Answer: Choice C) 1/4
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Work Shown:
Let's rearrange terms a bit to say the following:
[tex]y = 9\cos\left(\frac{\theta}{4}+\frac{3\pi}{2}\right)+4\\\\y = 9\cos\left(\frac{1}{4}\theta+\frac{3\pi}{2}\right)+4\\\\y = 9\cos\left(\frac{1}{4}\left(\theta+6\pi\right)\right)+4\\\\y = 9\cos\left(\frac{1}{4}\left(\theta-(-6\pi)\right)\right)+4\\\\[/tex]
The last equation is in the form [tex]y = A\cos\left(B\left(\theta-C\right)\right)+D\\\\[/tex]
where,
- |A| = amplitude
- B handles the period. Specifically T = 2pi/B, where T is the period
- C handles the phase shift, aka horizontal shifting
- D determines the midline and the vertical shifting
We only need to worry about the value of B.
In this case, B = 1/4
So,
[tex]T = \frac{2\pi}{B}\\\\T = 2\pi \div B\\\\T = 2\pi \div \frac{1}{4}\\\\T = 2\pi \times \frac{4}{1}\\\\T = 8\pi\\\\[/tex]
The period is 8pi. Every 8pi units, a full cycle is completed.
However, we're not going from 0 to 8pi, but instead from 0 to 2pi.
The given interval is 2pi units wide. This is (2pi)/(8pi) = 1/4 of a full cycle. This is why choice C is the answer.
