How do you use the half angle formulas to determine the exact values of sine, cosine, and tangent of the angle #pi/8#?

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Answer 1 · 2017-10-13T09:38:40

See the explanation below

We need

#cos(2x)=2cos^2x-1#

#cos(2x)=1-2sin^2x#

#cos(pi/4)=sqrt2/2#

Here,

#x=pi/8#

#cos^2x=(1+cos(2x))/2#

#cos^2(pi/8)=(1+sqrt2/2)/2=(2+sqrt2)/4#

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

#sin^2x=(1-cos(2x))/2#

#sin^2(pi/8)=(1-sqrt2/2)/2=(2-sqrt2)/4#

#sin(pi/8)=sqrt(2-sqrt2)/2#

#tan(pi/8)=sin(pi/8)/cos(pi/8)=sqrt((2-sqrt2)/(2+sqrt2))#

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

#=sqrt((4+2-4sqrt2)/(4-2))#

#tan(pi/8)=sqrt(3-2sqrt2)#