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Magazine Chapter 10: Problem 4
Complete the following table for the given functions and then plot the resulting graphs. $$\begin{array}{c|c|c|c|c|c|c|c|c|c} x & -\pi & -\frac{3 \pi}{4} & -\frac{\pi}{2} & -\frac{\pi}{4} & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3 \pi}{4} & \pi \\ \hline y & & & & & & & & & \end{array}$$ $$\begin{array}{c|c|c|c|c|c|c|c|c} x & \frac{5 \pi}{4} & \frac{3 \pi}{2} & \frac{7 \pi}{4} & 2 \pi & \frac{9 \pi}{4} & \frac{5 \pi}{2} & \frac{11 \pi}{4} & 3 \pi \\ \hline y & & & & & & & \end{array}$$ $$y=\cos x$$
Short Answer
Expert verified
First, compute the cosine for each given \( x \), fill in the table, then plot the cosine curve using these values.
Step by step solution
01
Understanding the Function
We are given the function \( y = \cos x \). This means for each value of \( x \), we need to calculate \( y \), which is the cosine of \( x \). The cosine function is periodic, with a period of \( 2\pi \). It ranges from -1 to 1.
02
Calculating \( y \) for \( x = -\pi \)
Calculate the cosine of \(-\pi\): \( \cos(-\pi) = -1 \). The function is symmetrical, so \( \cos(-x) = \cos(x) \).
03
Calculating \( y \) for \( x = -\frac{3\pi}{4} \)
Calculate the cosine of \(-\frac{3\pi}{4}\): \( \cos(-\frac{3\pi}{4}) = -\cos(\frac{3\pi}{4}) = -(-\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{2} \).
04
Calculating \( y \) for \( x = -\frac{\pi}{2} \)
Calculate the cosine of \(-\frac{\pi}{2}\): \( \cos(-\frac{\pi}{2}) = 0 \).
05
Calculating \( y \) for \( x = -\frac{\pi}{4} \)
Calculate the cosine of \(-\frac{\pi}{4}\): \( \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
06
Calculating \( y \) for \( x = 0 \)
Calculate the cosine at 0: \( \cos(0) = 1 \).
07
Calculating \( y \) for \( x = \frac{\pi}{4} \)
Calculate the cosine of \(\frac{\pi}{4}\): \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
08
Calculating \( y \) for \( x = \frac{\pi}{2} \)
Calculate the cosine of \(\frac{\pi}{2}\): \( \cos(\frac{\pi}{2}) = 0 \).
09
Calculating \( y \) for \( x = \frac{3\pi}{4} \)
Calculate the cosine of \(\frac{3\pi}{4}\): \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).
10
Calculating \( y \) for \( x = \pi \)
Calculate the cosine of \(\pi\): \( \cos(\pi) = -1 \).
11
Calculating \( y \) for Values from \( \frac{5\pi}{4} \) to \( 3\pi \)
Since the cosine function is periodic with a period of \(2\pi\), we repeat the cycle. Calculate: \( \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}, \cos\left(\frac{3\pi}{2}\right) = 0, \cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2}, \cos(2\pi) = 1, \cos\left(\frac{9\pi}{4}\right) = \frac{\sqrt{2}}{2}, \cos\left(\frac{5\pi}{2}\right) = 0, \cos\left(\frac{11\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \cos(3\pi) = -1 \).
12
Completing the Table
Fill in the table using the calculated values:\[\begin{array}{c|c|c|c|c|c|c|c|c|c}x & -\pi & -\frac{3 \pi}{4} & -\frac{\pi}{2} & -\frac{\pi}{4} & 0 & \frac{\pi}{4} & \frac{\pi}{2} & \frac{3 \pi}{4} & \pi \\hliney & -1 & \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 1 & \frac{\sqrt{2}}{2} & 0 & -\frac{\sqrt{2}}{2} & -1 \\hlinex & \frac{5 \pi}{4} & \frac{3 \pi}{2} & \frac{7 \pi}{4} & 2 \pi & \frac{9 \pi}{4} & \frac{5 \pi}{2} & \frac{11 \pi}{4} & 3 \pi \\hliney & -\frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} & 1 & \frac{\sqrt{2}}{2} & 0 & -\frac{\sqrt{2}}{2} & -1t\end{array}\]
13
Plotting the Graph
Use the completed table to plot the graph of \( y = \cos x \). The graph will show the characteristic wave-like pattern of the cosine function, oscillating between 1 and -1, completing one full cycle at \( 2\pi \) and repeating the cycle again onward.
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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 fundamental trigonometric functions. It is quite important in both mathematics and applications like physics and engineering. The function is typically represented as \( y = \cos x \). The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse. Another way to visualize the cosine function is by tracing it on the unit circle; the value of \( \cos x \) for any angle \( x \) corresponds to the x-coordinate of a point on the unit circle.
Let's delve into some of the core properties:
Let's delve into some of the core properties:
- **Range**: The output values of the cosine function are always between -1 and 1.
- **Symmetry**: The function is symmetric with respect to the y-axis, which means \( \cos(-x) = \cos(x) \).
- **Key Values**: For example, \( \cos(0) = 1 \), \( \cos(\pi) = -1 \), and \( \cos\left(\frac{\pi}{2}\right) = 0 \).
Periodic Functions
The cosine function belongs to a special category of functions called periodic functions. A function is considered periodic if it repeats its values at regular intervals or periods. The cosine function has a period of \( 2\pi \), which means it completes one full cycle of its values every \( 2\pi \) units.
Why does this matter?
Why does this matter?
- **Pattern Recognition**: By understanding periodicity, you can predict the behavior of the function beyond its initial cycle.
- **Simplicity in Calculations**: Calculations often become more straightforward since values are repeated over intervals. For instance, after calculating \( \cos(\pi) = -1 \), \( \cos(3\pi) \) is also \( -1 \) because \( 3\pi \) is \(2\pi\) plus \( \pi \).
- **Applications**: This property is widely used in signal processing where waves and signals are often modeled as periodic functions.
Graphing Trigonometric Functions
Graphing the cosine function can seem complex initially, but it's straightforward once you understand the function's properties. The graph of \( y = \cos x \) has a unique wave-like pattern:
Here鈥檚 how to plot it:
- The peaks of the graph touch the value 1, and the valleys go down to -1.
- The graph begins at 1 when \( x = 0 \) and repeats this pattern every \( 2\pi \).
- Between each \( \pi \), the values transition smoothly from peak to trough.
Here鈥檚 how to plot it:
- Identify key points such as \( (0,1) \), \( (\pi/2,0) \), \( (\pi,-1) \), and continue with these points using periodicity.
- Notice the symmetry around the y-axis during plotting.
- Be aware that for negative values of \( x \), the graph mirrors the positive side.
