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Math Help - Find polar coordinates

  1. #1
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    Find polar coordinates

    Hey I am not sure how to find the second coordinate of these polar coordinates.

    a) Find polar coordinates (r, θ) of the point (4, -4), where r > 0 and 0 ≤ θ ≤ 2π.

    So r= \sqrt{(4)^2+(4)^2}
    = \sqrt{32}
    =4\sqrt{2}

    I tried doing
    θ= tan^{-1}(\frac{y}{x})
    θ= tan^{-1}(\frac{-4}{4})
    θ= tan^{-1}(-1)
    θ= \frac{-\pi}{4}

    But this is incorrect?

    For r < 0 and 0 ≤ θ ≤ 2π.

    r=-4\sqrt{2}

    How do I find theta?
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  2. #2
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    Quote Originally Posted by mmattson07 View Post
    Hey I am not sure how to find the second coordinate of these polar coordinates.

    a) Find polar coordinates (r, θ) of the point (4, -4), where r > 0 and 0 ≤ θ ≤ 2π.

    So r= \sqrt{(4)^2+(4)^2}
    = \sqrt{32}
    =4\sqrt{2}

    I tried doing
    θ= tan^{-1}(\frac{y}{x})
    θ= tan^{-1}(\frac{-4}{4})
    θ= tan^{-1}(-1)
    θ= \frac{-\pi}{4}

    But this is incorrect?

    For r < 0 and 0 ≤ θ ≤ 2π.

    r=-4\sqrt{2}

    How do I find theta?
    Your calculation of r is correct.

    To find \theta, first note that the point (4, -4) is in the fourth quadrant. So you would need to find the focus angle, i.e. for the point (4, 4), and then subtract it from 2\pi.
    So

    \theta = 2\pi -\arctan{\frac{4}{4}}

     = 2\pi -\arctan{1}

     = 2\pi -\frac{\pi}{4}

     = \frac{7\pi}{4}.
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  3. #3
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    I see. So how would that differ for r<0?
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  4. #4
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    Can you ever possibly have a negative radius?
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    Perhaps not. I just added pi to account for -r.
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    r represents the distance between a point and the origin.

    A distance, in other words, a length, can never be negative.


    I don't see why you would add \pi.
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  7. #7
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    Quote Originally Posted by mmattson07 View Post
    Hey I am not sure how to find the second coordinate of these polar coordinates.

    a) Find polar coordinates (r, θ) of the point (4, -4), where r > 0 and 0 ≤ θ ≤ 2π.

    So r= \sqrt{(4)^2+(4)^2}
    = \sqrt{32}
    =4\sqrt{2}

    I tried doing
    θ= tan^{-1}(\frac{y}{x})
    θ= tan^{-1}(\frac{-4}{4})
    θ= tan^{-1}(-1)
    θ= \frac{-\pi}{4}

    But this is incorrect?
    That's incorrect only because you specifically said " 0\le \theta\le 2\pi" and -\pi/4 is less than 0. Since a complete circle is 2\pi, adding 2\pi, -\pi/4+ 2\pi= -\pi/4+ 8\pi/4= 7\pi/4 gives you exactly the same point and is now in the correct interval: 7\pi/4 is larger than 0 and less than 2\pi.

    For r < 0 and 0 ≤ θ ≤ 2π.

    r=-4\sqrt{2}

    How do I find theta?
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