Answer
(a) $\left( {r,\theta } \right) = \left( {3.606,0.983} \right)$
(b) $\left( {r,\theta } \right) = \left( {8.062, - 1.051} \right)$
(c) $\left( {r,\theta } \right) = \left( {8.544,4.354} \right)$
(d) $\left( {r,\theta } \right) = \left( {5.385,2.761} \right)$
Work Step by Step
Use the conversion formula from rectangular coordinates to polar coordinates given by
$r = \sqrt {{x^2} + {y^2}} $, ${\ \ \ }$ $\theta = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right)$.
(a) Since $\left( {x,y} \right) = \left( {2,3} \right)$, we have
$r = \sqrt {4 + 9} \simeq 3.605$, ${\ \ \ }$ $\theta = {\tan ^{ - 1}}\left( {\frac{3}{2}} \right) \simeq 0.983$.
So, $\left( {r,\theta } \right) = \left( {3.606,0.983} \right)$.
(b) Since $\left( {x,y} \right) = \left( {4, - 7} \right)$, we have
$r = \sqrt {16 + 49} \simeq 8.062$, ${\ \ \ }$ $\theta = {\tan ^{ - 1}}\left( {\frac{{ - 7}}{4}} \right) \simeq - 1.051$.
So, $\left( {r,\theta } \right) = \left( {8.062, - 1.051} \right)$.
(c) Since $\left( {x,y} \right) = \left( { - 3, - 8} \right)$, we have
$r = \sqrt {9 + 64} \simeq 8.544$, ${\ \ \ }$ $\theta = {\tan ^{ - 1}}\left( {\frac{{ - 8}}{{ - 3}}} \right) \simeq 4.354$.
So, $\left( {r,\theta } \right) = \left( {8.544,4.354} \right)$.
(d) Since $\left( {x,y} \right) = \left( { - 5,2} \right)$, we have
$r = \sqrt {25 + 4} \simeq 5.385$, ${\ \ \ }$ $\theta = {\tan ^{ - 1}}\left( {\frac{2}{{ - 5}}} \right) \simeq 2.761$.
So, $\left( {r,\theta } \right) = \left( {5.385,2.761} \right)$.
