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)−tan^3(θ))/(1−3*tan^2(θ))

4. Substitute θ=10^∘ into the formula to get tan(30^∘)=(3*tan(10^∘)−tan^3(10^∘))/(1−3*tan^2(10^∘))

5. Set x=tan(10^∘) and use the known value tan(30^∘)=1/√(,3) to form the cubic equation 1/√(,3)=(3*x−x^3)/(1−3*x^2)

6. Solve the cubic equation √(,3)*x^3−3*x^2−3√(,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)))

Want more problems? Check here!