In a distance star system a planet is 9 AU (astronomical unit) away from the sun, and has a circular orbit with period 12 earth years. The planet has a moon distanced 0.5 AU away, which circles the planet in a circular orbit
also with period 12 earth years. Suppose the orbital planes of the planet around the sun and the moon around
the planet coincide, and their motions are both counterclockwise. At midnight March 1 2021, the positions of
the sun, planet, and moon are observed to be (0, 0), (9, 0), (9, 0.5) on their common obital plane.
(a) Determine the position (xP (t), yP (t)) of the planet, t years after midnight March 1 2021.
(b) Determine the position (xM(t), yM(t)) of the moon, t years after midnight March 1 2021. Express your
answer in the form
xM(t) = A cos(kt + α) + xP (t)
yM(t) = A sin(kt + α) + yP (t)
where A, k > 0 and α ∈ (−π, π].
(c) Simplify your expressions for xM(t) and yM(t) to the form
xM(t) = B cos(kt + β)
yM(t) = C sin(kt + γ)
where B, C, k > 0 and β, γ ∈ (−π, π], leave your answers in exact form.
(d) Give TWO reasons why an attempt as in the previous parts to express our own moon’s position relative
to our sun as a simple trigonometric function would NOT work