What is the formula for sin(x/2)?

The formula for sin(x/2) is derived from the half-angle identities in trigonometry. Specifically, it states that the sine of half an angle can be expressed in terms of the cosine of the original angle. The correct relationship is:

[

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

]

This formula is particularly useful for calculating the sine of half of an angle when the cosine of the full angle is known. The presence of the square root indicates that the sine function can take on both positive and negative values depending on the quadrant in which the angle ( \frac{x}{2} ) lies.

The negative and positive signs reflect the periodic and even nature of the sine function. For instance, if ( x ) is in the first quadrant, ( \frac{x}{2} ) will also be in the first quadrant where sine is positive. Conversely, if ( x ) is in the third quadrant, ( \frac{x}{2} ) will be in the second quadrant where sine is still positive, but the expression accounts for all possible quadrants with the sign.

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