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.

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

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

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