Given that the exact value of sin pi10 1 squareroot4 for e

Given that the exact value of sin pi/10 = -1 + squareroot/4, for each question calculate the exact value of the given expression. You mat leave nested radicals in your answer, but rationalize any denominators. Sin pi/5 sin pi/20

Solution

Sin /10 = (-1+ 5)/4. We know that cosx = (1- sin2x) so that cos /10 = ( 1- sin2 /10 ) = [ 1 - (-1+ 5)/4.)2 ] = [ 1 - ( 1 -2 5 + 5)/16 ] = [ 1 - 6/16 + 2 5/16 ] = ( 10/16 - 5/8 ] =( 5/8 - 5/8) = [(5 - 5)/8 ] = (2)((5 - 5) /4. We also know that sin2x = (2sinx)(cos x). Therefore, sin /5 = sin (2*/10) = 2sin(/10)cos(/10) = 2*[ (-1+ 5)/4][( 2) [ (5 - 5) ]/4 = ( 2)[ (-1+ 5)] [ (5 - 5) ]/8 = [ (-1+ 5)] [ (10 - 2 5) ]/ 8

We also know that Sin (x/2) = ± [(1 -cosx)/2] . Therefore, sin (/20 ) = ± [ (1 – cos /10 )/2] = ± [1- (2)((5 - 5) /4 ] = ± ½ [ 4- 2((5 - 5)] = ± ½ [ 4- (10 -25)]