Find the Exact Value cot(pi/9)

Problem

cot(π/9)

Solution

1. Identify the angle in degrees to better understand the trigonometric ratio.

π/9* radians=20

2. Recognize that 20 is not a standard reference angle (like 30 45 or 60.

3. Determine if an exact radical form is required. The exact value of cot(20) cannot be expressed using simple square roots; it involves cube roots of complex numbers or the use of the triple-angle formula for cotangent.

4. Apply the triple-angle identity for cotangent, cot(3*θ)=(cot3(θ)−3*cot(θ))/(3*cot2(θ)−1) where θ=20 and 3*θ=60

cot(60)=(cot3(20)−3*cot(20))/(3*cot2(20)−1)

5. Substitute the known value cot(60)=1/√(,3)

1/√(,3)=(cot3(20)−3*cot(20))/(3*cot2(20)−1)

6. Conclude that since the value is requested as an "exact value" and it results from a cubic equation without a simpler radical form, the expression remains in its trigonometric form.

Final Answer

cot(π/9)=cot(π/9)

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