculations and visualizations of phenomena exhibiting periodic behavior, such as alternating current electricity or water waves.

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Most popular questions from this chapter

Solve the given problems. Find the function and graph it for a function of the form \(y=-2 \sin b x\) that passes through \((\pi / 4,-2)\) and for which \(b\) has the smallest possible positive value.

View at least two cycles of the graphs of the given functions on a calculator. $$y=0.5 \sec \left(0.2 x+\frac{\pi}{25}\right)$$

In Exercises \(47-50,\) graph the given functions. In Exercises 47 and \(48,\) first rewrite the function with a positive angle, and then graph the resulting function. $$y=0.4|\sin 6 x|$$

Solve the given problems. In Exercises 41 and 42 use a calculator to view the indicated curves. A wave traveling in a string may be represented by the equation \(y=A \sin 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right) .\) Here, \(A\) is the amplitude, \(t\) is the time the wave has traveled, \(x\) is the distance from the origin, \(T\) is the time required for the wave to travel one wavelength \(\lambda\) (the Greek letter lambda). Sketch three cycles of the wave for which \(A=2.00 \mathrm{cm}, T=0.100 \mathrm{s}, \lambda=20.0 \mathrm{cm},\) and \(x=5.00 \mathrm{cm}\)

Sketch the appropriate curves. A calculator may be used. The strain \(e\) (dimensionless) on a cable caused by vibration is \(e=0.0080-0.0020 \sin 30 t+0.0040 \cos 10 t,\) where \(t\) is mea- sured in seconds. Sketch two cycles of \(e\) as a function of \(t\)
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