The following graph shows at least one complete cycle of the graph of an equation containing a trigonometric function. Find an equation to match the graph. If you are using a graphing calculator, graph your equation to verify that it is correct. y = 20 20 4 20 20
The following graph shows at least one complete cycle of the graph of an equation containing a trigonometric function. Find an equation to match the graph. If you are using a graphing calculator, graph your equation to verify that it is correct. y = 20 20 4 20 20
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Instruction:
The following graph shows at least one complete cycle of the graph of an equation containing a trigonometric function. Find an equation to match the graph. If you are using a graphing calculator, graph your equation to verify that it is correct.
### Graph Description:
**Axes and Scale:**
- The vertical axis (y-axis) ranges from -6 to 6.
- The horizontal axis (x-axis) is labeled with multiples of \(\pi\): \(-\frac{\pi}{20}\), \(\frac{3\pi}{20}\), \(\frac{\pi}{4}\), \(\frac{7\pi}{20}\), and \(\frac{9\pi}{20}\).
**Graph Details:**
- The graph is a sinusoidal wave (resembling a cosine wave), starting from \((0, 6)\), descending to its lowest point at \((\frac{\pi}{4}, -6)\), and then ascending back to \((\frac{9\pi}{20}, 6)\).
- This represents one complete cycle of the trig function with an amplitude of 6.
### Explanation:
The graph likely represents a cosine function because it starts at its maximum point. The amplitude is 6, indicating the equation is of the form \(y = 6\cos(bx)\).
- **Period Determination:**
- The period of a cosine function is \( \frac{2\pi}{b} \).
- One full cycle is completed between \(x = 0\) and \(x = \frac{9\pi}{20}\), so \(\frac{9\pi}{20} = \frac{2\pi}{b}\).
- Solving for \(b\) gives \(b = \frac{40}{9}\).
Thus, the equation for the graph is:
\[ y = 6\cos\left(\frac{40}{9}x\right) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb49976cd-c3bf-478b-8ad4-67f602b8a4e2%2F5ae2c2c2-6346-4f85-8424-4f66df174e36%2Fb0yu9i_processed.png&w=3840&q=75)
Transcribed Image Text:### Instruction:
The following graph shows at least one complete cycle of the graph of an equation containing a trigonometric function. Find an equation to match the graph. If you are using a graphing calculator, graph your equation to verify that it is correct.
### Graph Description:
**Axes and Scale:**
- The vertical axis (y-axis) ranges from -6 to 6.
- The horizontal axis (x-axis) is labeled with multiples of \(\pi\): \(-\frac{\pi}{20}\), \(\frac{3\pi}{20}\), \(\frac{\pi}{4}\), \(\frac{7\pi}{20}\), and \(\frac{9\pi}{20}\).
**Graph Details:**
- The graph is a sinusoidal wave (resembling a cosine wave), starting from \((0, 6)\), descending to its lowest point at \((\frac{\pi}{4}, -6)\), and then ascending back to \((\frac{9\pi}{20}, 6)\).
- This represents one complete cycle of the trig function with an amplitude of 6.
### Explanation:
The graph likely represents a cosine function because it starts at its maximum point. The amplitude is 6, indicating the equation is of the form \(y = 6\cos(bx)\).
- **Period Determination:**
- The period of a cosine function is \( \frac{2\pi}{b} \).
- One full cycle is completed between \(x = 0\) and \(x = \frac{9\pi}{20}\), so \(\frac{9\pi}{20} = \frac{2\pi}{b}\).
- Solving for \(b\) gives \(b = \frac{40}{9}\).
Thus, the equation for the graph is:
\[ y = 6\cos\left(\frac{40}{9}x\right) \]
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