Conversion between Cartesian and Polar Coordinates in Plane/Examples/(-2, -pi over 4)

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Example of Use of Conversion between Cartesian and Polar Coordinates in Plane

The point $P$ defined in polar coordinates as:

$P = \polar {-2, -\dfrac \pi 4}$

can be expressed in the corresponding Cartesian coordinates as:

$P = \tuple {\sqrt 2, -\sqrt 2}$


Proof

Let $P$ be expressed in Cartesian coordinates as:

$P = \tuple {x, y}$


From Conversion between Cartesian and Polar Coordinates in Plane:

\(\ds x\) \(=\) \(\ds r \cos \theta\)
\(\ds y\) \(=\) \(\ds r \sin \theta\)

where in this case:

\(\ds r\) \(=\) \(\ds -2\)
\(\ds \theta\) \(=\) \(\ds -\dfrac \pi 4\)


Hence:

\(\ds x\) \(=\) \(\ds -2 \map \cos {-\dfrac \pi 4}\)
\(\ds \) \(=\) \(\ds -2 \times \dfrac {\sqrt 2} 2\) Cosine of $315 \degrees$
\(\ds \) \(=\) \(\ds \sqrt 2\)

and:

\(\ds y\) \(=\) \(\ds -2 \map \sin {-\dfrac \pi 4}\)
\(\ds \) \(=\) \(\ds -2 \times \paren {-\dfrac {\sqrt 2} 2}\) Sine of $315 \degrees$
\(\ds \) \(=\) \(\ds -\sqrt 2\)

Hence the result.

$\blacksquare$


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