To find the amplitude and period of the function
[tex]\[ y = \frac{3}{4} \cos \left( \frac{4}{3} x \right) \][/tex]
we will follow these steps:
1. Amplitude:
The amplitude of a cosine function [tex]\( y = A \cos(Bx) \)[/tex] is given by the absolute value of the coefficient [tex]\( A \)[/tex].
Here, the coefficient of the cosine function is [tex]\(\frac{3}{4}\)[/tex]. Therefore, the amplitude [tex]\(A\)[/tex] is:
[tex]\[
A = \left| \frac{3}{4} \right| = \frac{3}{4}
\][/tex]
2. Period:
The period of a cosine function [tex]\( y = A \cos(Bx) \)[/tex] is given by [tex]\(\frac{2\pi}{B}\)[/tex], where [tex]\(B\)[/tex] is the coefficient of [tex]\(x\)[/tex] inside the cosine function.
In this function, the coefficient [tex]\( B \)[/tex] is [tex]\(\frac{4}{3}\)[/tex]. Therefore, the period [tex]\(T\)[/tex] is:
[tex]\[
T = \frac{2\pi}{B} = \frac{2\pi}{\frac{4}{3}} = 2\pi \cdot \frac{3}{4} = \frac{6\pi}{4} = \frac{3\pi}{2}
\][/tex]
Thus, the exact values are:
- Amplitude: [tex]\( \frac{3}{4} \)[/tex]
- Period: [tex]\( \frac{3\pi}{2} \)[/tex]
