Graph one complete cycle. Label the axes accurately. y = cos
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Text for Educational Website:**
**Instruction:**
Graph one complete cycle. Label the axes accurately.
**Function:**
\[ y = \cos\left(\frac{1}{3}x\right) \]
**Explanation:**
This task involves graphing one complete cycle of the cosine function given by \( y = \cos\left(\frac{1}{3}x\right) \).
### Key Points:
- **Amplitude**: The amplitude of the cosine function remains 1, as there is no vertical scaling factor.
- **Period**: The period of the cosine function is modified by the factor \(\frac{1}{3}\). The standard period of \(\cos(x)\) is \(2\pi\). Thus, the period of \(y = \cos\left(\frac{1}{3}x\right)\) is:
\[
\text{Period} = \frac{2\pi}{1/3} = 6\pi
\]
- **X-Axis**: Mark the x-axis from \(0\) to \(6\pi\) to cover one complete cycle.
- **Y-Axis**: Range from -1 to 1, as the cosine function oscillates between these values.
**Graphing Steps:**
1. **Start** at \(x = 0\): \(y = 1\).
2. **Quarter Period** (\(x = \frac{1}{4}(6\pi) = \frac{3\pi}{2}\)): \(y = 0\).
3. **Halfway Through** (\(x = \frac{1}{2}(6\pi) = 3\pi\)): \(y = -1\).
4. **Three-Quarter Period** (\(x = \frac{3}{4}(6\pi) = \frac{9\pi}{2}\)): \(y = 0\).
5. **End** at \(x = 6\pi\): \(y = 1\).
By plotting these points and connecting them smoothly, you will graph one complete cycle of the function. Make sure to label the axes carefully to reflect these values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdeba73c6-02d1-47fd-8ecd-46654cff513f%2F2599f71a-ce47-4d37-8a83-7fd7d344ae98%2Fkxa06v_processed.png&w=3840&q=75)
Transcribed Image Text:**Text for Educational Website:**
**Instruction:**
Graph one complete cycle. Label the axes accurately.
**Function:**
\[ y = \cos\left(\frac{1}{3}x\right) \]
**Explanation:**
This task involves graphing one complete cycle of the cosine function given by \( y = \cos\left(\frac{1}{3}x\right) \).
### Key Points:
- **Amplitude**: The amplitude of the cosine function remains 1, as there is no vertical scaling factor.
- **Period**: The period of the cosine function is modified by the factor \(\frac{1}{3}\). The standard period of \(\cos(x)\) is \(2\pi\). Thus, the period of \(y = \cos\left(\frac{1}{3}x\right)\) is:
\[
\text{Period} = \frac{2\pi}{1/3} = 6\pi
\]
- **X-Axis**: Mark the x-axis from \(0\) to \(6\pi\) to cover one complete cycle.
- **Y-Axis**: Range from -1 to 1, as the cosine function oscillates between these values.
**Graphing Steps:**
1. **Start** at \(x = 0\): \(y = 1\).
2. **Quarter Period** (\(x = \frac{1}{4}(6\pi) = \frac{3\pi}{2}\)): \(y = 0\).
3. **Halfway Through** (\(x = \frac{1}{2}(6\pi) = 3\pi\)): \(y = -1\).
4. **Three-Quarter Period** (\(x = \frac{3}{4}(6\pi) = \frac{9\pi}{2}\)): \(y = 0\).
5. **End** at \(x = 6\pi\): \(y = 1\).
By plotting these points and connecting them smoothly, you will graph one complete cycle of the function. Make sure to label the axes carefully to reflect these values.
![**Identify the period for the graph.**
[Graph placeholder]
In this section, users are prompted to determine the period of a given graph. The period refers to the duration of one complete cycle of a repeating pattern in the graph. Understanding the period is essential for analyzing periodic functions, such as sine and cosine waves in trigonometry.
**Instructions:**
1. **Observe the Graph**: Look for repeating patterns or cycles.
2. **Measure the Duration**: Identify the start and end points of one cycle.
3. **Calculate the Period**: Determine the distance between these points.
Please enter your answer in the provided box. This task is important in applications involving sound waves, alternating current circuits, and other oscillating systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdeba73c6-02d1-47fd-8ecd-46654cff513f%2F2599f71a-ce47-4d37-8a83-7fd7d344ae98%2Fo2yb5jr_processed.png&w=3840&q=75)
Transcribed Image Text:**Identify the period for the graph.**
[Graph placeholder]
In this section, users are prompted to determine the period of a given graph. The period refers to the duration of one complete cycle of a repeating pattern in the graph. Understanding the period is essential for analyzing periodic functions, such as sine and cosine waves in trigonometry.
**Instructions:**
1. **Observe the Graph**: Look for repeating patterns or cycles.
2. **Measure the Duration**: Identify the start and end points of one cycle.
3. **Calculate the Period**: Determine the distance between these points.
Please enter your answer in the provided box. This task is important in applications involving sound waves, alternating current circuits, and other oscillating systems.
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