$(a)$ The earth-moon distance is about $60$ earth radii. What will be the angular diameter of the earth (approximately in degrees) as seen from the moon?
$(b)$ The moon is seen to have an angular diameter of $(1/2)^{\circ}$ from the earth. What is its relative size compared to the earth?
$(c)$ From parallax measurement,the sun is found to be at a distance of about $400$ times the earth-moon distance. Estimate the ratio of sun-earth diameters.

(N/A) The angle $\theta$ subtended at distance $r$ by an arc of length $l$ is given by $\theta = l/r$.
Here,$l = R_e$ (radius of earth) and $r = 60 R_e$.
$\theta = R_e / (60 R_e) = 1/60 \text{ rad}$.
Converting to degrees: $\theta = (1/60) \times (180^{\circ}/\pi) = 3/\pi \approx 0.955^{\circ} \approx 1^{\circ}$.
The angular diameter of the earth is $2\theta = 2 \times 1^{\circ} = 2^{\circ}$.
$(b)$ The angular diameter of the moon is $\alpha_m = (1/2)^{\circ}$ and the angular diameter of the earth as seen from the moon is $\alpha_e = 2^{\circ}$.
Since the angular diameter is proportional to the physical diameter $D$ for a fixed distance,the ratio of diameters is $D_e / D_m = \alpha_e / \alpha_m = 2^{\circ} / (1/2)^{\circ} = 4$.
Thus,the earth is $4$ times larger than the moon.
$(c)$ Let $d_s$ and $d_m$ be the distances of the sun and moon from the earth,and $D_s$ and $D_m$ be their diameters.
Given $d_s = 400 d_m$.
Since both the sun and moon subtend the same angular diameter $\alpha$ from the earth,$\alpha = D_s / d_s = D_m / d_m$.
Therefore,$D_s / D_m = d_s / d_m = 400$.
The ratio of the sun's diameter to the earth's diameter is $(D_s / D_m) \times (D_m / D_e) = 400 \times (1/4) = 100$.