Definition:Sine/Definition from Circle

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Definition

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


First Quadrant

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


SineFirstQuadrant.png


Let $P = \left({x, y}\right)$ 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 $AP$.


Second Quadrant

SineSecondQuadrant.png


Let $P = \left({x, y}\right)$ 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 $AP$.


Third Quadrant

SineThirdQuadrant.png


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 $AP$.


Fourth Quadrant

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 $AP$.