cos2APB

Cos2APB is a shorthand for the cosine of twice the angle APB, written as cos(2∠APB) or cos 2∠APB. Here ∠APB is the angle at point P formed by the lines PA and PB.

In trigonometry, the double-angle identity gives cos(2θ) = 2 cos^2 θ − 1. Applying this with θ = ∠APB gives cos(2∠APB) = 2 cos^2(∠APB) − 1. This expresses cos2APB in terms of the ordinary angle at P.

Cosine of ∠APB can be found from triangle APB using the law of cosines. If PA = c, PB = b, and AB = a, then cos ∠APB = (PA^2 + PB^2 − AB^2) / (2 · PA · PB) = (c^2 + b^2 − a^2) / (2bc). Substituting into the double-angle formula yields cos(2∠APB) = 2[(c^2 + b^2 − a^2)^2 / (4 b^2 c^2)] − 1, which simplifies to [ (c^2 + b^2 − a^2)^2 − 2 b^2 c^2 ] / (2 b^2 c^2).

In circle geometry, if A and B lie on a common circle and P also lies on the same circle (the circumcircle of triangle APB), then ∠AOB (with O the center) is twice ∠APB, and cos(2∠APB) equals cos ∠AOB.

Cos2APB is thus a useful quantity in solving geometric problems that involve the angle at P and its relation to side lengths or circular configurations.