Answer :
To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex], we'll follow these steps.
### Step-by-Step Solution:
#### 1. Amplitude:
The amplitude of a sinusoidal function of the form [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex] is given by the absolute value of [tex]\( A \)[/tex].
In the given equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex]:
- [tex]\( A = \frac{-5}{3} \)[/tex]
- The amplitude is [tex]\( |A| = \left|\frac{-5}{3}\right| = \frac{5}{3} \)[/tex].
So, the amplitude is:
[tex]\[ \frac{5}{3} \][/tex]
#### 2. Period:
The period of a sinusoidal function [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex] is calculated using the coefficient [tex]\( B \)[/tex] inside the function. The formula for the period is:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
In the given equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex]:
- [tex]\( B = \frac{\pi}{3} \)[/tex]
- The period is:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{\pi}{3}} = 2\pi \times \frac{3}{\pi} = 6 \][/tex]
So, the period is:
[tex]\[ 6 \][/tex]
#### 3. Phase Shift:
The phase shift of a sinusoidal function [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex] is determined from the horizontal shift within the function. The general formula to find the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{C}{B} \][/tex]
In the given equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex]:
- There is no [tex]\( C \)[/tex] term present (i.e., [tex]\( C = 0 \)[/tex])
Since [tex]\( C = 0 \)[/tex]:
[tex]\[ \text{Phase Shift} = -\frac{0}{\frac{\pi}{3}} = 0 \][/tex]
So, there is no phase shift.
### Summary:
- Amplitude: [tex]\( \frac{5}{3} \)[/tex]
- Period: [tex]\( 6 \)[/tex]
- Phase Shift: No phase shift.
Finally, we can conclude:
- Amplitude: [tex]\( 1.6666666666666667 \)[/tex]
- Period: [tex]\( 6.0 \)[/tex]
- Phase Shift: no phase shift
### Step-by-Step Solution:
#### 1. Amplitude:
The amplitude of a sinusoidal function of the form [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex] is given by the absolute value of [tex]\( A \)[/tex].
In the given equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex]:
- [tex]\( A = \frac{-5}{3} \)[/tex]
- The amplitude is [tex]\( |A| = \left|\frac{-5}{3}\right| = \frac{5}{3} \)[/tex].
So, the amplitude is:
[tex]\[ \frac{5}{3} \][/tex]
#### 2. Period:
The period of a sinusoidal function [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex] is calculated using the coefficient [tex]\( B \)[/tex] inside the function. The formula for the period is:
[tex]\[ \text{Period} = \frac{2\pi}{B} \][/tex]
In the given equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex]:
- [tex]\( B = \frac{\pi}{3} \)[/tex]
- The period is:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{\pi}{3}} = 2\pi \times \frac{3}{\pi} = 6 \][/tex]
So, the period is:
[tex]\[ 6 \][/tex]
#### 3. Phase Shift:
The phase shift of a sinusoidal function [tex]\( y = A \sin(Bx + C) \)[/tex] or [tex]\( y = A \cos(Bx + C) \)[/tex] is determined from the horizontal shift within the function. The general formula to find the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{C}{B} \][/tex]
In the given equation [tex]\( y=\frac{-5}{3} \sin \left(\frac{\pi}{3} x\right) \)[/tex]:
- There is no [tex]\( C \)[/tex] term present (i.e., [tex]\( C = 0 \)[/tex])
Since [tex]\( C = 0 \)[/tex]:
[tex]\[ \text{Phase Shift} = -\frac{0}{\frac{\pi}{3}} = 0 \][/tex]
So, there is no phase shift.
### Summary:
- Amplitude: [tex]\( \frac{5}{3} \)[/tex]
- Period: [tex]\( 6 \)[/tex]
- Phase Shift: No phase shift.
Finally, we can conclude:
- Amplitude: [tex]\( 1.6666666666666667 \)[/tex]
- Period: [tex]\( 6.0 \)[/tex]
- Phase Shift: no phase shift
