The sides of the triangle are 6, 7, 8. Find the cosine of the angle opposite the larger side.

Take advantage of the fact that in any triangle the largest of the angles lies opposite the largest of the sides of this triangle.

In the wording of the condition for this task, it is reported that the lengths of the sides of this triangle are 6, 7, and 8.

Therefore, the length of the longest side is 8.

Let’s denote the value of the angle opposite the side of length 8 through x.

Using the cosine theorem, we can write the following equation:

6 ^ 2 + 7 ^ 2 – 2 * 6 * 7 * cosx = 8 ^ 2,

solving which, we get:

36 + 49 – 84 * cosx = 64;

85 – 84 * cosx = 64;

84 * cosx = 85 – 64;

84 * cosx = 21;

cosx = 21/84 = 1/4;

x = arccos (1/4).

Answer: arccos (1/4).