Can the double-angle formula change a positive cosine to a negative result?

Question

Find sin 2(u), cos 2(u), and tan 2(u) given the coordinate point (2, -5). Why is my answer resulting in a negative value for cos 2(u)?

Answer 1

(2,-5) is in quadrant IV with positive cosines

angle u is tan^-1(-5/2) = about -68.2 degrees or 291.8 degrees

cos291.8 = about +0.371

cos(2u) = cos(583.6) = cos(583.6-360) = cos223.6= about -0.724

cosu>0, but cos2u<0 because 2u is in Quadrant III where all cosines are negative

cosu = adjacent side over hypotenuse = 2/sqr(29) = about +0.371

cos2u = cos^2(u) - sin^2(u) = [2/sqr29]^2 - [-5/sqr29]^2 = 4/29 -25/29 = - 21/29 = about -0.724