sinAPB

SinAPB denotes the sine of the angle APB, i.e., the angle at point P formed by the segments PA and PB in the plane containing A, P, and B. The quantity sin(APB) is a nonnegative real number between 0 and 1, since the angle ∠APB is defined to lie in the range 0 to π.

One common way to compute sin(APB) is via vectors. Let a = A − P and b = B − P be the directed vectors from P to A and from P to B. Then sin(APB) = |a × b| / (|a| |b|), where × denotes the cross product (in the plane this reduces to the magnitude of the z-component). Equivalently, if θ is the measure of ∠APB, then sin(APB) = sin θ.

Another relation involves the area of triangle APB. The area is given by Area(ΔAPB) = (1/2) |PA| |PB| sin(APB). Therefore sin(APB) = 2 · Area(ΔAPB) / (|PA| |PB|).

If the side lengths PA, PB, and AB are known, sin(APB) can also be found from the Law of Cosines. Let c = AB be the side opposite angle ∠APB. Then cos(APB) = (PA^2 + PB^2 − AB^2) / (2 · PA · PB), and sin(APB) = sqrt(1 − cos^2(APB)) with the angle convention 0 ≤ ∠APB ≤ π.

SinAPB is widely used in geometry and trigonometry to relate an angle at a vertex to side lengths, areas, and vector directions in the configuration A, P, B.