Conversion between Cartesian and Polar Coordinates in Plane/Examples/(4, pi over 3)

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

The point $P$ defined in polar coordinates as:

$P = \polar {4, \dfrac \pi 3}$

can be expressed in the corresponding Cartesian coordinates as:

$P = \tuple {2, 2 \sqrt 3}$


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:

\(\ds r\) \(=\) \(\ds 4\)
\(\ds \theta\) \(=\) \(\ds \dfrac \pi 3\)


Hence:

\(\ds x\) \(=\) \(\ds 4 \cos \dfrac \pi 3\)
\(\ds \) \(=\) \(\ds 4 \times \dfrac 1 2\) Cosine of $60 \degrees$
\(\ds \) \(=\) \(\ds 2\)

and:

\(\ds y\) \(=\) \(\ds 4 \sin \dfrac \pi 3\)
\(\ds \) \(=\) \(\ds 4 \times \dfrac {\sqrt 3} 2\) Sine of $60 \degrees$
\(\ds \) \(=\) \(\ds 2 \sqrt 3\)

Hence the result.

$\blacksquare$


Sources

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