Use an appropriate half-angle identity to find the exact value of sin(pi/12) = ?

Question

10688 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-17T20:31:02

#sqrt(2 - sqrt3)/2#

half-angle identity: #sin (x/2) = +- sqrt((1 - cos x)/2)#

here, #x/2 = pi/12#, so #x = pi/6#

substituting #pi/6# in for #x:#

#sqrt((1 - cos x)/2) = sqrt((1 - cos (pi/6))/2)#

#cos (pi/6) = (sqrt3)/2#

therefore

#sqrt((1 - cos (pi/6))/2) = sqrt((1 - (sqrt3)/2) /2#

simplifying the expression on the right-hand side:

#1 - (sqrt3)/2 = 2/2 - (sqrt3)/2 = (2 - sqrt3)/2#

#(1 - (sqrt3)/2)/2 = (2 - sqrt3)/4#

#sqrt((1 - (sqrt3)/2) /2) = sqrt((2 - sqrt3)/4)#

#= sqrt(2 - sqrt3)/2#

this is the simplest form of the exact value.

this value can be shown as #0.25882# to #5# significant figures.

#sin (pi/12)# is the same; #0.25882# to #5# s.f.,

however, as an exact value it is #= sqrt(2 - sqrt3)/2#.