Halfangle

Halfangle refers to an angle that is one half of another angle. In trigonometry, half-angle formulas relate trigonometric functions of an angle to those of half of the angle. These identities are derived from the double-angle formulas: sin x = 2 sin(x/2) cos(x/2), cos x = cos^2(x/2) - sin^2(x/2) = 1 - 2 sin^2(x/2) = 2 cos^2(x/2) - 1.

From these, the basic half-angle formulas are:

sin(x/2) = ± sqrt((1 - cos x)/2)

cos(x/2) = ± sqrt((1 + cos x)/2)

tan(x/2) = sin x / (1 + cos x) = (1 - cos x)/sin x = ± sqrt((1 - cos x)/(1 + cos x))

A commonly used substitution for integration and solving equations is t = tan(x/2), which gives sin x = 2 t /(1 + t^2), cos x = (1 - t^2)/(1 + t^2), and tan(x/2) = t.

Sign choices for sin(x/2) and cos(x/2) depend on the quadrant of x/2. Half-angle formulas are useful in simplifying expressions, evaluating trigonometric integrals, and solving trigonometric equations where the angle is in a known quadrant or where cos x or sin x is known.