Expert: Sherman D. - 3/16/2009

Question
find the exact value of tan pi/12 I do not know how to solve this problem so can you please solve this problem for me

Answer
the value pi is another way of saying 180 in radians form.

tan(pi/12) = tan(180/12) = tan(15)

tan(15) = tan(30/2)

tan(A/2) = sqrt(((1 - cos(A))/(1 + cos(A)))

tan(30/2) = sqrt(((1 - cos(30))/(1 + cos(30)))

tan(15) = sqrt(((1 - (sqrt(3)/2))/(1 + (sqrt(3)/2)))

tan(15) = sqrt((((2 - sqrt(3))/2)/((2 + sqrt(3))/2))

tan(15) = sqrt((2 - sqrt(3))/(2 + sqrt(3))

multiply top and bottom by (2 - sqrt(3))

tan(15) = sqrt(((2 - sqrt(3))^2 / (4 - 3))

tan(15) = 2 - sqrt(3)

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another way of doing this, which is also probably the way you wanted me to do this.

if you work it out like this

pi/4 + pi/x = pi/12
pi/4 - pi/12 = -ypi/x
(3pi - pi)/12 = -ypi/x
2pi/12
pi/6 = -ypi/x

i used pi/4 because when you do tan(pi/4) it gives you 1.

so

tan(pi/12) = tan((pi/4) - (pi/6)), or tan(45 - 30), but if you'd like, i'll work it out like tan((pi/4) - (pi/6))

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

tan(pi/12) = (1 - (sqrt(3)/3))/(1 + (sqrt(3)/3))

tan(pi/12) = ((3 - sqrt(3))/3)/((3 + sqrt(3))/3)

tan(pi/12) = (3 - sqrt(3))/(3 + sqrt(3))

multiply top and bottom by (3 - sqrt(3))

tan(pi/12) = ((3 - sqrt(3))(3 - sqrt(3))/(9 - 3)

tan(pi/12) = (9 - 3sqrt(3) - 3sqrt(3) + 3)/6

tan(pi/12) = (12 - 6sqrt(3))/6

tan(pi/12) = 2 - sqrt(3)

either way you still get the same answer.

info found at

http://home.alltel.net/okrebs/page103.html

and

http://math2.org/math/trig/identities.htm