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Math Help - Complex number easy inequality

  1. #1
    Member mabruka's Avatar
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    Complex number easy inequality

    I was looking through a proof and i stumbled upon this inequality :

     | e^{ia}-1 | \leq |a |

    a \in \mathbb{R}

    but i cannot convince myself why it is true.

    Can you think about any hint?


    thnx
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  2. #2
    Member mabruka's Avatar
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    I tried looking at it from this point but i am not sure how to involve |a|

    | e^{ia}-1 | = \sqrt{(cos(a) -1)^2 + sen^2(a) } = \sqrt{cos^2(a) - 2cos(a)+1 +sen^2(a)}
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mabruka View Post
    I was looking through a proof and i stumbled upon this inequality :

     | e^{ia}-1 | \leq |a |

    a \in \mathbb{R}

    but i cannot convince myself why it is true.

    Can you think about any hint?


    thnx
    For a second suppose that a\geqslant 0 then \left|e^{ia}-1\right|=\sqrt{\left(\cos(a)-1\right)^2+\sin^2(a)}= \sqrt{\cos^2(a)+\sin^2(a)+1-2\cos(a)}=\sqrt{2}\sqrt{1-\cos(a)}=2\sin\left(\frac{a}{2}\right)\leqslant 2\frac{a}{2}=a
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  4. #4
    Member mabruka's Avatar
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    Thank you! its clear now
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