Definition:Cosine/Definition from Circle
Definition
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 $y$-axis.
Then the cosine of $\theta$ is defined as the length of $AP$.
Hence in the first quadrant, the cosine 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 $y$-axis.
Then the cosine of $\theta$ is defined as the length of $AP$.
Hence in the second quadrant, the cosine is negative.
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 $y$-axis.
Then the cosine of $\theta$ is defined as the length of $AP$.
Hence in the third quadrant, the cosine 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 $y$-axis.
Then the cosine of $\theta$ is defined as the length of $AP$.
Hence in the fourth quadrant, the cosine is positive.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Definitions of the ratios
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosine