Question

Which equation below represents a cosine function with amplitude 1 , period \( \frac{2 \pi}{5} \), equation of center line \( y=-4 \). and horizontal phase shift \( \frac{t}{3} \) right? Selict ane: \

A) y = cos(2π/5(t-t/3))-4
B) y = cos(3/5(t-t/3))-4
C) y= cos(5/3(t-t/3))-4
D) y = cos(2π/3(t-t/3))-4

Answer 1

The correct equation representing a cosine function with an amplitude of 1, a period of
`\( (2 \pi)/(5) \)`, a center line equation of y = -4, and a horizontal phase shift
`\( (t)/(3) \)` to the right is Option D:
`\( y = \cos\left((2\pi)/(5)\left(t-(t)/(3)\right)\right) - 4 \)`.

Step-by-step explanation:

The general form of a cosine function is
`\( y = A \cos(B(t - C)) + D \)`, where:

- A is the amplitude,

- B affects the period
`(\( T = (2\pi)/(B) \))`,

- C indicates the horizontal phase shift, and

- D is the vertical shift.

Given:

- Amplitude A = 1,

- Period
`\( T = (2\pi)/(5) \)`,

- Center line equation y = -4 , which implies a vertical shift D = -4,

- Horizontal phase shift
`\( C = (t)/(3) \)` right, which means a positive C value.

From the period formula,
`\( T = (2\pi)/(B) \)`, we solve for B:

`\[ (2\pi)/(5) = (2\pi)/(B) \implies B = (2\pi)/((2\pi)/(5)) = 5 \]`

The correct equation becomes
`\( y = \cos\left(5\left(t-(t)/(3)\right)\right) - 4 \)`. Simplifying the inner term yields
`\( y = \cos\left((10t)/(3)\right) - 4 \)`, which matches the given equation structure in Option D. Therefore, Option D is the accurate representation of the cosine function meeting all the specified criteria.