Initial Side
TrigonometryThe initial side of an angle in standard position is the fixed ray along the positive x-axis from which the angle is measured.
Definition
The initial side is the starting ray of an angle in standard position. It always points to the right along the positive x-axis.
Example
When you draw any angle in standard position, the initial side is the fixed arm pointing to the right. The other arm (the one that moves) is the terminal side.
Key Insight
"Initial" means beginning. The initial side is where the rotation starts before you measure the angle.
Definition
The initial side is the fixed ray of an angle in standard position, lying along the positive $x$-axis (pointing from the origin toward positive $x$). Rotation from the initial side to the terminal side, measured counterclockwise positive, defines the angle.
Example
For a $120^\circ$ angle: initial side $=$ positive $x$-axis direction. Terminal side $=$ rotated $120^\circ$ counterclockwise from the initial side. The angle between them is $120^\circ$.
Key Insight
By convention, all angles in standard position share the same initial side. This consistency is what makes coterminal angles possible: different rotations can land on the same terminal side.
Definition
The initial side is the reference ray corresponding to angle $0$, identified with the positive $x$-axis in the Cartesian plane. Formally, it corresponds to the identity element of the rotation group $SO(2)$: zero rotation from the standard orientation. The choice of initial side is a gauge choice, and changing it amounts to an overall phase shift in the angle parametrization.
Example
In navigation, the initial side might be North instead of East (positive $x$-axis). Converting between math-convention (East $= 0^\circ$, CCW positive) and compass-convention (North $= 0^\circ$, CW positive) requires the transformation: compass bearing $= 90^\circ - \text{math angle}$.
Key Insight
The initial side defines the branch cut for the argument function in complex analysis. The principal argument $\text{Arg}(z)$ in $(-\pi, \pi]$ uses the negative real axis as a branch cut, which is equivalent to choosing a different "initial side" for measuring angles.