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Thread: Deduce the value in surd form

  1. #1
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    Deduce the value in surd form

    If $\displaystyle A=36^o$, show that $\displaystyle \sin 3A= \sin 2A$, and deduce that $\displaystyle \cos 36^o=\frac{\sqrt{5}+1}{4}$

    I have already shown $\displaystyle \sin 3A= \sin 2A$ when $\displaystyle A=36^o$, since $\displaystyle 108^o=180^o-72^o$.
    But I don't know how to deduce that $\displaystyle \cos 36^o=\frac{\sqrt{5}+1}{4}$.
    Thanks!
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  2. #2
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    Hello arze
    Quote Originally Posted by arze View Post
    If $\displaystyle A=36^o$, show that $\displaystyle \sin 3A= \sin 2A$, and deduce that $\displaystyle \cos 36^o=\frac{\sqrt{5}+1}{4}$

    I have already shown $\displaystyle \sin 3A= \sin 2A$ when $\displaystyle A=36^o$, since $\displaystyle 108^o=180^o-72^o$.
    But I don't know how to deduce that $\displaystyle \cos 36^o=\frac{\sqrt{5}+1}{4}$.
    Thanks!
    Four steps:

    • Use the identities $\displaystyle \sin2A=2\sin A \cos A$ and $\displaystyle \sin3A=3\sin A -4\sin^3A$ in the equation $\displaystyle \sin 3A= \sin 2A$


    • Divide through by $\displaystyle \sin A$ (it's non-zero, so that's OK)


    • Replace $\displaystyle \sin^2A$ by $\displaystyle 1 - \cos^2A$


    • Solve the quadratic for $\displaystyle \cos A$, taking the positive root.

    Can you complete it now?

    Grandad
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  3. #3
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    Yes! i can complete it now! thanks
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