Find the Exact Value cot(pi/7)

Problem

cot(π/7)

Solution

1. Identify the expression as the cotangent of the angle π/7 radians.

2. Recognize that π/7 is not one of the standard angles (like π/6 π/4 or π/3 for which the trigonometric values are typically expressed in simple radicals.

3. Determine the exact value using the properties of the roots of unity or the heptagonal triangle. The value can be expressed as a root of a cubic equation or using a finite sum.

4. Express the value in its most common exact form using the relationship cot(θ)=cos(θ)/sin(θ)

5. Note that while numerical approximations exist, the exact value is often represented using the radical expression derived from the roots of the polynomial x6+x5+x4+x3+x2+x+1=0

Final Answer

cot(π/7)=√(,(1+cos((2*π)/7))/(1−cos((2*π)/7)))

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