How do you find the exact values of cos 195° using the half angle formula?

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Answer 1 · 2015-08-30T19:49:26

#color(red)(cos195 = -sqrt(2 +sqrt3)/2)#

The cosine half-angle formula is

#cos(x/2) = ±sqrt((1 + cos x) / 2)#

The sign is positive if #x/2# is in the first or fourth quadrant and negative if #x/2# is in the second or third quadrant.

∴ #cos195 = cos(180+15) = -cos15#

#15°# is in the first quadrant, so we use the positive sign.

#cos15 =cos(30/2) = sqrt((1+cos30)/2)#

#cos15 = sqrt((1 + sqrt3/2)/2) = sqrt((2 + sqrt3)/4)#

#cos195 = -cos15 =-sqrt(2 +sqrt3)/2#