Question

HELP!! What is the exact value of Tangent (pi/12)

A. StartFraction 1 Over 2 + StartRoot 3 EndRoot EndFraction
B. StartFraction 1 Over StartRoot 3 EndRoot EndFraction
C. StartFraction 3 Over StartRoot 3 EndRoot EndFraction
D. StartFraction 1 Over 2 minus StartRoot 3 EndRoot EndFraction

Answer 1

D: 1/2 - √3.

Explanation:

The Unit Circle is a circle with radius 1 that is used to determine the value of sine and cosine of any angle.

But that table do not directly give the exact value so to solve for its exact value, express π/12 as a sum or difference of two angles where the values of the six trigonometric functions are known.

tan π/12 = tangent (π/4 - π/6)

Using the the difference of angles identity we get,

tan (π/4 - π/6) = (1 - √3 / 3) / (1 + 1 x √3 / 3)

Multiply by 3,

tan (π/4 - π/6) = (3 - √3) / (3 + √3)

Multiply by 3 - √3

tangent (π/4 - π/6) = (3 - √3)(3 - √3) / (3 - √3)(3 + √3)

FOIL and simplify.

tangent (π/4 - π/6) = 9 - 6√3 + 3 / 9 -3

tangent (π/4 - π/6) = 1/2 - √3

Answer 2

2 - sqrt(3)

Explanation:

Split pi/12 into two angles where the values of the six trigonometric functions are known.

tan (pi/4 - pi/6)

Apply the difference of angles identity

`(tan(pi/4) - tan(pi/6))/(1 + tan(pi/4)tan(pi/6))`

tan(pi/4) = 1 , tan(pi/6) = (sqroot3)/3

Plug in and Simplify

`(1-(√(3) )/(3) )/(1+1(√(3) )/(3) )`

`((3-√(3) )/(3) )/((3+√(3) )/(3) )`

`(3-√(3) )/(3) *(3)/(3+√(3) )`

`(3-√(3) )/(3+√(3))` Need to multiply this by
`(3+√(3) )/(3+√(3) )`

Expand and simplify numerator:
`(6)/((3+√(3) )^(2) )`

Expand and simplify denominator:
`(6)/(12+6√(3))`

Cancel the common factor:
`(1)/(2+√(3))`