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)=(cot^3(θ)−3*cot(θ))/(3*cot^2(θ)−1) where θ=20^∘ and 3*θ=60^∘

cot(60^∘)=(cot^3(20^∘)−3*cot(20^∘))/(3*cot^2(20^∘)−1)

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

1/√(,3)=(cot^3(20^∘)−3*cot(20^∘))/(3*cot^2(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)

Want more problems? Check here!