Find polar coordinates of each point whose rectangular coordinates are:

(a) $(4,-4)$

(b) $( -1,-\sqrt{3})$

(c) $( 4,-1)$

Figure 35

Solution (a) The point $( 4,-4)$, plotted in Figure 35, is in quadrant IV. The distance from the pole to the point $(4,-4)$ is $$\begin{equation*} r=\sqrt{x^{2}+y^{2}}=\sqrt{4^{2}+( -4) ^{2}}=\sqrt{32}=4\sqrt{2} \end{equation*}$$

Since the point $( 4,-4)$ is in quadrant IV, $-\dfrac{\pi }{2} <\theta <0.$ So, $$\begin{equation*} \theta =\tan ^{-1}\left( \dfrac{y}{x}\right) =\tan ^{-1}\left( \dfrac{-4}{4}\right) =\tan ^{-1}( -1) =-\dfrac{\pi }{4} \end{equation*}$$

666

A pair of polar coordinates for this point is $\left( 4\sqrt{2},-\dfrac{\pi }{4}\right) .$ Other possible representations include $\left( -4\sqrt{2}, \dfrac{3\pi }{4}\right)$ and $\left( 4\sqrt{2},\dfrac{7\pi }{4}\right)$.

Figure 36

(b) The point $( -1,-\sqrt{3}) ,$ plotted in Figure 36, is in quadrant III. The distance from the pole to the point $( -1,-\sqrt{3})$ is $$\begin{equation*} r=\sqrt{( -1) ^{2}+( -\sqrt{3}) ^{2}}=\sqrt{1+3}=2 \end{equation*}$$

Since the point $( -1,-\sqrt{3})$ lies in quadrant III and the inverse tangent function gives an angle in quadrant I, we add $\pi$ to $\tan ^{-1}\left( \dfrac{y}{x}\right)$ to obtain an angle in quadrant III. $$\begin{equation*} \theta =\tan ^{-1}\left( \dfrac{-\sqrt{3}}{-1}\right) +\pi =\tan ^{-1}( \sqrt{3}) +\pi =\dfrac{\pi }{3}+\pi =\dfrac{4\pi }{3} \end{equation*}$$

A pair of polar coordinates for the point is $\left( 2,\dfrac{4\pi }{3} \right)$. Other possible representations include $\left( -2,\dfrac{\pi }{3} \right)$ and $\left( 2,-\dfrac{2\pi }{3}\right)$.

Figure 37

(c) The point $\left( 4,-1\right) ,$ plotted in Figure 37, lies in quadrant IV. The distance from the pole to the point $( 4,-1)$ is $$\begin{equation*} r=\sqrt{x^{2}+y^{2}}=\sqrt{4^{2}+( -1) ^{2}}=\sqrt{17} \end{equation*}$$

Since the point $( 4,-1)$ is in quadrant IV, $-\dfrac{\pi }{2}<\theta <0.$ So, $$\begin{equation*} \theta =\tan ^{-1}\left( \dfrac{y}{x}\right) =\tan ^{-1}\left( \dfrac{-1}{4} \right) \approx -0.245\hbox{ radians} \end{equation*}$$

A pair of polar coordinates for this point is $( \sqrt{17} ,-0.245) .$ Other possible representations for the point include $\left( \sqrt{17} ,\tan ^{-1}\left( -\dfrac{1}{4}\right) +2\pi \right) \approx ( \sqrt{17} ,6.038)$ and $\left( -\sqrt{17},\tan ^{-1}\left( -\dfrac{1}{4}\right) +\pi \right) \approx ( -\sqrt{17},2.897)$.