Answer
$1.119AU$ and $0.427AU$
Work Step by Step
Step 1. Use the figure given in the Exercise to identity the quantities: distance between the Earth (E) and the Sun (S) $ES=1 AU$, distance between the Sun (S) and Venus (V) $SV=0.723 AU$, elongation angle $\alpha=39.4^\circ$
Step 2. In the triangle of ESV, use the law of sines, we have $\frac{sin\angle EVS}{1}=\frac{sin\alpha}{0.723}$ which gives $sin\angle EVS=\frac{sin39.4^\circ}{0.723}=0.8770$ and $\angle EVS=sin^{-1}0.878=61.39^\circ, 118.61^\circ$
Step 3. Based on the results above, there are two possible distances between the Earth and Venus, one corresponds to an angle of $\angle ESV1=180-39.4-61.39=79.21^\circ$, another corresponds to an angle of $\angle ESV2=180-39.4-118.61=21.99^\circ$
Step 4. In the triangle of EVS, use the Law of Sines for the above two cases: $\frac{EV1}{sin79.21^\circ,}=\frac{0.723}{sin39.4^\circ}$ which gives $EV1\approx1.119AU$, and $\frac{EV2}{sin21.99^\circ,}=\frac{0.723}{sin39.4^\circ}$ which gives $EV2\approx0.427AU$
