Find the Exact Value cot(10)

Problem

cot(10^∘)

Solution

1. Identify the expression as the cotangent of 10 degrees, which is a trigonometric function representing the ratio of the adjacent side to the opposite side in a right triangle, or cos(10^∘)/sin(10^∘)

2. Determine if 10^∘ is a standard angle. Since 10^∘ is not one of the common angles (0^∘,30^∘,45^∘,60^∘,90^∘, its exact value is typically expressed in terms of radicals involving nested square roots or by using the triple-angle formula for 30^∘

3. Relate to the triple-angle identity tan(3*θ)=(3*tan(θ)−tan^3(θ))/(1−3*tan^2(θ)) Setting θ=10^∘ gives tan(30^∘)=(3*tan(10^∘)−tan^3(10^∘))/(1−3*tan^2(10^∘))

4. Substitute the known value tan(30^∘)=1/√(,3) into the identity to form the cubic equation 1/√(,3)=(3*tan(10^∘)−tan^3(10^∘))/(1−3*tan^2(10^∘))

5. Conclude that while the value can be represented as a root of a cubic polynomial, the simplest "exact value" form is the trigonometric expression itself or its equivalent in terms of sine and cosine.

Final Answer

cot(10^∘)=cot(10^∘)

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