How to find cos(π/8) value,not exact values but in rational numbers?

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Answer 1 · 2018-05-09T08:33:48

#cos(pi/8)=1/2sqrt(2+sqrt2)#

We can use the double angle identity #cos2A=2cos^2A-1#.

Let #A=pi/8#, then #2A=pi/4# and identity becomes

#cos(pi/4)=2cos^2(pi/8)-1#

or #1/sqrt2=2cos^2(pi/8)-1#

or #2sqrt2cos^2(pi/8)=sqrt2+1#

and #cos^2(pi/8)=(sqrt2+1)/(2sqrt2)*sqrt2/sqrt2#

Now as #pi/8# lies in #Q1#, we can only have positive value of #cos(pi/8)#

and #cos(pi/8)=sqrt((2+sqrt2)/4)#

= #1/2sqrt(2+sqrt2)#