Evaluate cos^2(pi/12).

Question

a)1/4
b)(2+sqrt3)/4**
c)3/2
d)(2-sqrt3)/4
Please help :(

Answer 1

recall cos(2A) = 2cos^2 A - 1
cos^2 A = (cos(2A) + 1)/2
now π/12 = 15°
so we can consider 2A to be 30° and A = 15°

cos^2 (π/12) = (cos π/6 + 1)/2 = (cos 30° + 1)/2
= ...

yup, you are correct

Answer 2

π rad = 180°

π / 12 = 180° / 12 = 15°

15° = 45° - 30°

cos ( A - B ) = sin A ∙ sin B + cos A ∙ cos B

sin 45° = cos 45° = √2 / 2 = 1 / √2

cos ( 45° - 30° ) = cos 15° = sin 45° ∙ sin 30° + cos 45° ∙ cos 30° =

( 1 / √2 ) ∙ 1 / 2 + ( 1 / √2 ) ∙ √3 / 2 =

( 1 / √2 ) ∙ ( 1 / 2 + √3 / 2 ) = ( 1 / √2 ) ∙ ( 1 / 2 ) ∙ ( 1 + √3 ) =

1 / ( √2 ∙ 2 ) ( √3 + 1 ) = ( √3 + 1 ) / ( 2 ∙ √2 ) = ( √3 + 1 ) / 2 √2

cos² 15° = [ ( √3 + 1 ) / 2 √2 ]² = ( √3 + 1 )² / ( 2 √2 )² = [ ( √3 )² + 2 ∙ √3 ∙1 + 1² ] / [ 2² ∙ ( √2 )² ] =

( 3 + 2 √3 + 1 ) / ( 4 ∙ 2 ) = ( 4 + 2 √3 ) / ( 4∙ 2 ) = 2 ∙ ( 2 + √3 ) / ( 4 ∙ 2 ) = ( 2 + √3 ) / 4

cos² ( π / 12 ) = cos² 15° = ( 2 + √3 ) / 4