9.Find the complex number with the given modulus and argument ,in the form x + iy.  a) Express z= -2-2(2¡)^1/2 [it's root] in polar form.?

2 Answers

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  • z = -2 - 2 * sqrt(2i)

    z = -2 - 2 * sqrt(2) * sqrt(i)

    Let's find the square root of i first

    i^(1/2) =>

    (0 + i)^(1/2) =>

    (cos(pi/2 + 2pi * k) + i * sin(pi/2 + 2pi * k))^(1/2) =>

    (cos((pi/2) * (1 + 4k)) + i * sin((pi/2) * (1 + 4k)))^(1/2) =>

    cos((pi/2) * (1/2) * (1 + 4k)) + i * sin((1/2) * (pi/2) * (1 + 4k)) =>

    cos((pi/4) * (1 + 4k)) + i * sin((pi/4) * (1 + 4k))

    k is an integer.  Let's go with the principal value, so let k = 0

    cos(pi/4) + i * sin(pi/4) =>

    (sqrt(2)/2) + i * (sqrt(2)/2) =>

    (sqrt(2)/2) * (1 + i)

    z = -2 - 2 * sqrt(2) * sqrt(i)

    z = -2 - 2 * sqrt(2) * (sqrt(2)/2) * (1 + i)

    z = -2 - 2 * (1 + i)

    z = -2 - 2 - 2i

    z = -4 - 2i

    Modulus

    sqrt((-4)^2 + (-2)^2) = sqrt(16 + 4) = sqrt(20) = 2 * sqrt(5)

    Argument

    2 * sqrt(5) * cos(t) = -4

    2 * sqrt(5) * sin(t) = -2

    sin(t) = -2 / (2 * sqrt(5))

    cos(t) = -4 / (2 * sqrt(5))

    tan(t) = 1/2

    t = arctan(1/2) + pi * k.  Again, k is an integer.

    2 * sqrt(5) and the angle is arctan(1/2) + pi * k

  • rotchm
    Lv 7
    4 weeks ago

    Ambiguous. What does " z= -2-2(2¡)^1/2 [it's root] " mean? 

    Take your time and use the keyboard correctly. 

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