sinAOB2

SinAOB2 is a notational shorthand that is sometimes used to denote the square of the sine of the angle AOB, where O is the vertex and OA and OB are the two rays forming the angle. Equivalently, sinAOB2 represents sin^2(∠AOB), with θ = ∠AOB and sinAOB2 = sin^2 θ.

To compute sin^2 θ, one can use the cosine of the angle obtained from the dot product of the vectors OA and OB. If a = OA and b = OB, then cos θ = (a · b) / (|a||b|). Therefore sin^2 θ = 1 − cos^2 θ. In vector form, sin^2 θ = |a × b|^2 / (|a|^2 |b|^2); in two dimensions, |a × b| is the magnitude of the scalar cross product, equal to |a||b| sin θ.

A common geometric context is when O is the center of a circle and OA = OB = R. Here cos θ = (a · b) / R^2 and sin^2 θ = 1 − (a · b)^2 / R^4. Related identities include sin^2 θ = (1 − cos 2θ)/2. For central angle θ, the chord length AB is AB = 2R sin(θ/2), which yields sin^2(θ/2) = AB^2 / (4R^2). The quantity sin^2 θ also appears in expressions for areas, since the area of triangle OAB equals (1/2)|OA||OB| sin θ, and its square is (1/4)|OA|^2|OB|^2 sin^2 θ.

Note: without formatting, sinAOB2 can be ambiguous; this article treats it as sin^2(∠AOB). See also sin, cos, dot product, cross product, central angle, chord length.