Question

As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction F of the lunar disc that is lit. When the angle between the sun, the earth, and the moon is θ(0<=θ<=360∘), then
F=1/2(1−cosθ)
Determine the angles θ, that correspond to the following phases:

(a) F = 0 (new moon)
(b) F = 0.25 (a crescent moon)
(c) F = 0.5 (first or last quarter)
(d) F = 1 (full moon)

Answer 1

Answer:

a)   0∘ and  360∘

b)   300∘ and  60∘

c)   90∘ and  270∘

d)   180∘

Step-by-step explanation:

θ(0<=θ<=360∘)

then

a)

F=1/2(1−cosθ)

Putting F= 0

we get

0 = 1/2 (1−cosθ)

(1−cosθ) = 0

cosθ = 1

θ = $cos^-^1$ (1)

 = 0∘ and  360∘

b) If F = 0.25

then putting values

1/4 = 1/2 (1−cosθ)

(1−cosθ) = 1/2

cosθ = 1/2

θ = $cos^-^1$ (1/2)

 = 300∘ and  60∘

c) If F = 0.5

then  putting values we get

1/2 = 1/2 (1−cosθ)

(1−cosθ) = 1

cosθ = 0

θ = $cos^-^1$ (0)

 = 90∘ and  270∘

d) If F = 1

then  putting values we get

1 = 1/2 (1−cosθ)

(1−cosθ) = 2

cosθ = -1

θ = $cos^-^1$ (-1)

 = 180∘

So after these calculation we determine the vales of angles that are given uper.