Trigonometric functions on a coordinate system

Given a coordinate system, an angle (a radian $ \theta$ on the unit circle) always starts from the positive $x$-axis, that is, from the point $ (1,0)$. We have the radian  $ \theta \ge 0$ if the angle with the terminal side $ OP$ is constructed counterclockwise.

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We have $ \theta \le 0$ if it is clockwise.

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In this setting of coordinate system, the cosine and the sine functions of $ \theta$ are defined by

    $\displaystyle \cos\theta =$   $\displaystyle \mbox{the $x$-coordinate of $P$; }$  
    $\displaystyle \sin\theta =$   $\displaystyle \mbox{the $y$-coordinate of $P$. }$  

That is, the point $ P$ on the unit circle with the radian $ \theta$ is given by

$\displaystyle P = (\cos\theta,\: \sin\theta) .
$

The other trigonometric functions of $ \theta$ are then determined by (1-2) and (1-3).



Department of Mathematics
Last modified: 2005-09-29