Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Certainly! Let's determine the amplitude and period of the function [tex]\( f(x) = -8 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) \)[/tex].
### Step 1: Simplify the Function Using Trigonometric Identities
First, we'll simplify the given function. We can use the double-angle identity for sine, which states:
[tex]\[ \sin(2a) = 2 \sin(a) \cos(a) \][/tex]
Let [tex]\( a = \frac{3x}{2} \)[/tex]. Then we have:
[tex]\[ \sin(2a) = \sin \left( 2 \cdot \frac{3x}{2} \right) = \sin(3x) \][/tex]
Thus,
[tex]\[ 2 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) = \sin(3x) \][/tex]
Given the function [tex]\( f(x) = -8 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) \)[/tex], we can rewrite it as:
[tex]\[ f(x) = -8 \left( \frac{1}{2} \sin(3x) \right) \][/tex]
[tex]\[ f(x) = -4 \sin(3x) \][/tex]
### Step 2: Determine the Amplitude
The amplitude of a sine or cosine function of the form [tex]\( y = A \sin(Bx) \)[/tex] or [tex]\( y = A \cos(Bx) \)[/tex] is the absolute value of the coefficient [tex]\( A \)[/tex].
For the function [tex]\( f(x) = -4 \sin(3x) \)[/tex]:
- The coefficient [tex]\( A \)[/tex] is [tex]\(-4\)[/tex].
- The amplitude is the absolute value of [tex]\( A \)[/tex], which is [tex]\( | -4 | = 4 \)[/tex].
### Step 3: Determine the Period
The period of a sine or cosine function [tex]\( y = \sin(Bx) \)[/tex] or [tex]\( y = \cos(Bx) \)[/tex] is given by [tex]\( \frac{2\pi}{B} \)[/tex].
For the function [tex]\( f(x) = -4 \sin(3x) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] is [tex]\( 3 \)[/tex].
- Therefore, the period is [tex]\( \frac{2\pi}{3} \)[/tex].
### Summary
- Amplitude: [tex]\( 4 \)[/tex]
- Period: [tex]\( \frac{2\pi}{3} \approx 2.0943951023931953 \)[/tex]
Therefore, the amplitude and period of the graph of [tex]\( f(x) = -8 \sin \frac{3x}{2} \cos \frac{3x}{2} \)[/tex] are [tex]\( 4 \)[/tex] and [tex]\( \frac{2\pi}{3} \)[/tex], respectively.
### Step 1: Simplify the Function Using Trigonometric Identities
First, we'll simplify the given function. We can use the double-angle identity for sine, which states:
[tex]\[ \sin(2a) = 2 \sin(a) \cos(a) \][/tex]
Let [tex]\( a = \frac{3x}{2} \)[/tex]. Then we have:
[tex]\[ \sin(2a) = \sin \left( 2 \cdot \frac{3x}{2} \right) = \sin(3x) \][/tex]
Thus,
[tex]\[ 2 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) = \sin(3x) \][/tex]
Given the function [tex]\( f(x) = -8 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{3x}{2} \right) \)[/tex], we can rewrite it as:
[tex]\[ f(x) = -8 \left( \frac{1}{2} \sin(3x) \right) \][/tex]
[tex]\[ f(x) = -4 \sin(3x) \][/tex]
### Step 2: Determine the Amplitude
The amplitude of a sine or cosine function of the form [tex]\( y = A \sin(Bx) \)[/tex] or [tex]\( y = A \cos(Bx) \)[/tex] is the absolute value of the coefficient [tex]\( A \)[/tex].
For the function [tex]\( f(x) = -4 \sin(3x) \)[/tex]:
- The coefficient [tex]\( A \)[/tex] is [tex]\(-4\)[/tex].
- The amplitude is the absolute value of [tex]\( A \)[/tex], which is [tex]\( | -4 | = 4 \)[/tex].
### Step 3: Determine the Period
The period of a sine or cosine function [tex]\( y = \sin(Bx) \)[/tex] or [tex]\( y = \cos(Bx) \)[/tex] is given by [tex]\( \frac{2\pi}{B} \)[/tex].
For the function [tex]\( f(x) = -4 \sin(3x) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] is [tex]\( 3 \)[/tex].
- Therefore, the period is [tex]\( \frac{2\pi}{3} \)[/tex].
### Summary
- Amplitude: [tex]\( 4 \)[/tex]
- Period: [tex]\( \frac{2\pi}{3} \approx 2.0943951023931953 \)[/tex]
Therefore, the amplitude and period of the graph of [tex]\( f(x) = -8 \sin \frac{3x}{2} \cos \frac{3x}{2} \)[/tex] are [tex]\( 4 \)[/tex] and [tex]\( \frac{2\pi}{3} \)[/tex], respectively.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.