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Math Help - How do you write 2+3i in polar form?

  1. #1
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    How do you write 2+3i in polar form?

    How do you write 2+3i in polar form?
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  2. #2
    MHF Contributor Unknown008's Avatar
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    The polar form has the general form

    r(\cos\theta + i\sin\theta)

    Where r is the modulus of the complex number and theta the argument of the complex number.
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  3. #3
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    The complex number 2+ 3i is represented by the point (2, 3) in the complex plane. The "modulus" unknown008 refers to is the distance from (0, 0) to (2, 3) which is \sqrt{(2- 0)^2+ (3- 0)^2} and the "argument" is the angle the line from (0, 0) to (2, 3) makes with the positive x-axis. That line forms the hypotenuse of a right triangle with "opposite side" the y value, 3, and "near side" the x value, 2. The angle, \theta, is given by tan(\theta)= \frac{3}{2}.

    You can derive those relations directly from 2+ 3i= r(cos(\theta)+ i sin(\theta). We must have rcos(\theta)= 2 and rsin(\theta)= 3. Square each and add and you get r^2cos^2(\theta)+ r^2sin^2(\theta)= r^2= 2^2+ 3^2. Divide the second equation by the first and you get \frac{r sin(\theta)}{r cos(\theta)}= \frac{sin(\theta)}{cos(\theta)}= tan(\theta)= \frac{3}{2}.
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