Standard Position
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis.
Definition
An angle is in standard position when its vertex sits at the origin and its initial side lies along the positive $x$-axis, with the terminal side rotated from there; positive angles rotate counterclockwise, negative angles clockwise. This canonical representation enables the unit-circle definitions of trig functions: the angle $\theta$ in standard position corresponds to the arc of length $|\theta|$ on the unit circle from $(1, 0)$, traveled counterclockwise ($\theta > 0$) or clockwise ($\theta < 0$), tying angle measure directly to the group structure of $S^1$.
Example
Draw a $45^\circ$ angle by putting the vertex at $(0,0)$, starting one side along the positive $x$-axis, and rotating the other side $45^\circ$ counterclockwise; a $210^\circ$ angle lands its terminal side in the third quadrant. In standard position, $-\pi/2$ gives the terminal point $(0, -1)$, the same as $3\pi/2$; these coterminal angles make it clear that trig functions are well-defined by terminal side location, not angle magnitude.
Key Insight
Standard position is a shared starting point: if everyone places angles the same way, it is easy to compare and discuss them, and the quadrant of the terminal side determines the signs of the trig functions (for example, in quadrant II, $\sin > 0$ but $\cos < 0$). The standard-position convention is the coordinate bridge between abstract angle values and geometric positions on the circle, the interface between the additive group $(\mathbb{R}, +)$ of angle values and the multiplicative group $(S^1, \cdot)$ of complex numbers of modulus $1$.