Question

Find the exact value of tan pi/12.

A.) 2 - sqrt3
B.) 2 + sqrt3
C.) 3 - sqrt2
D.) 3 + sqrt2

Answer 1

A.) 2 -√3

Explanation:

If you know a little bit about the tangent function, you know tan(π/4) = 1. Then the tangent of π/12 will definitely be less than 1. There is only one answer choice that has a value less than 1:

2 -√3

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Your calculator can help you choose the correct answer, too.

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If you want to figure this out, you can use either a half-angle formula or a difference of angles formula.

tan(α -β) = (tan(α) -tan(β))/(1 +tan(α)tan(β))

tan(π/12) = tan(π/3 -π/4) = (tan(π/3) -tan(π/4))/(1 +tan(π/3)tan(π/4))

= (√3 -1)/(1 +√3) = (√3 -1)²/((√3+1)(√3-1)) = (3 -2√3 +1)/(3 -1)

tan(π/12) = 2-√3

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Half-angle formula

tan(α/2) = (1 -cos(α))/sin(α)

tan(π/12) = (1 -cos(π/6))/sin(π/6) = (1 -(√3)/2)/(1/2)

tan(π/12) = 2-√3