Answer
$$r \approx 1,076 \space mi$$
Work Step by Step
1. Dividing the given angle by 2, we will get the angle $\angle A$ of a triangle with $opp. = r$ and $adj. = 236,900 \space mi + r$ (r = radius of the moon), which is a Right Triangle
$$\angle A = \frac{0.518 ^o}{2} = 0.259 ^o$$
Tangent definition: $$tan \space \theta = \frac{opp.}{adj.} \longrightarrow opp. = tan \space \theta \times adj.$$
$$r = (tan \space 0.259^o)(236,900 \space mi + r) $$ $$r = (tan \space 0.259^o)(236,900 \space mi) + r \space tan \space 0.259 ^o $$ $$r - r \space tan \space 0.259 ^o = (tan \space 0.259^o)(236,900 \space mi)$$ $$r(1 - tan \space 0.259^o )= (tan \space 0.259^o)(236,900 \space mi)$$ $$r= \frac{(tan \space 0.259^o)(236,900 \space mi)}{(1 - tan \space 0.259^o )}$$ $$r \approx 1,076 \space mi$$
