Find the exact value of sin(pi/12) , cos (11pi/12) , and tan (7pi/12)?

1 Answer
Sep 25, 2015

Find sin (pi/12); cos (pi/12) and tan ((7pi)/12)

Explanation:

Call sin pi/12 = sin t. Use trig identity: cos 2a = 1 - sin^2 a
cos 2pi/12 = cos pi/6 = sqrt3/2 = 1 - sin^2 t.
sin^2 t = 1 - sqrt3/2 = (2 - sqrt3)/4
sin t = +- sqrt(2 - sqrt3)/2. Since pi/12 has its sin positive, then
sin (pi/12) = sin t = sqrt(2 - sqrt3)/2

Call cos (pi/2) = cos t
cos ((2pi)/12) = sqrt3/2 = 2cos^2 t - 1
cos^2 t = (2 + sqrt3)/4
cos t = +- sqrt(2 + sqrt3)/2. Since cos (pi/12) is positive, then
cos (pi/12) = cos t = sqrt(2 + sqrt3)/2

tan ((7pi)/12) = tan (pi/12 + pi) = tan (pi/12) = sin/(cos) =
= (sqrt(2 - sqrt3))/(sqrt(2 + sqrt3))