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Magazine Chapter 10: Problem 26
Although units of \(\pi\) are convenient, we must remember that \(\pi\) is only a number. Numbers that are not multiples of \(\pi\) may be used. Plot the indicated graphs by finding the values of \(y\) that correspond to values of \(x\) of 0,1,2,3,4,5,6, and 7 on a calculator. (Remember, the numbers \(0,1,2,\) and so on represent radian measure.) $$y=2 \cos x$$
Short Answer
Expert verified
Plot y for x values from 0 to 7 to get a cosine wave scaled by 2.
Step by step solution
01
Understand the Function
We need to plot the graph of the function \( y = 2 \cos x \) for \( x \) values ranging from 0 to 7. This means evaluating the function for each integer value in this range.
02
Calculate \(y\) for \(x = 0\)
Substitute \( x = 0 \) into the equation: \( y = 2 \cos(0) \). Since \( \cos(0) = 1 \), we have \( y = 2 \times 1 = 2 \).
03
Calculate \(y\) for \(x = 1\)
Substitute \( x = 1 \) into the equation: \( y = 2 \cos(1) \). Using a calculator, we find \( \cos(1) \approx 0.5403 \), so \( y \approx 2 \times 0.5403 = 1.0806 \).
04
Calculate \(y\) for \(x = 2\)
Substitute \( x = 2 \) into the equation: \( y = 2 \cos(2) \). Using a calculator, \( \cos(2) \approx -0.4161 \), so \( y \approx 2 \times -0.4161 = -0.8322 \).
05
Calculate \(y\) for \(x = 3\)
Substitute \( x = 3 \) into the equation: \( y = 2 \cos(3) \). Using a calculator, \( \cos(3) \approx -0.98999 \), so \( y \approx 2 \times -0.98999 = -1.97998 \).
06
Calculate \(y\) for \(x = 4\)
Substitute \( x = 4 \) into the equation: \( y = 2 \cos(4) \). Using a calculator, \( \cos(4) \approx -0.65364 \), so \( y \approx 2 \times -0.65364 = -1.30728 \).
07
Calculate \(y\) for \(x = 5\)
Substitute \( x = 5 \) into the equation: \( y = 2 \cos(5) \). Using a calculator, \( \cos(5) \approx 0.28366 \), so \( y \approx 2 \times 0.28366 = 0.56732 \).
08
Calculate \(y\) for \(x = 6\)
Substitute \( x = 6 \) into the equation: \( y = 2 \cos(6) \). Using a calculator, \( \cos(6) \approx 0.96017 \), so \( y \approx 2 \times 0.96017 = 1.92034 \).
09
Calculate \(y\) for \(x = 7\)
Substitute \( x = 7 \) into the equation: \( y = 2 \cos(7) \). Using a calculator, \( \cos(7) \approx 0.7539 \), so \( y \approx 2 \times 0.7539 = 1.5078 \).
10
Plot the Points
Now we have the following pairs of coordinates to plot: - \((0, 2)\)- \((1, 1.0806)\)- \((2, -0.8322)\)- \((3, -1.97998)\)- \((4, -1.30728)\)- \((5, 0.56732)\)- \((6, 1.92034)\)- \((7, 1.5078)\) These points can be plotted on a graph with the x-axis ranging from 0 to 7 and the y-axis showing the resulting values.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosine Function
The cosine function is one of the key trigonometric functions, often written as \(\cos(\theta)\). It measures the horizontal component of the angle \(\theta\) in a right-angled triangle. This function is especially useful in cyclic or periodic phenomena, like waves. For instance, consider the function \(y = 2 \cos(x)\), where each \(x\) represents an angle measured in radians and the output \(y\) represents a scaled version of this angle's horizontal component.
This scaling means the graph will stretch vertically by a factor of 2.
A few properties of the cosine function include:
This scaling means the graph will stretch vertically by a factor of 2.
A few properties of the cosine function include:
- It has a maximum value of 1 and a minimum value of -1, if not scaled or shifted.
- The function is periodic with a period of \(2\pi\).
- \(\cos(x)\) is an even function, symmetric about the y-axis.
Radian Measure
Radian measure is a way of expressing angles. In trigonometry, using radians is often more natural than degrees. A radian is defined as the angle subtended by an arc that is equal in length to the radius of the circle.
This results in a circle having \(2\pi\) radians in total, which equals 360 degrees.
Why do we use radians?
Here are a few reasons:
This results in a circle having \(2\pi\) radians in total, which equals 360 degrees.
Why do we use radians?
Here are a few reasons:
- The radian measure leads to simpler and more natural formulas, especially in calculus.
- Trigonometric functions have simpler derivatives and integrals in radians.
- Many periodic phenomena in physics and engineering are naturally expressed in radians.
Graphing Functions
Graphing functions involves plotting a set of points that represent the relationship defined by the function equation. For \(y = 2 \cos(x)\), this means computing the cosine of various angle measures and multiplying by 2, essentially creating vertical stretches of the original cosine curve.
When graphing such functions, keep a few tips in mind:
When graphing such functions, keep a few tips in mind:
- Choose several values of \(x\) across one cycle to capture the essential shape of the graph.
- Consider transformations such as vertical stretches or phase shifts that modify the basic curve.
- Accurate calculator use is crucial for non-standard angles.
Calculator Use in Mathematics
Calculators are invaluable in handling complex calculations and revealing insights when dealing with trigonometric expressions. In our exercise with \(y = 2 \cos(x)\), calculators help us find the precise cosine values for specific radian measures.
Here's how they can aid your mathematical journey:
Here's how they can aid your mathematical journey:
- The calculator provides quick conversions and precise computations, guaranteeing accuracy when finding values like \(\cos(1) \approx 0.5403\).
- For non-obvious angles, such as 1 radian, the calculator shows results swiftly, aiding in plotting graphs accurately.
- It can handle functions, derivatives, and integrals beyond simple trigonometric operations when advanced modes like graphing are used.
