Angles

The Player's yaw (facing) is a float (from ${\textstyle -180.0}$ to ${\textstyle 180.0}$ )

An angle is an integer (from ${\textstyle 0}$ to ${\textstyle 2^{16}-1}$ )

Angles and Half Angles

Minecraft uses angles for movement calculations, which means the Player's yaw has to be converted before being used.

This conversion induces some imprecision, as information is lost when casting a float to an int.

This also means two close (but different) facings can result in the same angle being used (a single angle spans across ~Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.0055°} ).

Sin() and Cos() source code (from MathHelper):

public class MathHelper
{
    private static final float[] SIN_TABLE = new float[65536];


    public static float sin(float value)
    {
        return SIN_TABLE[(int)(value * 10430.378F) & 65535];
    }

    public static float cos(float value)
    {
        return SIN_TABLE[(int)(value * 10430.378F + 16384.0F) & 65535];
    }


    static
    {
        for (int i = 0; i < 65536; ++i)
        {
            SIN_TABLE[i] = (float)Math.sin((double)i * Math.PI * 2.0D / 65536.0D);
        }
    }
}

Note: & 65535 is the same as % 65536, which is the remainder of a division by 65536 ( $2^{16}$ )

By analyzing the sin() and cos() function from the source code, we can notice that the yaw-to-angle conversion is not exactly the same for both functions.

Because floats are rather imprecise for larger values, this means

Half angles are specific values for which the corresponding angle are interpreted differently by cos() and sin().

These half angles are found between angles (hence the name) and can be used to slightly increase jump distance in Tool-Assisted Parkour.

Note: Fast Math is an Optifine feature that reduces the number of angles to 4096 ( $2^{12}$ ). With Fast Math, half angles are fewer in number, but are up to 16x more effective. For example, it makes a no-sprint 3b jump possible with only flat momentum.