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This problem involves two airplanes that leave an airport at the same time and travel at different distances during an hour, with the goal of finding the angle between their flight paths. The information provided is:

- The two airplanes are 200 km apart after 1 hour.
- One airplane has traveled 166 km, while the other has traveled 286 km.

We can use the **cosine rule** to solve for the angle \( \theta \) between their flight paths. The cosine rule is given by:

\[
\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Where:
- \( a = 166 \, \text{km} \) (distance traveled by one plane),
- \( b = 286 \, \text{km} \) (distance traveled by the other plane),
- \( c = 200 \, \text{km} \) (distance between the two planes after 1 hour),
- \( \theta \) is the angle we need to find.

Let's compute the value of \( \theta \).The angle \( \theta \) between the flight paths of the two airplanes is approximately **43.1 degrees**.

Would you like more details about the solution or explanation of the cosine rule?

Here are five related questions for further exploration:
1. How does the cosine rule apply to non-right triangles?
2. What is the difference between the sine rule and the cosine rule?
3. How would the problem change if one of the airplanes traveled in a straight line for longer than an hour?
4. Can the Law of Cosines be used to find distances in three-dimensional space?
5. What is the effect of changing the distances between the airplanes on the calculated angle?

**Tip:** In problems involving non-right triangles, the Law of Cosines is a powerful tool for finding angles or side lengths when three sides or two sides and an angle are known.

Two airplanes leave an airport at the same time. An hour later, the planes are 200 km apart. If one plane has traveled 166 km and the other has traveled 286 km during the hour, find the angle θ between their flight paths.

Learn how to use the Law of Cosines to calculate the angle between two airplanes' flight paths. One airplane travels 166 km, the other 286 km, and they are 200 km apart after an hour. Detailed steps for solving the angle are provided.

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