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

1 Answer
Aug 21, 2015

frac{sqrt 6 - sqrt 2}{4}

frac{- sqrt 2 - sqrt 6}{4}

-2 - sqrt 3

Explanation:

a = sin 15º = sin (60 - 45) = sin 60 cos 45 - cos 60 sin 45

a = sqrt 3 / 2 sqrt 2 / 2 - 1/2 sqrt 2 / 2 = frac{sqrt 6 - sqrt 2}{4}

b = cos frac{11 pi}{12} = - cos frac{pi}{12} = - cos (60º - 45º)

b = - cos 60 cos 45 - sin 60 sin 45

b = - 1/2 sqrt 2/2 - sqrt 3/2 sqrt 2 /2

b = frac{- sqrt 2 - sqrt 6}{4}

c = tan 105º = frac{sin 75}{- cos 75} = frac{sin 30 cos 45 + cos 30 sin 45}{-cos 30 cos 45 + sin 30 sin 45}

c = frac{sin 30 + cos 30}{sin 30 - cos 30} = frac{1/2 + sqrt 3 / 2}{1/2 - sqrt 3 / 2} = frac{1 + sqrt 3}{1 - sqrt 3} = frac{(1 + sqrt 3)^2}{1 - 3}