Coordinate Geometry - Straight Line

Find the points whose polar coordinates are:

  1. $\left(2, 45^\circ\right)$

  2. $\left(5, \dfrac{-\pi}{2}\right)$


  1. The polar coordinates are $\;$ $\left(r, \theta\right) = \left(2, 45^\circ\right)$

    Let the corresponding Cartesian coordinates be $\left(x, y\right)$

    Now, $\;$ $x = r \cos \theta = 2 \cos 45^\circ = \dfrac{2}{\sqrt{2}} = \sqrt{2}$

    and $\;$ $y = r \sin \theta = 2 \sin 45^\circ = \dfrac{2}{\sqrt{2}} = \sqrt{2}$

    $\therefore \;$ The corresponding Cartesian coordinates are $\;$ $\left(x, y\right) = \left(\sqrt{2}, \sqrt{2}\right)$


  2. The polar coordinates are $\;$ $\left(r, \theta\right) = \left(5, \dfrac{-\pi}{2}\right)$

    Let the corresponding Cartesian coordinates be $\left(x, y\right)$

    Now, $\;$ $x = r \cos \theta = 5 \cos \left(\dfrac{-\pi}{2}\right) = 0$

    and $\;$ $y = r \sin \theta = 2 \sin \left(\dfrac{-\pi}{2}\right) = -2$

    $\therefore \;$ The corresponding Cartesian coordinates are $\;$ $\left(x, y\right) = \left(0, -2\right)$

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