Calculus/problem

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Expert: - 3/10/2009


Question
I have a problem that i can not solve it says use a doule or half angle formula to find the exact value of: tan pi/12

Answer
The tan(π/12) can be found with the formula
tan(Θ/2) = (1-cos(Θ))/sin(Θ).

Note that here, Θ = π/6, so Θ/2 = π/12.

Putting π/12 into the equation, we get
tan(π/12) = (1-cos(π/6))/sin(π/6).

Now cosē(Θ/2) = (1+cos(Θ))/2 and sinē(Θ/2) = (1-cos(Θ))/2.

If we let Θ = π/3 this time, we get
cosē(π/6) = (1+cos(π/3))/2 and sinē(π/6) = (1-cos(π/3))/2.

Putting these back into the formula
tan(π/12) = (1-cos(π/6))/sin(π/6), we get
tan(π/12) = (1 - sqrt[{1+cos(π/3)}/2)/sqrt[{(1-cos(π/3)}/2].

Since the cos(π/3) is known  to be √3/2,
that can be put in and the tan(π/12) known exactly.

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