Write the equation of a sine function that has the following characteristics. 1 Amplitude: 8 Period: 7 Phase shift: 7 ... Type the appropriate values to complete the sine function. y=sin(x-) (Use integers or fractions for any numbers in the expression. Simplify your answers.)
Write the equation of a sine function that has the following characteristics. 1 Amplitude: 8 Period: 7 Phase shift: 7 ... Type the appropriate values to complete the sine function. y=sin(x-) (Use integers or fractions for any numbers in the expression. Simplify your answers.)
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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![### Sine Function Characteristics and Equation Completion
**Problem Statement:**
Write the equation of a sine function that has the following characteristics:
- **Amplitude:** 8
- **Period:** \( 7\pi \)
- **Phase shift:** \( \frac{1}{7} \)
**Equation Form Completion:**
Type the appropriate values to complete the sine function:
\[ y = \_ \sin \left( \_ x - \_ \right) \]
*(Use integers or fractions for any numbers in the expression. Simplify your answers.)*
---
**Explanation:**
To build the sine function using the given characteristics, we need to fill in the general form of the sine function:
\[ y = A \sin(Bx - C) \]
Where:
- \( A \) is the amplitude.
- \( B \) is related to the period (\( \text{Period} = \frac{2\pi}{B} \)).
- \( C \) is the phase shift (\( \text{Phase shift} = \frac{C}{B} \)).
**Steps to Find the Values:**
1. **Amplitude (A):**
Given: Amplitude = 8
\[ A = 8 \]
2. **Period (B):**
Given: Period = \( 7\pi \)
\[ B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{7\pi} = \frac{2}{7} \]
3. **Phase Shift (C):**
Given: Phase shift = \( \frac{1}{7} \)
Using \( \text{Phase shift} = \frac{C}{B} \):
\[ \frac{C}{\frac{2}{7}} = \frac{1}{7} \]
Solving for \( C \):
\[ C = \frac{2}{7} \times \frac{1}{7} = \frac{2}{49} \]
Therefore, the completed sine function is:
\[ y = 8 \sin \left( \frac{2}{7} x - \frac{2}{49} \right) \]
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Transcribed Image Text:### Sine Function Characteristics and Equation Completion
**Problem Statement:**
Write the equation of a sine function that has the following characteristics:
- **Amplitude:** 8
- **Period:** \( 7\pi \)
- **Phase shift:** \( \frac{1}{7} \)
**Equation Form Completion:**
Type the appropriate values to complete the sine function:
\[ y = \_ \sin \left( \_ x - \_ \right) \]
*(Use integers or fractions for any numbers in the expression. Simplify your answers.)*
---
**Explanation:**
To build the sine function using the given characteristics, we need to fill in the general form of the sine function:
\[ y = A \sin(Bx - C) \]
Where:
- \( A \) is the amplitude.
- \( B \) is related to the period (\( \text{Period} = \frac{2\pi}{B} \)).
- \( C \) is the phase shift (\( \text{Phase shift} = \frac{C}{B} \)).
**Steps to Find the Values:**
1. **Amplitude (A):**
Given: Amplitude = 8
\[ A = 8 \]
2. **Period (B):**
Given: Period = \( 7\pi \)
\[ B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{7\pi} = \frac{2}{7} \]
3. **Phase Shift (C):**
Given: Phase shift = \( \frac{1}{7} \)
Using \( \text{Phase shift} = \frac{C}{B} \):
\[ \frac{C}{\frac{2}{7}} = \frac{1}{7} \]
Solving for \( C \):
\[ C = \frac{2}{7} \times \frac{1}{7} = \frac{2}{49} \]
Therefore, the completed sine function is:
\[ y = 8 \sin \left( \frac{2}{7} x - \frac{2}{49} \right) \]
---
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