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Math Help - CIS problems

  1. #1
    Member wiseguy's Avatar
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    Thumbs down CIS problems

    A small portion of the last chapter in my textbook is devoted to cis problems. It is very unclear about the cis trigometric function, providing no examples whatsoever.

    Here are two problems from my textbook I attempted to work on:

    "Write in a + bi form: 5 cis 300
    Round the value of a to the hundredth.

    Leave your answer in the form ___+___i"

    and

    "Multiply: (6 cis 100) (2 cis 70)
    Leave your answer in the form ___cis___"


    If I could get an explanation on these two problems, I'll (hoprefully) be able to work though the rest, and I'll have a good study session. I'm guaranteed at least one question on cis on my final.


    At first I thought the folks who wrote the book made a typo for cos, given the "I" key on the keyboard is so close to the "O" key.

    Many thanks in advance to quick help!
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  2. #2
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    Quote Originally Posted by wiseguy View Post
    A small portion of the last chapter in my textbook is devoted to cis problems. It is very unclear about the cis trigometric function, providing no examples whatsoever.

    Here are two problems from my textbook I attempted to work on:

    "Write in a + bi form: 5 cis 300
    Round the value of a to the hundredth.

    Leave your answer in the form ___+___i"

    and

    "Multiply: (6 cis 100) (2 cis 70)
    Leave your answer in the form ___cis___"


    If I could get an explanation on these two problems, I'll (hoprefully) be able to work though the rest, and I'll have a good study session. I'm guaranteed at least one question on cis on my final.


    At first I thought the folks who wrote the book made a typo for cos, given the "I" key on the keyboard is so close to the "O" key.

    Many thanks in advance to quick help!

    You have to (1) understand that cis(\theta):=\cos\theta+i\sin\theta , (2) you have to know pretty well your trigonometry and (3) you have to remember the definition a+ib=\sqrt{a^2+b^2}\,cis\theta\,,\,\,\theta=\arcta  n\frac{b}{a}\,,\,\,a,b\in\mathbb{R}\,,\,a\neq 0 , and if a=0\,,\,\,\theta=\frac{\pi}{2}+k\pi\,,\,\,k\in\mat  hbb{Z}.

    So let's do one, and you do the other one: 5\,cis(300^\circ)=5(\cos 300^\circ+i\sin 300^\circ)=5\left(\frac{1}{2}-\frac{\sqrt{3}}{2}i\right) = \frac{5}{2}-\frac{5\sqrt{3}}{2}i

    Tonio
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  3. #3
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    Quote Originally Posted by wiseguy View Post
    A small portion of the last chapter in my textbook is devoted to cis problems. It is very unclear about the cis trigometric function, providing no examples whatsoever.

    Here are two problems from my textbook I attempted to work on:

    "Write in a + bi form: 5 cis 300
    Round the value of a to the hundredth.

    Leave your answer in the form ___+___i"

    and

    "Multiply: (6 cis 100) (2 cis 70)
    Leave your answer in the form ___cis___"


    If I could get an explanation on these two problems, I'll (hoprefully) be able to work though the rest, and I'll have a good study session. I'm guaranteed at least one question on cis on my final.


    At first I thought the folks who wrote the book made a typo for cos, given the "I" key on the keyboard is so close to the "O" key.

    Many thanks in advance to quick help!
    Hi Wiseguy...

    "cis" means "cos+isin"

    so cis(\theta)=cos\theta+isin(\theta)

    For a complex number a+bi..."a" is the real part and "bi" is the imaginary part.

    5cis300^o=5\left(cos300^o+isin300^o\right)

    hence ... a=5cos300^o and b=5sin300^o


    6cis100^o2cis670^o=6\left(cos100^o+isin100^o\right  )2\left(cos70^o+isin70^o\right)

    =12\left(cos100^ocos70^o+icos100^osin70^o+icos70^o  sin100^o-sin100^osin70^o\right)

    Now combine the real parts to get "a" and the imaginary parts to get "b"
    after applying the appropriate trigonometric identities to them.
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  4. #4
    Member wiseguy's Avatar
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    Quote Originally Posted by tonio View Post
    You have to (1) understand that cis(\theta):=\cos\theta+i\sin\theta , (2) you have to know pretty well your trigonometry and (3) you have to remember the definition a+ib=\sqrt{a^2+b^2}\,cis\theta\,,\,\,\theta=\arcta  n\frac{b}{a}\,,\,\,a,b\in\mathbb{R}\,,\,a\neq 0 , and if a=0\,,\,\,\theta=\frac{\pi}{2}+k\pi\,,\,\,k\in\mat  hbb{Z}.

    So let's do one, and you do the other one: 5\,cis(300^\circ)=5(\cos 300^\circ+i\sin 300^\circ)=5\left(\frac{1}{2}-\frac{\sqrt{3}}{2}i\right) = \frac{5}{2}-\frac{5\sqrt{3}}{2}i

    Tonio
    Okay, I ended up with 12cis(-0.984807753)
    ...this is the form they were asking for in the second question: "Leave your answer in the form ___cis___"
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  5. #5
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    Quote Originally Posted by wiseguy View Post
    Okay, I ended up with 12cis(-0.984807753)
    ...this is the form they were asking for in the second question: "Leave your answer in the form ___cis___"
    From 12\left(cos100^ocos70^o-sin100^osin70^o+i[cos100^osin70^o+cos70^osin100^o]\right)

    you should use cosAcosB-sinAsinB=cos(A+B)

    and cosAsinB+cosBsinA=sin(A+B)

    where A=100^0 and B=70^o

    to get 12cis(A+B)
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  6. #6
    Member wiseguy's Avatar
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    Which equals 12cis170...?
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  7. #7
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    Quote Originally Posted by wiseguy View Post
    Which equals 12cis170...?
    Yes,

    you evaluated cos170^o=-0.9848

    However cis170^0=cos170^0+isin170^o
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