Angle Of Rotation

The angles considered in Euclidean Geometry are all less than two right angles

, but for the purpose of Trigonometry it is necessary to extend the conception of angular magnitude so as to include angles of all magnitudes, positive or negative.

Suppose that the ray $overrightarrow{OP}$ in the figure is capable of revolving about the point $O$, and suppose that in this way it has passed successively from the position $overrightarrow{OA}$ to the positions occupied by $overrightarrow{OB}$, $overrightarrow{OC}$, $overrightarrow{OD}$, $ldots$, then the angle between $overrightarrow{OA}$ and any position such as $overrightarrow{OC}$ is measured by the amount of revolution which the ray $overrightarrow{OP}$ has undergone in passing from its initial position $overrightarrow{OA}$ into its final position $overrightarrow{OC}$. We denote this angle by $angle AOC$.

The point $O$ is called the $ext{origin}$, $overrightarrow{OA}$ the $ext{initial line}$ and $overrightarrow{OC}$ the $ext{terminal line}$; the revolving line $overrightarrow{OP}$ is known as the $ext{generating line}$ or the $ext{radius vector}$. The lines $overrightarrow{OA}$ and $overrightarrow{OC}$ are called the boundary lines of the angle $angle AOC$.

For a complete revolution $overrightarrow{OP}$ makes four right angles; and moreover, the radius vector $overrightarrow{OP}$ may make any number of complete revolutions through the original position $overrightarrow{OA}$ before taking up its final position $overrightarrow{OC}$. It will thus be seen that in Trigonometry angles are not restricted as in Euclid, but may be of any magnitude.

Moreover, $overrightarrow{OP}$ may revolve about the point $O$, either in clockwise direction or counter-clockwise direction. We adopt the convention that :

When the revolution of the radius vector $overrightarrow{OP}$ is counter-clockwise, the angle measured is positive.

When the revolution of the radius vector $overrightarrow{OP}$ is clockwise, the angle measured is negative.

In accordance with our convention then, whenever $overrightarrow{OP}$ makes a complete counter-clockwise revolution, it has turned through four right angles reckoned positive, and whenever it makes a complete clockwise revolution, it has turned through four right angles taken negatively.

by: jeri

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