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Find the amplitude and period of the function.

[tex]\[ y=\frac{2}{3} \cos 5x \][/tex]

Give the exact values, not decimal approximations.

Amplitude: [tex]\(\boxed{\frac{2}{3}}\)[/tex]

Period: [tex]\(\boxed{\frac{2\pi}{5}}\)[/tex]

Sagot :

GinnyAnsweridn
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]
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