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Chapter 5: Problem 1

a) State the five key points for \(y=\sin x\) that occur in one complete cycle from \(\mathbf{0}\) to \(2 \boldsymbol{\pi}\) b) Use the key points to sketch the graph of \(y=\sin x\) for \(-2 \pi \leq x \leq 2 \pi .\) Indicate the key points on your graph. c) What are the \(x\) -intercepts of the graph? d) What is the \(y\) -intercept of the graph? e) What is the maximum value of the graph? the minimum value?

Short Answer

Expert verified
Key points: 0, \pi/2, \pi, 3\pi/2, 2\pi. X-intercepts: 0,\pi, 2\pi, -\pi, -2\pi. Y-intercept: 0. Max value: 1. Min value: -1.

Step by step solution

01

Identify Key Points of One Complete Cycle for \(y=\sin x\) from \(0\) to \(2\pi\)

The five key points in one complete cycle of the sine function are: \(0, \pi/2, \pi, 3\pi/2, \ and 2\pi\). At these points, the values of \(y=\sin x\) are as follows:- At \(x=0\), \(y=sin(0)=0\)- At \(x=\pi/2\), \(y=sin(\pi/2)=1\)- At \(x=\pi\), \(y=sin(\pi)=0\)- At \(x=3\pi/2\), \(y=sin(3\pi/2)=-1\)- At \(x=2\pi\), \(y=sin(2\pi)=0\)
02

Sketch the Graph of \(y=\sin x\) for \(-2\pi \leq x \leq 2\pi\)

To sketch the graph of \(y=\sin x\), plot the key points from step 1 for the range \(-2\pi \leq x \leq 2\pi\). The pattern repeats periodically. Connect these points smoothly to form a sine wave. Ensure that the wave crosses the x-axis at multiples of \(\pi\), reaches its maximum value of 1 at \(\pi/2\) and \(-3\pi/2\), and its minimum value of -1 at \(3\pi/2\) and \(-\pi/2\).
03

Find the x-intercepts of the Graph

The x-intercepts are where the graph crosses the x-axis. For \(y=\sin x\), this occurs when \(y=0\). Hence, the x-intercepts are at \(x=0, \pi, 2\pi, -\pi, \ and -2\pi\).
04

Find the y-intercept of the Graph

The y-intercept is where the graph crosses the y-axis, i.e., at \(x=0\). For \(y=\sin x\), \(y(0)=\sin(0)=0\). Thus, the y-intercept is 0.
05

Determine the Maximum and Minimum Values of the Graph

The maximum value of \(y=\sin x\) is 1, which occurs at \(\pi/2 + 2n\pi\) (where \(n\) is an integer). The minimum value is -1, which occurs at \(3/2\pi + 2n\pi\) (where \(n\) is an integer).

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Key Concepts

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

sine wave key points
In one complete cycle of the sine function from 0 to 2蟺, there are five key points to remember. The function starts at 0, rises to its maximum value, comes back to zero, dips to its minimum value, and then returns to zero. These points occur at:
- At x = 0, y = sin(0) = 0
- At x = 蟺/2, y = sin(蟺/2) = 1
- At x = 蟺, y = sin(蟺) = 0
- At x = 3蟺/2, y = sin(3蟺/2) = -1
- At x = 2蟺, y = sin(2蟺) = 0
These points outline the basic shape of the sine wave, which oscillates between -1 and 1.
x-intercepts of sin function
The x-intercepts of the sine function are the points where the graph crosses the x-axis. For y = sin(x), this occurs when the function value y = 0. In one full cycle from 0 to 2蟺, these x-intercepts occur at:
- x = 0
- x = 蟺
- x = 2蟺
Since sine is a periodic function, it repeats this pattern indefinitely. Therefore, additional x-intercepts also exist at intervals of 蟺, such as -蟺 and -2蟺. These intercepts help divide the graph into segments, showing where the sine wave crosses zero.
y-intercepts of sin function
The y-intercept of a function is the point where the graph crosses the y-axis. For the sine function y = sin(x), this occurs at x = 0. Therefore, the y-intercept is:
- At x = 0, y = sin(0) = 0
This is the starting point of the sine wave. Unlike x-intercepts, there is only one y-intercept for the sine function's basic period because the function is periodic, and the y-axis is the only place where x = 0 within this function's cycle.
maximum and minimum values in trig functions
In the sine function y = sin(x), the maximum and minimum values it can take are constrained between -1 and 1. The maximum value of 1 is achieved where:
- At x = 蟺/2 + 2n蟺, where n is an integer
This means the peak occurs at every half cycle plus a multiple of 2蟺.
The minimum value of -1 is achieved where:
- At x = 3蟺/2 + 2n蟺, where n is an integer
Thus, the trough occurs at every three-halves cycle plus a multiple of 2蟺.
These values demonstrate the oscillating nature of the sine wave.
graphing sine function
Graphing the sine function y = sin(x) over a range from -2蟺 to 2蟺 involves plotting its key points and connecting them smoothly. Here鈥檚 a step-by-step approach:
- Identify key points: start at 0, rise to 1 at 蟺/2, return to 0 at 蟺, dip to -1 at 3蟺/2, and return to 0 at 2蟺.
- Extend this pattern in both directions to cover -2蟺 to 2蟺.
- Mark x-intercepts where the graph crosses the x-axis: at multiples of 蟺 (0, 卤蟺, 卤2蟺).
- Note the maximum and minimum values: 1 at 蟺/2, -1 at 3蟺/2, within each cycle.
- The smooth, wave-like shape of the sine function is created by connecting these points in a consistent manner.
This graph represents the fundamental periodic nature of the sine wave, showing its continuous oscillation.

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Most popular questions from this chapter

A security camera scans a long straight fence that encloses a section of a military base. The camera is mounted on a post that is located \(5 \mathrm{m}\) from the midpoint of the fence. The camera makes one complete rotation in 60 s. a) Determine the tangent function that represents the distance, \(d\), in metres, along the fence from its midpoint as a function of time, \(t,\) in seconds, if the camera is aimed at the midpoint of the fence at \(t=0\) b) Graph the function in the interval \(-15 \leq t \leq 15\) c) What is the distance from the midpoint of the fence at \(t=10 \mathrm{s},\) to the nearest tenth of a metre? d) Describe what happens when \(t=15 \mathrm{s}\)

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