Which of the following expressions correctly represents the half-angle formula for sin(x/2)?

The half-angle formula for sine is derived from the double-angle formulas and provides a way to express the sine of half an angle in terms of cosine. Specifically, the half-angle formula for sine states that:

[

\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{2}}

]

This formula is particularly useful in trigonometry when you want to find the sine of an angle that is half of a known angle. The formula captures the relationship between the sine of the half angle and the cosine of the full angle.

The square root part of the formula indicates that the sign (positive or negative) depends on the quadrant in which ( \frac{x}{2} ) lies. When ( x ) is in certain ranges, ( \sin\left(\frac{x}{2}\right) ) will be positive, while in other ranges it may be negative.

In this context, the expression that correctly represents the half-angle formula for ( \sin\left(\frac{x}{2}\right) ) is square root ((1 - cos(x))/2). This clearly shows how the sine of half an angle relates to the cosine of the original angle