To find the amplitude and period of the function [tex]\( y = \frac{2}{3} \cos 5x \)[/tex], we follow these steps:
### Amplitude
The amplitude of a cosine function [tex]\( y = A \cos(Bx) \)[/tex] is given by the absolute value of the coefficient of the cosine function. In this case, the coefficient [tex]\( A \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
So, the amplitude is:
[tex]\[ \boxed{\frac{2}{3}} \][/tex]
### Period
The period of a cosine function [tex]\( y = A \cos(Bx) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex] of [tex]\( x \)[/tex] inside the cosine function. The formula for the period [tex]\( T \)[/tex] of [tex]\( y = \cos(Bx) \)[/tex] is:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
Here, the coefficient [tex]\( B \)[/tex] is [tex]\( 5 \)[/tex]. Thus, the period is:
[tex]\[ T = \frac{2\pi}{5} \][/tex]
So, the period is:
[tex]\[ \boxed{\frac{2\pi}{5}} \][/tex]
In summary, for the function [tex]\( y = \frac{2}{3} \cos 5x \)[/tex]:
- The amplitude is [tex]\( \boxed{\frac{2}{3}} \)[/tex]
- The period is [tex]\( \boxed{\frac{2\pi}{5}} \)[/tex]
