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Math Help - Complex Numbers

  1. #1
    Member classicstrings's Avatar
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    Complex Numbers

    Hello, im a bit stuck!

    Let z = cis theta

    Show that a. Modz = 1

    and b. 1/z = cis(-theta)

    Thanks for your help!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by classicstrings View Post
    Hello, im a bit stuck!

    Let z = cis theta

    Show that a. Modz = 1

    and b. 1/z = cis(-theta)

    Thanks for your help!
    a) You are given:

    z=cos(theta)+i.sin(theta)

    so:

    |z|=sqrt[cos^2(theta)+sin^2(theta)]

    but cos^2(theta)+sin^2(theta)=1, so |z|=1.

    b) 1/z=1/[cos(theta)+i.sin(theta)]

    now multiply top and bottom on the right by cos(theta)-i.sin(theta) to get:

    1/z=(cos(theta)-i.sin(theta)/[(cos(theta)+i.sin(theta))(cos(theta)-i.sin(theta))]

    .....=(cos(theta)-i.sin(theta)/[cos^2(theta)+sin^2(theta))]

    .....=cos(theta)-i.sin(theta)

    Now cos(theta)=cos(-theta), and sin(theta)=-sin(-theta), so putting these into the last equation above gives:

    1/z=cos(-theta)+i.sin(-theta)

    as required.

    RonL
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  3. #3
    Member classicstrings's Avatar
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    Thanks for your help CB!
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