Find the exact value of the expression: tan pi/12?

Answer 1

tan x/2 = (1 - cos x) / sinx
tan pi/12 = (1 - cos pi/6) / sin pi/6) = (1 - sqrt(3)/2) / (1/2) = 2 - sqrt(3)

Answer 2

OK, the others here say that tan pi/12 = 2 - sqrt(3), but why?

Use the half-angle formulas:

cos (pi/12) = sqrt((cos(pi/6)+1)/2)
= sqrt((sqrt(3)/2+1)/2

sin (pi/12) = sqrt(-cos(pi/6)+1)/2
= sqrt(-sqrt(3)/2+1)/2


So tan (pi/12) = sin(pi/12)/cos(pi/12)
= (sqrt(-sqrt(3)/2+1)/2 ) / (sqrt(sqrt(3)/2+1)/2)
= sqrt ((-sqrt(3)+2)/2) / ((sqrt(3)+2)/2))
= sqrt ((-sqrt(3)+2)) / ((sqrt(3)+2)))

Rationalize the denominator inside the outer sqrt:

= sqrt((-sqrt(3)/2+1)^2) /(4-3))
= sqrt((-sqrt(3)+2)^2
= 2 - sqrt(3),

since the square root of the square of a positive quantity is that quantity back again.

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

Answer 3

tan Л/12= tan (Л/6)/2=√((1-cos Л/6)/(1+cos Л/6)
cos Л/6=√3 /2
tan Л/12=√((1-√3 /2)/(1+√3 /2))

Answer 4

tan(π/12) = √{[1 - cos(π/6)][1 + cos(π/6)]}
= √{(1 - √3/2)(1 + √3/2)} = (1 - √3/2)/√(1 - 3/4)
= (1 - √3/2)(1/2) = 2(1 - √3/2) = 2 - √3

Answer 5

2 - Sqrt(3)