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)−tan3(θ))/(1−3*tan2(θ)) Setting θ=10 gives tan(30)=(3*tan(10)−tan3(10))/(1−3*tan2(10))

4. Substitute the known value tan(30)=1/√(,3) into the identity to form the cubic equation 1/√(,3)=(3*tan(10)−tan3(10))/(1−3*tan2(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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