Definition:Sine/Definition from Circle
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.
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
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
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
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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Definitions of the ratios
- 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $1$. Functions: $1.5$ Trigonometric or Circular Functions: $1.5.2$ Sine Function
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): sine