Find the Exact Value sec(pi/7)

Problem

sec(π/7)

Solution

1. Identify the angle π/7 as a non-standard angle in trigonometry. Unlike angles such as π/6 π/4 or π/3 the value of sec(π/7) cannot be expressed using simple square roots or basic radicals.

2. Relate the secant function to the cosine function using the reciprocal identity.

sec(π/7)=1/cos(π/7)

3. Determine if an algebraic form exists. The value cos(π/7) is a root of a cubic equation derived from the heptagonal triangle, specifically 8*x3−4*x2−4*x+1=0 However, expressing this root in radicals requires complex numbers (the "casus irreducibilis").

4. Conclude that the exact value is most simply represented in its trigonometric form, as any radical expression would be significantly more complex and less practical.

Final Answer

sec(π/7)=sec(π/7)

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