find the minimum coaltitude which can directly receive a signal from a geostationary satellite.

Answer

To find the minimum coaltitude which can directly receive a signal from a geostationary satellite, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Problem**: We need to find the minimum angle of elevation (coaltitude) at which a signal can be received from a geostationary satellite. A geostationary satellite is located at a fixed position relative to the Earth's surface.

2. **Identify the Key Distances**:
- The radius of the Earth (R) is approximately 6400 km.
- The distance of the geostationary satellite from the center of the Earth is approximately 42,000 km.

3. **Visualize the Geometry**:
- Draw a diagram with the Earth at the center, the geostationary satellite in orbit, and a line representing the signal from the satellite to a point on the Earth's surface.
- The line from the center of the Earth to the satellite is the radius of the satellite's orbit, and the line from the center of the Earth to the point on the surface is the radius of the Earth.

4. **Set Up the Right Triangle**:
- The line from the satellite to the point on the Earth's surface forms a right triangle with the radius of the Earth and the radius of the satellite's orbit.
- The height (h) of the triangle is the radius of the Earth (6400 km), and the base (b) is the distance from the center of the Earth to the satellite (42,000 km).

5. **Use Trigonometry**:
- We can use the sine function to find the angle of elevation (θ) from the point on the Earth's surface to the satellite.
- The sine of the angle θ can be expressed as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{R}{d} \]
where \( R = 6400 \) km (radius of the Earth) and \( d = 42000 \) km (distance from the center of the Earth to the satellite).

6. **Calculate the Sine of the Angle**:
- Plugging in the values:
\[ \sin(\theta) = \frac{6400}{42000} \]
- Simplifying this gives:
\[ \sin(\theta) = \frac{64}{420} = \frac{16}{105} \]

7. **Find the Angle**:
- To find θ, we take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{16}{105}\right) \]

8. **Final Result**:
- The angle θ represents the minimum coaltitude required to receive a signal from the geostationary satellite.