Sure, let's break this down step-by-step:
To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\(y = 8\cos(9x)\)[/tex], we need to follow these guidelines:
### Amplitude
The amplitude of a cosine function [tex]\(y = A\cos(Bx + C)\)[/tex] is given by the coefficient [tex]\(A\)[/tex] in front of the cosine function.
In this case, the equation is [tex]\(y = 8\cos(9x)\)[/tex]. The coefficient [tex]\(A\)[/tex] is 8, so the amplitude is:
[tex]\[ \text{Amplitude} = 8 \][/tex]
### Period
The period of a cosine function [tex]\(y = A\cos(Bx + C)\)[/tex] is calculated using the coefficient [tex]\(B\)[/tex] inside the cosine function, which affects the frequency. The formula for the period is:
[tex]\[ \text{Period} = \frac{2\pi}{|B|} \][/tex]
For the given function [tex]\(y = 8\cos(9x)\)[/tex], the coefficient [tex]\(B\)[/tex] is 9. Plugging this into the formula, we get:
[tex]\[ \text{Period} = \frac{2\pi}{9} \approx 0.6981317007977318 \][/tex]
### Phase Shift
The phase shift of a cosine function [tex]\(y = A\cos(Bx + C)\)[/tex] is determined by the term [tex]\( C \)[/tex]. It is calculated as:
[tex]\[ \text{Phase Shift} = -\frac{C}{B} \][/tex]
In the given function [tex]\(y = 8\cos(9x)\)[/tex], there is no additional term inside the argument of the cosine function beyond [tex]\(9x\)[/tex] (meaning [tex]\(C = 0\)[/tex]). Therefore, the phase shift is:
[tex]\[ \text{Phase Shift} = -\frac{0}{9} = 0 \][/tex]
So, the phase shift is:
[tex]\[ \text{Phase Shift} = 0 \][/tex]
### Summary
- Amplitude: 8
- Period: [tex]\( \approx 0.6981317007977318 \)[/tex]
- Phase Shift: 0 (No phase shift)
Therefore, the detailed solution yields these results:
- Amplitude: 8
- Period: Approx. 0.6981317007977318
- Phase Shift: No phase shift
