What is the value of Cos(pi/8)?

2 Answers
Alan P.
Oct 18, 2016

cos(pi/8)=sqrt(2+sqrt(2))/2 ~~0.92388

Explanation:

Version 1
Things to remember:
color(white)("XXX")pi/8 = (pi/4)/2

color(white)("XXX")cos(pi/4)= sqrt(2)/2

color(white)("XXX")cos(theta/2)=+-sqrt((1+cos(theta))/2

rArr cos(pi/8) = sqrt((1+sqrt(2)/2)/2) (note the negative possibility can be ignored since pi/8 is in Q I)

color(white)("XXXXXXX")=sqrt(2+sqrt(2))/2

using a calculator
color(white)("XXXXXXX")~~0.92388

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Version 2 (if you're going to use a calculator anyway.
Use calculator to evaluate:
color(white)("XXX")cos(pi/8)~~0.92388

Shwetank Mauria ยท Zor Shekhtman
Oct 18, 2016

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

Explanation:

Let us assume pi/8=A and then 2A=pi/4

Now as cos2A=2cos^2A-1,

we have cos(pi/4)=2cos^2A-1

or 2cos^2A=1+1/sqrt2=(sqrt2+1)/sqrt2

or 2cos^2A=(sqrt2+1)/2sqrt2

Multiplying numerator and denominator by sqrt2 on RHS

cos^2A=(2+sqrt2)/4

or cosA=sqrt((2+sqrt2)/4)

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

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