Terminal Side
TrigonometryThe terminal side of an angle in standard position is the rotating ray that ends at the angle's final position after rotation from the initial side.
Definition
The terminal side is the moving ray of an angle. Starting from the initial side (positive x-axis), you rotate to the terminal side to create the angle.
Example
For a $90^\circ$ angle, the terminal side ends up pointing straight up along the positive $y$-axis. For $180^\circ$, it points left along the negative $x$-axis.
Key Insight
"Terminal" means ending. The terminal side is where the rotation stops.
Definition
The terminal side is the ray that has been rotated from the initial side by the angle's measure. Its position determines the values of all trig functions: the terminal side of $\theta$ intersects the unit circle at $(\cos(\theta), \sin(\theta))$.
Example
The terminal side of $225^\circ$ lies in quadrant III (both $x$ and $y$ negative). It intersects the unit circle at $(-\sqrt{2}/2, -\sqrt{2}/2)$, confirming $\cos(225^\circ) = \sin(225^\circ) = -\sqrt{2}/2$.
Key Insight
Two different angles can have the same terminal side (coterminal angles). The terminal side position is all that matters for computing trig values, which is why $\sin(\theta) = \sin(\theta + 2\pi)$ for any $\theta$.
Definition
The terminal side of angle $\theta$ in standard position is the ray from the origin through the point $(\cos(\theta), \sin(\theta))$ on the unit circle. The terminal side parametrizes points on $S^1$ as $\theta$ varies. Coterminal angles share a terminal side, reflecting the periodicity of the map $\theta \to e^{i\theta}$ from $\mathbb{R}$ to $S^1$ with period $2\pi$.
Example
The terminal side of $7\pi/4$ passes through $(\cos(7\pi/4), \sin(7\pi/4)) = (\sqrt{2}/2, -\sqrt{2}/2)$, in quadrant IV. This is coterminal with $-\pi/4$ and $15\pi/4$, all sharing the same terminal side.
Key Insight
The terminal side is the geometric realization of the equivalence class of angles modulo $2\pi$. Formally, two angles are equivalent if their difference is a multiple of $2\pi$, and the terminal side represents the unique element of each class on the unit circle.