Definition:Sine/Definition from Circle/Third Quadrant

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


Let $P = \left({x, y}\right)$ be the point on $C$ in the third 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 third quadrant, the sine of such an angle is negative.