**Given information:**
- The angle θ subtended by the two directions of observation at the moon is 2 degrees.
- The diameter of the Earth is 3.14 * 10^7 m.

**Approach:**
To solve this problem, we can use the concept of trigonometry and the properties of circles to find the distance of the moon from the Earth. We will consider a right triangle formed by the Earth's center, point A, point B, and the moon. Using this triangle, we can find the distance of the moon from the Earth.

**Solution:**

Let's consider the following diagram:
```
Earth (O)
/ \
/ \
/ \
/ \
A -------- B
\ /
\ /
\ /
\ /
Moon (M)
```

**Step 1: Finding the angle at the center of the Earth**
- Since the angle subtended by the two directions of observation at the moon is 2 degrees, the angle at the center of the Earth, O, is twice that, i.e., 4 degrees.
- The angle at the center of a circle is twice the angle at the circumference subtended by the same arc.
- Therefore, the angle at the center of the Earth, O, is 4 degrees.

**Step 2: Finding the length of the arc AB**
- The length of the arc AB can be calculated using the formula:
Length of arc = (θ/360) * 2πr, where θ is the angle in degrees and r is the radius of the circle.
- In this case, the radius of the Earth is half its diameter, so r = (3.14 * 10^7) / 2.
- Substituting the values, we get:
Length of arc AB = (4/360) * 2π * (3.14 * 10^7) / 2.

**Step 3: Finding the distance of the moon from the Earth**
- In the right triangle formed by the Earth's center, point A, point B, and the moon, we have the following:
- Angle AOB = 90 degrees (angle in a semicircle).
- Angle AOM = 2 degrees (given).
- We can use trigonometry to find the distance AM (distance of the moon from the Earth).
- The tangent of angle AOM is given by the ratio of the opposite side (AM) to the adjacent side (OM).
- tan(AOM) = AM / OM.
- Rearranging the equation, we get:
- AM = OM * tan(AOM).
- We know that the length of arc AB is equal to the length of the circumference of the Earth, i.e., 2πr.
- Therefore, the length of arc AB = 2πr = (4/360) * 2π * (3.14 * 10^7) / 2.
- The length of the arc AB is also equal to the difference between the distances AO and BO.
- Therefore, the length of arc AB = AO - BO.
- Since AO = OM + AM and BO = OM - AM, we can substitute these values in the equation above to get:
- (4/360) * 2π *