Find the Exact Value tan(10)

Problem

tan(10)

Solution

1. Identify the task as finding the exact trigonometric value of tan(10)

2. Recognize that 10 is not a standard angle (like 30 45 or 60 and cannot be expressed simply using standard radical forms without complex nested radicals or imaginary numbers.

3. Relate the angle to a known identity, such as the triple angle formula for tangent: tan(3*θ)=(3*tan(θ)−tan3(θ))/(1−3*tan2(θ))

4. Substitute θ=10 into the formula to get tan(30)=(3*tan(10)−tan3(10))/(1−3*tan2(10))

5. Set x=tan(10) and use the known value tan(30)=1/√(,3) to form the cubic equation 1/√(,3)=(3*x−x3)/(1−3*x2)

6. Solve the cubic equation √(,3)*x3−3*x2−3√(,3)*x+1=0 using the cubic formula or trigonometric methods, which yields the exact form involving cube roots of complex numbers.

Final Answer

tan(10)=(i*(1+i√(,3))√(3,1+i√(,3))−i*(1−i√(,3))√(3,1−i√(,3)))/(√(3,1+i√(,3))+√(3,1−i√(,3)))

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