Definition:Sine/Definition from Circle

From ProofWiki
Jump to navigation Jump to search

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 plane.


SineFirstQuadrant.png


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 $AP$.

Hence in the first quadrant, the sine is positive.


Second Quadrant

SineSecondQuadrant.png


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 $AP$.

Hence in the second quadrant, the sine is positive.


Third Quadrant

SineThirdQuadrant.png


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 $AP$.

Thus by definition of third quadrant, the sine is negative.


Fourth Quadrant

SineFourthQuadrant.png


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 $AP$.

Hence in the fourth quadrant, the sine is negative.


Also presented as

Some sources use a general radius of $r$, thereby defining the sine as:

$\sin \theta = \dfrac y r$


Sources