Question

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Answer 1

`\displaystyle 2`

Explanation:

`\displaystyle y = Acos(Bx - C) + D`

When working with a trigonometric equation like this, always remember the information below:

`\displaystyle Vertical\:Shift \hookrightarrow D \\ Horisontal\:[Phase]\:Shift \hookrightarrow (C)/(B) \\ Wavelength\:[Period] \hookrightarrow (2)/(B)\pi \\ Amplitude \hookrightarrow |A|`

So, the first procedure is to find the period of this graph, and when calculated, you should arrive at this:

`\displaystyle \boxed{(1)/(2)} = (2)/(4\pi)\pi`

You will then plug this into the frequency formula,
`\displaystyle T^(-1) = F.` Look below:

`\displaystyle (1)/(2)^(-1) = F \\ \\ \boxed{2 = F}`

Therefore, the frequency of motion is two hertz.

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Answer 2

There you go:

`y = 4 \cos(4\pi \: t + (\pi)/(4) )`

`y = a\cos(w\: t + \alpha )`

`w = 4\pi`

`2\pi \: f = 4\pi`

`f = 2`