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Chapter 5: Problem 2

a) State the five key points for \(y=\cos x\) that occur in one complete cycle from 0 to \(2 \pi\) b) Use the key points to sketch a graph of \(y=\cos x\) 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 are at 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), 2 \( 2\pi \). The x-intercepts are at \( -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \). The y-intercept is at (0,1). The maximum value is 1, and the minimum value is -1.

Step by step solution

01

State the Five Key Points

For the function \( y = \cos x \), the key points in one complete cycle from 0 to \( 2 \pi \) occur at \(x = 0 \), \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). At these points, the values of \( y = \cos x \) are 1, 0, -1, 0, and 1, respectively.
02

Sketch the Graph and Key Points

To sketch the graph of \( y = \cos x \) for \( -2 \pi \leq x \leq 2 \pi \), plot the key points found in the previous step for one cycle and replicate this pattern for the range. The key points are: \((-2\pi, 1)\), \((-\frac{3\pi}{2}, 0)\), \((-\pi, -1)\), \((-\frac{\pi}{2}, 0)\), \((0, 1)\), \((\frac{\pi}{2}, 0)\), \((\pi, -1)\), \((\frac{3\pi}{2}, 0)\), and \((2\pi, 1)\). Connect these points with a smooth curve to get the graph.
03

Determine the x-Intercepts

The \( x \)-intercepts, where the graph crosses the x-axis, are the points where \( \cos x = 0 \). These occur at \( x = -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2} \) within the given range.
04

Determine the y-Intercept

The \( y \)-intercept is the point where the graph crosses the y-axis. For \( y = \cos x \), this occurs at \( x = 0 \), so the \( y \)-intercept is the point \( (0, 1) \).
05

Determine the Maximum and Minimum Values

The maximum value of \( \cos x \) is 1, which occurs at \( x = 2k\pi \) for any integer \( k \). The minimum value of \( \cos x \) is -1, which occurs at \( x = (2k+1)\pi \) for any integer \( k \). Within the given range, the maximum value is 1 and the minimum value is -1.

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Key Concepts

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

Key Points of Cosine Function
The cosine function, denoted as \(y=\cos x\), has specific key points that occur in one complete cycle from 0 to \(2\pi\). These key points are essential for understanding how the function behaves over its period.
For \(y=\cos x\), the key points occur at:
  • \(x=0\)
  • \(x=\frac{\pi}{2}\)
  • \(x=\pi\)
  • \(x=\frac{3\pi}{2}\)
  • \(x=2\pi\)

At these points, the corresponding values of \(y=\cos x\) are:
  • 1
  • 0
  • -1
  • 0
  • 1

Understanding these key points helps to recognize the periodical nature of the cosine function and forms the basis for graphing it effectively.
Graphing Trigonometric Functions
Graphing trigonometric functions like \(y=\cos x\) involves plotting the key points and then drawing a smooth curve. To sketch the graph of the function for \(-2\pi \leq x \leq 2\pi\), follow these steps:
  • Plot the key points from one complete cycle: \((0,1)\), \((\frac{\pi}{2},0)\), \((\pi,-1)\), \((\frac{3\pi}{2},0)\), and \((2\pi,1)\).
  • Replicate the pattern for the given range past these initial points: \((-2\pi,1)\), \((-\frac{3\pi}{2},0)\), \((-\pi,-1)\), \((-\frac{\pi}{2},0)\), and \((0,1)\).

Connecting these points with a smooth wave-like curve will give you the graph. Understanding how to plot these can visually demonstrate how the \(\cos x\) function behaves over its period, alternating between its maximum and minimum values.
Trigonometric Intercepts
Intercepts are points where the graph crosses the axes. For the function \(y=\cos x\), both x- and y-intercepts are significant.
The x-intercepts are the points where the graph crosses the x-axis, which happens when \(\cos x = 0\). For the range \(-2\pi \leq x \leq 2\pi\), this occurs at:
  • \(-\frac{3\pi}{2}\)
  • \(-\frac{\pi}{2}\)
  • \((\frac{\pi}{2}\)
  • \((\frac{3\pi}{2}\)

These are the points where the graph touches the x-axis.

The y-intercept is the point where the graph crosses the y-axis. For \(y=\cos x\), this occurs at \(x=0\). Hence, the y-intercept is \((0,1)\). Intercepts are helpful in plotting the graph and understanding where the function equals zero or intersects the axes.
Maximum and Minimum Values in Trigonometry
In trigonometry, the maximum and minimum values of functions tell us about their range and behavior.
For the cosine function \(y=\cos x\), these extreme values are crucial.
The maximum value of \(\cos x\) is:\[y=1\]
This occurs at points where \(x=2k\pi\) for any integer \(k\).
The minimum value of \(\cos x\) is:\[y=-1\]
This happens at points where \(x=(2k+1)\pi\) for any integer \(k\).
Within the range \(-2\pi \leq x \leq 2\pi\), the function oscillates between these maximum and minimum values, defining its amplitude. Understanding the maxima and minima is vital for analyzing the complete behavior of the trigonometric functions.

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Most popular questions from this chapter

A point on an industrial flywheel experiences a motion described by the function \(h(t)=13 \cos \left(\frac{2 \pi}{0.7} t\right)+15\) where \(h\) is the height, in metres, and \(t\) is the time, in minutes. a) What is the maximum height of the point? b) After how many minutes is the maximum height reached? c) What is the minimum height of the point? d) After how many minutes is the minimum height reached? e) For how long, within one cycle, is the point less than \(6 \mathrm{m}\) above the ground? f) Determine the height of the point if the wheel is allowed to turn for \(1 \mathrm{h}\) 12 min.

Determine the period, the sinusoidal axis, and the amplitude for each of the following. a) The first maximum of a sine function occurs at the point \(\left(30^{\circ}, 24\right),\) and the first minimum to the right of the maximum occurs at the point \(\left(80^{\circ}, 6\right).\) b) The first maximum of a cosine function occurs at \((0,4),\) and the first minimum to the right of the maximum occurs at \(\left(\frac{2 \pi}{3},-16\right).\) c) An electron oscillates back and forth 50 times per second, and the maximum and minimum values occur at +10 and \(-10,\) respectively.

A rotating light on top of a lighthouse sends out rays of light in opposite directions. As the beacon rotates, the ray at angle \(\theta\) makes a spot of light that moves along the shore. The lighthouse is located \(500 \mathrm{m}\) from the shoreline and makes one complete rotation every 2 min. a) Determine the equation that expresses the distance, \(d,\) in metres, as a function of time, \(t,\) in minutes. b) Graph the function in part a). c) Explain the significance of the asymptote in the graph at \(\theta=90^{\circ}\).

Determine the period (in degrees) of each function. Then, use the language of transformations to describe how each graph is related to the graph of \(y=\cos x\) a) \(y=\cos 2 x\) b) \(y=\cos (-3 x)\) c) \(y=\cos \frac{1}{4} x\) d) \(y=\cos \frac{2}{3} x\)

The graph of \(y=\tan \theta\) appears to be vertical as \(\theta\) approaches \(90^{\circ}\) a) Copy and complete the table. Use a calculator to record the tangent values as \(\theta\) approaches \(90^{\circ}\). b) What happens to the value of \(\tan \theta\) as \(\theta\) approaches \(90^{\circ} ?\) c) Predict what will happen as \(\theta\) approaches \(90^{\circ}\) from the other direction.
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