Question

How do you evaluate `cos (pi/8)`?

Answer 1

`cos(pi/8) = sqrt(1/2+sqrt(2)/4)`

Explanation:

`"Use the double-angle formula for cos(x) : "`
`cos(2x) = 2 cos^2(x) - 1`
`=> cos(x) = pm sqrt((1 + cos(2x))/2)`
`"Now fill in x = "pi/8`
`=> cos(pi/8) = pm sqrt((1 + cos(pi/4))/2)`
`=> cos(pi/8) = sqrt((1+sqrt(2)/2)/2)`
`=> cos(pi/8) = sqrt(1/2+sqrt(2)/4)`

`"Remarks : "`
`"1) "cos(pi/4) = sin(pi/4) = sqrt(2)/2" is a known value"`
`"because "sin(x) = cos(pi/2-x)," so "`
`sin(pi/4)=cos(pi/4)" and "sin^2(x)+cos^2(x) = 1`
`=> 2 cos^2(pi/4) = 1 => cos(pi/4) = 1/sqrt(2) = sqrt(2)/2.`
`"2) because "pi/8" lies in the first quadrant, "cos(pi/8) > 0", so"`
`"we need to take the solution with the + sign."`