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Converting standard to polar form

  1. Apr 14, 2017 #1
    1. The problem statement, all variables and given/known data
    you are given the standard form z = 3 - 3i

    2. Relevant equations


    3. The attempt at a solution
    so to convert this to polar form, i know that ##r = 3√2## but how do i find theta here? There are so many mixed answers it seems online that I can't tell... i know that ##(3,-3)## is in the last quadrant and that ##tan^-1(-3/3) = -45##.

    But how can I do this all without a calculator first of all? I have a final where no calculators are allowed. Some sites are telling me that theta is just -45 or -pi/4 here. Others are telling me that its 360 - (-45) or 360 + -45.
    What the heck is the right answer???

    Also, just for my understanding here. say I have a different standard form where ##z=-8i## and I want to find its cubed roots. Would theta be 270 here? or ##3pi/2##? Because ##tan^-1(-8/0)## is undefined.
     
    Last edited: Apr 14, 2017
  2. jcsd
  3. Apr 14, 2017 #2
    Hi Arnoldjavs3:

    What is the difference between the two answers: (a) -45, and (b) 360-45=315.

    BTW: I don't know what your teacher requires, but in general it is better to include a symbol like "o" or "deg" for an angle using degrees as a unit rather than omit it.

    Regards,
    Buzz
     
  4. Apr 14, 2017 #3
    Oh... right. I didn't know how to add the degree symbol with latex. I feel stupid now.

    How about the degree for ##z=-8i##? Am I right to think that it is 270o?
     
  5. Apr 14, 2017 #4

    LCKurtz

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    Draw a line from the origin to ##(3,-3)##. Label it ##r##. Then draw an arc counterclockwise from the positive ##x## axis to ##r##. That arc subtends the angle you want. Don't use any inverse trig formula, just look at it. You should see that it is ##180^\circ + 45^\circ## or ##\pi +\frac \pi 4 =\frac{5
    \pi} 4##. Just draw a quick picture for this kind of problem.
    [Edit, corrected] As Mark44 points out in post #6, I meant
    ##270^\circ + 45^\circ## or ##\frac{3\pi} 2 +\frac \pi 4 =\frac{7
    \pi} 4##.
    Again, don't use inverse trig functions here. You want$$
    r^3e^{i3\theta} = 8e^{\frac {3\pi i} 2}$$ So ##r=2## and ##3\theta = \frac {3\pi} 2 + 2n\pi##.
     
    Last edited: Apr 14, 2017
  6. Apr 14, 2017 #5
    Hi Arnoldjavs3:

    What do you think the answer is?

    BTW: How to represent the value of an angle in the third or fourth quadrant is an arbitrary convention. The two choices are
    (a) 180 < θ < 360, or
    (b) 0 > θ > - 180.
    You might want to notice which convention your teacher usually uses, and do the same.

    Another BTW re
    There are many useful symbols available by selecting "∑" on the formatting option bar.

    Regards,
    Buzz
     
    Last edited: Apr 14, 2017
  7. Apr 14, 2017 #6

    Mark44

    Staff: Mentor

    @LCKurtz, I'm sure you really mean ##270^\circ + 45^\circ## or ##\frac {3\pi} 2 + \frac \pi 4 = \frac{7\pi} 4##.
     
  8. Apr 14, 2017 #7

    LCKurtz

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    Yes, of course. For some reason I copied his point as ##(-3,-3)##.
     
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