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min max cos in triangle with sides 1-x (III Soros Olympiad 1996-97 R3 9.8)

Source:

May 31, 2024

trigonometryalgebrageometric inequality

Problem Statement

The two sides of the triangle are equal to

1

and

x

, and

x \ge 1

. The values

a

and

b

are the largest and smallest angles of this triangle, respectively. Find the greatest value of

\cos a

and the smallest value of

\cos b