Question

Use the half angle formula to determine the exact values of the sine, cosine, and tangent of the angle:
pi/8

Answer 1

cos 2x = cos² x - sin² x = cos² x - (1 - cos² x) = 2 cos² x - 1

<=>

cos [ 2 (pi/8) ] = 2 cos² [ pi/8 ] - 1

cos [ pi/4 ] = 2 cos² [ pi/8 ] - 1

cos² [ pi/8 ] = [ 1 + cos [ pi/4 ] ] / 2

cos² [ pi/8 ] = [ 1 + (sqr (2)/2) ] / 2

cos² [ pi/8 ] = [ 2 + sqr (2) ] / 4


As pi/8 belongs to [ 0 ; pi/2 ], its cosine is positive

<=>

cos [ pi/8 ] = sqr [ [ 2 + sqr (2) ] / 4 ]

cos [ pi/8 ] = sqr [ 2 + sqr (2) ] / 2 <=== answer

*******

Verification :

sqr [ 2 + sqr (2) ] / 2 = 0.9238795...

arccos 0.9238795 = 0.392699... radians = 3.1415926535897... / 8 = pi/8 ===> OK.