Definition:Sine/Definition from Circle/Fourth Quadrant

Definition
Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.


 * SineFourthQuadrant.png

Let $P = \left({x, y}\right)$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let $AP$ be the perpendicular from $P$ to the $x$-axis.

Then the sine of $\theta$ is defined as the length of $AP$.

Thus by definition of fourth quadrant, the sine of such an angle is negative.