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Math Help - Identity

  1. #1
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    Identity

    sinA / (1+cosA) = tan 0.5A
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  2. #2
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    Hello, Punch!

    We need a few identities: . \begin{array}{cccccccccc}\sin^2\!\frac{A}{2} &=& \dfrac{1-\cos A}{2} && \Rightarrow && \sin\frac{A}{2} &=& \sqrt{\dfrac{1-\cos A}{2}} \\ \\[-3mm]<br />
\cos^2\!\frac{A}{2} &=& \dfrac{1+\cos A}{2}  && \Rightarrow && \cos\frac{A}{2} &=& \dfrac{1+\cos A}{2}\end{array}


    Prove: . \frac{\sin A}{1+\cos A}\: =\: \tan \tfrac{A}{2}

    The right side is: . \tan\frac{A}{2} \;=\;\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} \;=\;\frac{\sqrt{\dfrac{1-\cos A}{2}}}{\sqrt{\dfrac{1+\cos A}{2}}}  \;=\;\sqrt{\frac{1-\cos A}{1+\cos A}}

    Multiply by \frac{1+\cos A}{1 + \cos A}:\quad\sqrt{\frac{1-\cos A}{1+\cos A}\cdot\frac{1+\cos A}{1+\cos A}} \;=\;\sqrt{\frac{1-\cos^2\!A}{(1+\cos A)^2}} \;=\;\sqrt{\frac{\sin^2\!A}{(1+\cos A)^2}}

    Therefore: . \tan\frac{A}{2} \;=\;\frac{\sin A}{1 + \cos A}

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, Punch!

    We need a few identities: . \begin{array}{cccccccccc}\sin^2\!\frac{A}{2} &=& \dfrac{1-\cos A}{2} && \Rightarrow && \sin\frac{A}{2} &=& \sqrt{\dfrac{1-\cos A}{2}} \\ \\[-3mm] \cos^2\!\frac{A}{2} &=& \dfrac{1+\cos A}{2} && \Rightarrow && \cos\frac{A}{2} &=& \dfrac{1+\cos A}{2}\end{array}" alt="
    \cos^2\!\frac{A}{2} &=& \dfrac{1+\cos A}{2} && \Rightarrow && \cos\frac{A}{2} &=& \dfrac{1+\cos A}{2}\end{array}" />


    The right side is: . \tan\frac{A}{2} \;=\;\frac{\sin\frac{A}{2}}{\cos\frac{A}{2}} \;=\;\frac{\sqrt{\dfrac{1-\cos A}{2}}}{\sqrt{\dfrac{1+\cos A}{2}}} \;=\;\sqrt{\frac{1-\cos A}{1+\cos A}}

    Multiply by \frac{1+\cos A}{1 + \cos A}:\quad\sqrt{\frac{1-\cos A}{1+\cos A}\cdot\frac{1+\cos A}{1+\cos A}} \;=\;\sqrt{\frac{1-\cos^2\!A}{(1+\cos A)^2}} \;=\;\sqrt{\frac{\sin^2\!A}{(1+\cos A)^2}}

    Therefore: . \tan\frac{A}{2} \;=\;\frac{\sin A}{1 + \cos A}
    Hello! Thanks a lot thank you very much!
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  4. #4
    Super Member 11rdc11's Avatar
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    lol Soroban, guess you can explain it better than me.

    http://www.mathhelpforum.com/math-he...-identity.html
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  5. #5
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    Just a question, do we need the memorize all the formulas, or the formulas are given in the formula sheet during the examination?
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  6. #6
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by Punch View Post
    Just a question, do we need the memorize all the formulas, or the formulas are given in the formula sheet during the examination?
    It up to the teacher. He or she may let you use a formula sheet or you may have to memorize them. You could also learn to derive them.
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