Without using a calculator find the exact value of Cos 𝜋 /12

Answer 1

cosπ/12 = √(1 + cosπ/6)/2 = 1/2√(2 + √3)

Answer 2

cos(pi/12) = cos(15) = about 0.9659 by a calculator or about .97

for an exact value of cos15 use the half angle formula

pi/3 = 60

pi/6 = 30

cos30 = sqr3/2

cos15 = cos(30/2)

the half angle formula is:

cos(x/2) = + or - sqr((1+cosx)/2)

cos15 = + or -sqr[(1+cos30)/2 = + or - sqr[1+sqr3/2)/2] = + or - sqr(1/2 +sqr3/2)

ignore the negative square root

cos15 = exactly +sqr(1/2 + (sqr3)/4]

sqr3/4 = 1.732/4 = 0.433

sqr3/4 +.5 = 0.933

sqr.933 = .9659 the same as the calculator answer to 4 decimal places