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Math Help - [SOLVED] How can i show this, trig identity

  1. #1
    Junior Member maths900's Avatar
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    [SOLVED] How can i show this, trig identity

    \frac{sin\theta}{1-cos\theta}=\frac{1}{tan(\theta/2)}
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by maths900 View Post
    \frac{sin\theta}{1-cos\theta}=\frac{1}{tan(\theta/2)}
    Do you know about Weierstrass substitution ?

    Let t=\tan (\theta/2)

    Then \sin(\theta)=\frac{2t}{1+t^2} and \cos(\theta)=\frac{1-t^2}{1+t^2}



    Otherwise... :
    \frac{1}{\tan(\theta/2)}=\frac{1}{\frac{\sin(\theta/2)}{\cos(\theta/2)}}=\frac{\cos(\theta/2)}{\sin(\theta/2)}

    Now the RHS :
    \frac{\sin \theta}{1-\cos \theta}=\frac{2 \sin(\theta/2)\cos(\theta/2)}{1-(1-2 \sin^2(\theta/2))}=\frac{2 \sin(\theta/2)\cos(\theta/2)}{2 \sin^2(\theta/2)}=\dots
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  3. #3
    Junior Member maths900's Avatar
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    Moo, for the last bit, did you use the double angle formulas

    sin(2\theta)=2sin(\theta)cos(\theta)

    and cos(2\theta)=1-2sin^2(\theta)
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  4. #4
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    Hello, maths900!


    We're expected to know the Half-Angle Identities:

    . . \sin^2\tfrac{\theta}{2} \:=\:\frac{1-\cos\theta}{2} \quad\Rightarrow\quad 1 - \cos \:=\:2\sin^2\!\tfrac{\theta}{2}

    . . \sin\theta \:=\:2\sin\tfrac{\theta}{2} \cos\tfrac{\theta}{2}



    \frac{\sin\theta}{1-\cos\theta}\:=\:\frac{1}{\tan\frac{\theta}{2}}

    The left side is: . \frac{\sin\theta}{1-\cos\theta} \;=\;\frac{2\sin\frac{\theta}{2}\cos\frac{\theta}{  2} }{2\sin^2\frac{\theta}{2}} \;=\;\frac{\cos\frac{\theta}{2}}{\sin\frac{\theta}  {2}}<br />

    . . . . . = \;\frac{1}{\left(\dfrac{\sin\frac{\theta}{2}}{\cos  \frac{\theta}{2}}\right)} \;=\;\frac{1}{\tan\frac{\theta}{2}}

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  5. #5
    Junior Member maths900's Avatar
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    oh ok lol thanks
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  6. #6
    Moo
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    Quote Originally Posted by maths900 View Post
    Moo, for the last bit, did you use the double angle formulas

    sin(2\theta)=2sin(\theta)cos(\theta)

    and cos(2\theta)=1-2sin^2(\theta)
    Yes :P

    I prefer remembering them this way rather than "half-angle identities" :P
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  7. #7
    Junior Member maths900's Avatar
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    Quote Originally Posted by Moo View Post
    Yes :P

    I prefer remembering them this way rather than "half-angle identities" :P
    lol me too.
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