Find the exact value of sin (5π/8)

Question

Sum and different identities: Find the exact value of sin (5π/8)

Answer 1

use the half angle formula for sin(x/2)

sinx/2 = + or - sqr((1-cosx)/2)

sin5pi/8 = sin(5pi/4/2)

sin5pi/4 = -(1/2)sqr2, 5pi/4 = 225 degrees, sin225 = -sin45 =-sqr2/2

which also = cos5pi/4

sin(225/2) = sin112.5 = sin67.5 = about 0.923879532

=sqr((1-cos225)/2) = sqr((1+sqr2/2)/2) = sqr(1/2 + sqr2/4) = exact value of sin5pi/8

= about 0.923879532

exact value of sin(5pi/8) = sqr(1/2 + (1/4)sqr2)

sum of angles formula is

sin(A+B) = sinAcosB + sinBcosA

if A=B = 5pi/8 then

sin(A+B)= sin2A = 2sinAcosA = -(1/2)sqr2

cosA = (sqr2)/4sinA

cos(A+B) = cosAcosB - sinAsinB

if A=B = 5pi/8 then

cos(A+B) = cos(5pi/4) = -sqr2/2 = cos^2A - sin^2A = (-sqr2/4sinA) -sin^2A

sin^2A +1/8sin^2A - sqr2/2 = 0, multiply by 8sin^2A

8sin^4A + 1 - 4sqr2sin^2A = 0 use quadratic formula

sin^2(5pi/8) = 4sqr2/16 + (1/16)sqr(32+32) = sqr2/4 + 1/2

sin(5pi/8) = sqr((sqr2)/4 + 1/2)

= about 0.924