Aircraft Distance
1. **Problem statement:** We need to find the distance of an aircraft from a point K on the horizontal ground given the angle of elevation.
2. **Understanding the problem:** The angle of elevation is the angle between the horizontal ground and the line of sight to the aircraft.
3. **Formula used:** If $h$ is the height of the aircraft above the ground and $d$ is the horizontal distance from point K to the point directly below the aircraft, then the distance $D$ from K to the aircraft is given by the hypotenuse of the right triangle formed:
$$D = \frac{h}{\sin(\theta)}$$
where $\theta$ is the angle of elevation.
4. **Explanation:** The sine of the angle of elevation is the ratio of the opposite side (height $h$) to the hypotenuse (distance $D$). Rearranging gives the formula above.
5. **Intermediate work:** If the height $h$ and angle $\theta$ are known, substitute them into the formula to find $D$.
6. **Example:** If the aircraft is 500 meters high and the angle of elevation is 30 degrees,
$$D = \frac{500}{\sin(30^\circ)} = \frac{500}{0.5} = 1000 \text{ meters}$$
So, the aircraft is 1000 meters from point K.
**Final answer:** The distance from point K to the aircraft is $$D = \frac{h}{\sin(\theta)}$$ where $h$ is the height and $\theta$ is the angle of elevation.