## By using the Law of cosines, we can find the distance between Earth and Venus. Let's denote the distance between the Sun and Venus as VS and the distance between the Sun and Earth as ES. We also know that the angle between the lines Sun-Earth and Sun-Venus is 30 degrees.

The Law of cosines states that in a triangle with sides a, b, and c, and the angle opposite to side c denoted as C, the equation is:

c^2 = a^2 + b^2 - 2abcos(C)

In our case, we want to find the length of the side VE, which is the distance between Earth and Venus. The equation becomes:

(VE)^2 = (VS)^2 + (ES)^2 - 2(VS)(ES)cos(30)

Substituting the given values, we have:

(VE)^2 = (1.1 * 10^8)^2 + (1.5 * 10^8)^2 - 2(1.1 * 10^8)(1.5 * 10^8)cos(30)

Simplifying the equation, we get:

(VE)^2 = 1.21 * 10^16 + 2.25 * 10^16 - 3.3 * 10^16 * (sqrt(3)/2)

(VE)^2 = 3.46 * 10^16 - 2.86 * 10^16 * sqrt(3)

Taking the square root of both sides, we find:

VE = sqrt(3.46 * 10^16 - 2.86 * 10^16 * sqrt(3))

By calculating this expression, we get two possible solutions:

VE = 2.0 * 10^8 km and VE = 0.5 * 10^8 km

These are the distances between Earth and Venus for the two possible positions of Venus given the angle and the known distances from the Sun.