Find the Exact Value cos((2pi)/7)

Problem

cos((2*π)/7)

Solution

1. Identify the expression as the cosine of an angle that is not a standard multiple of π/6 or π/4

2. Recognize that the value of cos((2*π)/7) is a root of a specific cubic polynomial derived from the cyclotomic equation z7−1=0

3. Relate the angle to the roots of the equation 8*x3+4*x2−4*x−1=0 where x=cos((2*π)/7)

4. Determine that while the value can be expressed using radicals through Cardano's formula, it involves the cube roots of complex numbers (casus irreducibilis).

5. Conclude that the simplest exact form is the trigonometric expression itself, as it cannot be simplified into real nested radicals using basic arithmetic.

Final Answer

cos((2*π)/7)=cos((2*π)/7)

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