Question

Find the distance between the pair of points on the earth surface A(43°N,50°E) B(20°N,20°E) use the formula theta÷360 ×2π6400

Answer 1

To find the distance between two points on the Earth's surface given their geographical coordinates (latitude and longitude), we can use the formula you provided, which is based on the angular distance between the two points.

### Step 1: Convert Latitude and Longitude to Radians

First, we'll need to find the angular distance (θ) between points A(43°N, 50°E) and B(20°N, 20°E).

1. **Calculate the difference in latitude and longitude:**
- Latitude difference: \( |43° - 20°| = 23° \)
- Longitude difference: \( |50° - 20°| = 30° \)

2. **Place the differences in radians:**
- \(\Delta \text{lat} = 23° \times \frac{\pi}{180} \approx 0.401 \, \text{radians}\)
- \(\Delta \text{long} = 30° \times \frac{\pi}{180} \approx 0.524 \, \text{radians}\)

### Step 2: Use Spherical Coordinates

To find the angular distance between the two points, we need to apply the spherical distance formula:

\[
\Delta \sigma = \arccos(\sin(\text{lat1}) \sin(\text{lat2}) + \cos(\text{lat1}) \cos(\text{lat2}) \cos(\Delta \text{long}))
\]

Where:
- \(\text{lat1} = 43° \text{N} = 0.750 \, \text{radians}\)
- \(\text{lat2} = 20° \text{N} = 0.349 \, \text{radians}\)

3. **Calculate using the formula:**
\[
\Delta \sigma = \arccos(\sin(0.750) \sin(0.349) + \cos(0.750) \cos(0.349) \cos(0.524))
\]
- Calculate:
- \(\sin(0.750) ≈ 0.681\)
- \(\sin(0.349) ≈ 0.340\)
- \(\cos(0.750) ≈ 0.731\)
- \(\cos(0.349) ≈ 0.940\)
- \(\cos(0.524) ≈ 0.866\)

Substitute these values into the formula:
\[
\Delta \sigma = \arccos(0.681 \cdot 0.340 + 0.731 \cdot 0.940 \cdot 0.866)
\]
\[
= \arccos(0.231 + 0.613)
\]
\[
= \arccos(0.844) ≈ 0.150\, \text{radians}
\]
(approximately)

### Step 3: Calculate the Distance

4. **Use the distance formula:**
Now, use the formula to convert the angular distance (in radians) to distance.

\[
\text{Distance} = \frac{\theta}{360} \times 2\pi \times R
\]

Where R (the Earth's radius) is approximately 6400 km.

Using \(\theta\) in degrees:
\[
\theta ≈ 8.6° \ \text{(converting from radians)} = 0.150 * \frac{180}{\pi}
\]

Substituting into the equation:
\[
\text{Distance} = \frac{8.6}{360} \times 2\pi \times 6400 \approx |0.0239| \times 2\pi \times 6400
\]
Calculating that gives,
\[
\text{Distance} \approx 0.0239 \cdot 40212 \approx 962.8\, \text{km}
\]

So, the distance between point A and point B is approximately **962.8 km**.