Find the Exact Value tan(pi/9)

Problem

tan(π/9)

Solution

1. Identify the angle π/9 in degrees. Since π radians equals 180 the angle is 20

2. Recognize that 20 is not a standard angle with a simple radical form for its tangent.

3. Apply the triple-angle formula for tangent, which relates tan(3*θ) to tan(θ)

tan(3*θ)=(3*tan(θ)−tan3(θ))/(1−3*tan2(θ))

4. Substitute θ=π/9 into the formula. Since 3*θ=π/3 we know tan(π/3)=√(,3)

√(,3)=(3*tan(π/9)−tan3(π/9))/(1−3*tan2(π/9))

5. Define x=tan(π/9) to form a cubic equation.

√(,3)*(1−3*x2)=3*x−x3

6. Rearrange the equation into standard cubic form.

x3−3√(,3)*x2−3*x+√(,3)=0

7. Conclude that while the value can be expressed as a root of this cubic equation, it does not simplify into a basic combination of square roots. The exact value is conventionally left as tan(π/9)

Final Answer

tan(π/9)=tan(π/9)

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