A marathon swim follows a triangular course marked with three buoys, A, B, and C. The distance from buoy A to B is 400 meters, B to C is 500 meters, and C to A is 600 meters. What is the largest angle the swimmers must turn between the buoys?

To find the largest angle in the triangular course formed by the buoys A, B, and C, we can use the **cosine rule**. The cosine rule states that for any triangle with sides \( a \), \( b \), and \( c \), and opposite angles \( A \), \( B \), and \( C \) respectively:

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

In this case:

- \( a = 500 \) meters (distance from B to C)
- \( b = 600 \) meters (distance from C to A)
- \( c = 400 \) meters (distance from A to B)

We are interested in finding the angle opposite to the longest side, which is angle \( \theta \) opposite side \( c = 600 \) meters (distance from C to A).

So, we calculate \( \cos(\theta) \) as follows:

\[
\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab} = \frac{500^2 + 400^2 - 600^2}{2 \times 500 \times 400}
\]

Let's compute that:

\[
500^2 = 250000
\]
\[
400^2 = 160000
\]
\[
600^2 = 360000
\]
\[
\cos(\theta) = \frac{250000 + 160000 - 360000}{400000} = \frac{50000}{400000} = 0.125
\]

Finally, to find the angle \( \theta \), take the inverse cosine (arccos) of 0.125:

\[
\theta = \arccos(0.125)
\]

Calculating \( \theta \):

\[
\theta \approx 82.82^\circ
\]

This is the largest angle in the triangle since it is opposite the longest side (600 meters).

So, the largest angle the swimmers must turn is approximately **82.82 degrees**.

Would you like further details on this solution, or do you have any additional questions?

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**Related Questions:**
1. What is the smallest angle in this triangle?
2. How would the course change if one of the distances were increased?
3. Can the cosine rule be applied to other geometric shapes?
4. What if the course was quadrilateral instead of triangular?
5. How do the distances between buoys affect the overall difficulty of the swim?
6. What is the perimeter of the triangular course?
7. Can the angle calculation be done using the sine rule instead?
8. How would a change in one of the buoy positions affect the angles?

**Tip:** When dealing with triangles, always check which angle corresponds to the longest side, as this will be the largest angle.

Learn how to calculate the largest angle in a triangular swim course using the cosine rule and inverse trigonometric functions. In this problem, calculate the angle opposite the longest side in a triangle formed by buoys A, B, and C with distances 400m, 500m, and 600m respectively. Suitable for high school students studying trigonometry.

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