The sides of the triangle are 5, 7, and 8. Find the angle opposite the middle side.

For any triangle, the statement is true: the square of the side is equal to the sum of the squares of the other two sides minus the double product of the other two sides by the cosine of the angle between them: c ^ 2 = a ^ 2 + b ^ 2-2 * a * b * cosC, where cosC is the cosine corner opposite side c. Hence cosC = (a ^ 2 + b ^ 2-c ^ 2) / 2 * a * b. In our case, angle C lies against the side equal to 7.cosC = (5 ^ 2 + 8 ^ 2-7 ^ 2) / 2 * 5 * 8 = (25 + 64-49) / 10 * 8 = 40/80 = 1/2. Knowing the cosine of the angle, we can find its degree measure: C = arccos0.5 = 60 degrees.