How do you evaluate cos (pi/8)?

1 Answer
MattyMatty
Mar 2, 2018

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."

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