What is the half-angle identity for sine?

The half-angle identity for sine is indeed correctly stated as sin(x/2) = ±√[(1 - cos(x))/2]. This identity is derived from the double angle formulas in trigonometry, where we start with the cosine double angle formula: cos(2θ) = 1 - 2sin²(θ). By setting θ = x/2, we can rearrange this to find sin²(θ) and ultimately express sin(θ) in terms of cos(x).

To arrive at the half-angle identity for sine, we look at sin²(x/2) = (1 - cos(x))/2. Taking the square root of both sides provides us with sin(x/2) = ±√[(1 - cos(x))/2]. The positive and negative signs account for the fact that sine can take on both positive and negative values depending on the quadrant in which x/2 is located.

Understanding this identity is important in many areas of trigonometry, especially when solving problems that involve angles that are half of known angles. The relationship helps in evaluating trigonometric functions where the angle is split in two, easing calculations in various applications including integration and solving trigonometric equations.