Question
half angle formula to find exact value
cos 112.5 degree
cos 112.5 degree
Answers
Reiny
recall cos 2x = 2cos^2 x - 1
2cos^2 x = cos 2x + 1
cos^2 x = (cos 2x + 1)/2
cos^2 (112.5°) = (cos 225° + 1)/2
now cos 225 = cos(180 + 45)
= cos180cos45 - sin180sin45
= -1(√2/2) - 0
= -√2/2
cos^2 112.5 = (-√2/2 + 1)/2= (2-√2)/4
cos(112.5) = ± √(2-√2)/2
but 112.5 is in quadrant II
so
cos(112.5°) = -(√(2-√2) )/2
2cos^2 x = cos 2x + 1
cos^2 x = (cos 2x + 1)/2
cos^2 (112.5°) = (cos 225° + 1)/2
now cos 225 = cos(180 + 45)
= cos180cos45 - sin180sin45
= -1(√2/2) - 0
= -√2/2
cos^2 112.5 = (-√2/2 + 1)/2= (2-√2)/4
cos(112.5) = ± √(2-√2)/2
but 112.5 is in quadrant II
so
cos(112.5°) = -(√(2-√2) )/2