Write the equation of a sine function that has the following characteristics. Amplitude: 9 Period: 3 Phase shift: 8 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. Amplitude: 9 Period: 3 Phase shift: 8 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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Question
![**Title: Constructing a Specific Sine Function**
**Introduction:**
Understanding how to write the equation of a sine function based on its characteristics is essential in trigonometry. This exercise involves constructing such an equation given specific properties, including amplitude, period, and phase shift.
**Problem Statement:**
_Write the equation of a sine function that has the following characteristics:_
- **Amplitude:** 9
- **Period:** \( 3\pi \)
- **Phase shift:** \( -\frac{1}{8} \)
----------------------
**Step-by-Step Instructions:**
1. **Identify the general form of the sine function:**
The general form of a sine function is:
\[
y = A \sin(Bx + C) + D
\]
where:
- \( A \) represents amplitude.
- \( \frac{2\pi}{B} \) is the period.
- \( \frac{-C}{B} \) is the phase shift.
- \( D \) represents vertical shift (not needed for this problem).
2. **Substitute the given amplitude**:
\[
A = 9
\]
3. **Calculate \( B \) using the period**:
Given the period \( P = 3\pi \),
\[
B = \frac{2\pi}{P} = \frac{2\pi}{3\pi} = \frac{2}{3}
\]
4. **Determine \( C \) using the phase shift**:
Given the phase shift \( -\frac{1}{8} \),
\[
\text{Phase shift} = \frac{-C}{B} \Rightarrow -\frac{1}{8} = \frac{-C}{\frac{2}{3}} \Rightarrow C = \frac{2}{3} \cdot \frac{1}{8} = \frac{1}{12}
\]
5. **Construct the equation**:
\[
y = 9 \sin\left(\frac{2}{3}x + \frac{1}{12}\right)
\]
**Completion Exercise:**
_Type the appropriate values to complete the sine function:_
\[
y = \boxed{9} \sin \left( \boxed{\frac{2}{3}} x + \boxed{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87573c1f-e99e-41d1-a108-161ab6bfa656%2Fc3adad22-60f4-43dc-8a6c-7c3245657df1%2Fxz0xw3m_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Constructing a Specific Sine Function**
**Introduction:**
Understanding how to write the equation of a sine function based on its characteristics is essential in trigonometry. This exercise involves constructing such an equation given specific properties, including amplitude, period, and phase shift.
**Problem Statement:**
_Write the equation of a sine function that has the following characteristics:_
- **Amplitude:** 9
- **Period:** \( 3\pi \)
- **Phase shift:** \( -\frac{1}{8} \)
----------------------
**Step-by-Step Instructions:**
1. **Identify the general form of the sine function:**
The general form of a sine function is:
\[
y = A \sin(Bx + C) + D
\]
where:
- \( A \) represents amplitude.
- \( \frac{2\pi}{B} \) is the period.
- \( \frac{-C}{B} \) is the phase shift.
- \( D \) represents vertical shift (not needed for this problem).
2. **Substitute the given amplitude**:
\[
A = 9
\]
3. **Calculate \( B \) using the period**:
Given the period \( P = 3\pi \),
\[
B = \frac{2\pi}{P} = \frac{2\pi}{3\pi} = \frac{2}{3}
\]
4. **Determine \( C \) using the phase shift**:
Given the phase shift \( -\frac{1}{8} \),
\[
\text{Phase shift} = \frac{-C}{B} \Rightarrow -\frac{1}{8} = \frac{-C}{\frac{2}{3}} \Rightarrow C = \frac{2}{3} \cdot \frac{1}{8} = \frac{1}{12}
\]
5. **Construct the equation**:
\[
y = 9 \sin\left(\frac{2}{3}x + \frac{1}{12}\right)
\]
**Completion Exercise:**
_Type the appropriate values to complete the sine function:_
\[
y = \boxed{9} \sin \left( \boxed{\frac{2}{3}} x + \boxed{\
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