Definition:Sine/Definition from Circle

Definition

The sine of an angle in a right triangle can be extended to the full circle as follows:

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

Let $P = \tuple {x, y}$ be the point on $C$ in the first 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 $AP$.

Let $P = \tuple {x, y}$ be the point on $C$ in the second 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 $AP$.

Let $P = \tuple {x, y}$ 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 $AP$.

Let $P = \tuple {x, y}$ 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 $AP$.