In primary school, an angle was defined as the amount by which two line segments spread apart from their common point (vertex). For example, if you have to rays starting from point $O$, the “angle” is just how wide the space between those two rays is.

Before diving into trigonometry, we need to expand the definition of an angle.

Expanding the Definition for Trigonometry

Why do we change something so straightforward? Because the basic definition limits us. In trigonometry, we often need to talk about:

• Negative Angles:
  Imagine rotating one of the lines in the opposite direction. Instead of a positive opening, we’d call that a negative angle.

• Angles Beyond 360°:
  What if you keep rotating past a full circle? With the basic definition, you’d be stuck at 360°. But by defining an angle as the amount of rotation from a starting position, we can handle multiple rotations, like 450° or even more.

This new approach means we see an angle not just as a static “opening” but as a dynamic amount of rotation.

Key Terms for Understanding a Plane Angle

• vertex : the point $O$
• sides of the angle : the rays $OA$ and $OB$
• initial side : the ray $OA$
• terminal side : the ray $OB$

It represents the amount of rotation from OA to OB.

Why this matters

By redefining an angle as the amount of rotation from one line to another, we break free the limited range taught in primary school. This means we can now handle:

• Negative angles : Use for rotations in the opposite direction
• Angles over $360^{\circ}$ : Use for rotations over 360 degrees