How do you convert the cartesian coordinate (−2, −9) into polar coordinates?

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How do you convert the cartesian coordinate (−2, −9) into polar coordinates?

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Answer 1 · 2015-09-26T20:22:11

$(-2,-9)$ [Cartesian] $=(sqrt(85),pi+arctan(9/2))$ [Polar]

Explanation:

Polar coordinates are in the form $(r,theta)$
where $r$ is the radial distance from the origin
and $theta$ is the angle measured (in a counter clockwise direction) from the positive X-axis.
enter image source here
The radius can be calculated using the Pythagorean Theorem
$color(white)("XXXX")r = sqrt((-2)^2+(-9)^2) = sqrt(85)$

The angel $theta$ is $pi$ (half a rotation) plus and additional (reference) angle $hattheta$
where $tan(hattheta) = (-9)/(-2) = 9/2$
which implies
$color(white)("XXXX")hattheta=arctan(9/2)$
and
$color(white)("XXXX")theta=pi+arctan(9/2)$

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How do you convert the cartesian coordinate (−2, −9) into polar coordinates?

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