Wondering about ancient methods of estimating the relative planetary distances

Question

Regarding https://physics.stackexchange.com/a/156155/349202 and estimating with geometry the relative distance of Venus to the sun prior to the Venus transit of 1769, I follow the logic of @viktor-blasjo. The relevant quote is:

Distance from Venus (or Mercury) to the sun: continually measure the angle VES; when it is at a maximum the angle EVS will be right, and we know ES so we can find VS. (Since Venus and Mercury move much faster than the earth, the earth can be considered stationary for the purposes of this demonstration.)

A diagram of my understanding of the geometry here is:

However, I also read that Venus's maximum elongation (angular separation) from the Sun is 47°. Based on the above method, wouldn't that put Venus farther from the sun than Earth, since the tangent of 47° is greater than 1?

I'm curious how this is accounted for in the above method.

Answer 1

Your diagram of Venus's maximum elongation is wrong.
Actually it looks like this:

The right angle is at Venus, not at the sun.
From the right triangle you get $$\sin 47° = \frac{r_\text{Venus}}{r_\text{Earth}}$$

Answer 2

Your picture is simply wrong. ES is the hypotenuse. It is $\angle$EVS that is $\perp$. You should be the sine function rather than the tangent function. This then guarantees that VS < ES.