What is the polar form of ( - 4 , 32 ) ?

Question

What is the polar form of             (      -      4      ,      32      )       ?

ran1suel23 · Accepted Answer

Step 1The rectangular point             (      -      4      ,      32      )       is in the form             (      x      ,      y      )      Polar points are in the form             (      r      ,      θ      )       .To find r, we effectively need to find the hypotenuse of the right triangle with legs of x and y.Thus,       r    =                            x          2                +                  y          2                    . Here, this becomes      r    =                                        (            -            4            )                    2                +                              (            32            )                    2                      =                            4          2                +                  32          2                      =                            4          2                +                  4          2                          (                      8            2                    )                      =                            4          2                          (          1          +                      8            2                    )                      =    4          65      Even though the point             (      -      4      ,      32      )       is in Quadrant II, the value of r is a magnitude and is still positive.To find       θ   , we first need to write some statement involving       θ   given the information that we know.Looking at the image, we have       θ   , the side opposite       θ   , and the side adjacent to       θ   in a right triangle. Thus, we can say            tan      θ        =                  opposite                    adjacent              =          y      x      Solving for       θ   :      θ    =                  tan                  -          1                            (                  y          x                )            Using our known values:      θ    =                  tan                  -          1                            (                  32                      -            4                          )              =                  tan                  -          1                            (        -        8        )              =    -    1.44644133  Note, however, that this is a negative value and that       -    1.44644133    &amp;gt;    -          π      2       , so this is really an angle in Quadrant IV.To find the value of this angle in Quadrant II, we know that it will be       π   minus the magnitude of the angle we determined.That is,      θ    =    π    -    1.44644133    =    1.69515132  So, our point is:            (      r      ,      θ      )        =          (      4              65            ,      1.69515132      )

What is the polar form of <mstyle displaystyle="true"> ( - 4 ,

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