Find the value of tan(π/7)·tan(2π/7)·tan(3π/7).
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Step-by-Step Solution
Step 1: Define the angle and use trigonometric identities
Let's define θ=7π to simplify the expression. The problem then becomes finding the value of tan(θ)tan(2θ)tan(3θ). We know that 7θ=π, which implies 4θ=π−3θ.
Step 2: Apply tangent to both sides
Taking the tangent of both sides of the equation 4θ=π−3θ, we get tan(4θ)=tan(π−3θ). Since tan(π−x)=−tan(x), this simplifies to tan(4θ)=−tan(3θ).
Step 3: Expand using tangent multiple angle formulas
We use the multiple angle formulas for tangent: tan(2x)=1−tan2x2tanx, tan(3x)=1−3tan2x3tanx−tan3x, and tan(4x)=1−tan2(2x)2tan(2x). Substituting these into the equation tan(4θ)=−tan(3θ) gives the expanded form.
Step 4: Simplify the equation
Let t=tanθ. After cross-multiplication and simplification, we get a polynomial equation in terms of t. We can factor out t from the left side of the equation tan(4θ)=−tan(3θ) if tanθ=0. Since θ=π/7, tanθ=0. The simplified equation is tan6θ−21tan4θ+35tan2θ−7=0.
Step 5: Relate to roots of a polynomial
Let x=tan2θ. The equation becomes x3−21x2+35x−7=0. The roots of this cubic equation are tan2(π/7), tan2(2π/7), and tan2(3π/7).
Step 6: Apply Vieta's formulas
According to Vieta's formulas, for a cubic equation ax3+bx2+cx+d=0, the product of the roots is -d/a. In our case, the product of the roots tan2(π/7), tan2(2π/7), and tan2(3π/7) is −(−7)/1=7.
Step 7: Calculate the final product
Since π/7, 2π/7, and 3π/7 are all in the first quadrant, their tangent values are positive. Therefore, the product tan(π/7)⋅tan(2π/7)⋅tan(3π/7) must be positive. Taking the square root of the product of the squares of the tangents, we get 7.