JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Conjugate, Modulus and Argument of complex number

  • question_answer
    If \[z=1-\cos \alpha +i\sin \alpha \], then amp\[z\]=

    A)   \[\frac{\alpha }{2}\]

    B)  \[-\frac{\alpha }{2}\]

    C)   \[\frac{\pi }{2}+\frac{\alpha }{2}\]

    D)  \[\frac{\pi }{2}-\frac{\alpha }{2}\]

    Correct Answer: D

    Solution :

    amp\[(z)={{\tan }^{-1}}\frac{\sin \alpha }{1-\cos \alpha }\] \[={{\tan }^{-1}}\left( \cot \frac{\alpha }{2} \right)={{\tan }^{-1}}\left\{ \tan \left( \frac{\pi }{2}-\frac{\alpha }{2} \right) \right\}=\frac{\pi }{2}-\frac{\alpha }{2}\].


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