Find the Exact Value cos(pi/9)

Problem

cos(π/9)

Solution

1. Identify the angle in degrees to understand the context. Since π radians is 180 the angle is 180/9=20

2. Recognize that 20 is not a standard angle with a simple radical form. Finding an exact value requires solving a cubic equation derived from the triple-angle formula for cosine.

3. Apply the triple-angle formula cos(3*θ)=4*cos3(θ)−3*cos(θ)

4. Substitute θ=π/9 into the formula.

cos(3⋅π/9)=4*cos3(π/9)−3*cos(π/9)

5. Simplify the left side using the known value cos(π/3)=1/2

1/2=4*cos3(π/9)−3*cos(π/9)

6. Formulate the cubic equation by letting x=cos(π/9)

8*x3−6*x−1=0

7. Conclude that because this cubic equation has no rational roots and its solutions involve the cube roots of complex numbers (the "casus irreducibilis"), the most simplified exact form is the expression cos(π/9) itself or its representation as a root of the cubic equation.

Final Answer

cos(π/9)=cos(π/9)

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