Question

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

Answer 1

`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`

Answer 2

`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`