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Math Help - Trig Identity

  1. #1
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    Trig Identity

    Verify:
    Sin(α/3)Cos(α/3)=.5Sin(2α/3)
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  2. #2
    Senior Member eumyang's Avatar
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    Use the sine double-angle identity.
    \begin{aligned}<br />
\tfrac{1}{2}\sin \left(\tfrac{2\alpha}{3} \right) &= \tfrac{1}{2}\sin \left(2 \cdot \tfrac{\alpha}{3} \right) \\<br />
&= \tfrac{1}{2} \cdot 2 \sin \left(\tfrac{\alpha}{3} \right) \cos \left(\tfrac{\alpha}{3} \right) \\<br />
&= \sin \left(\tfrac{\alpha}{3} \right) \cos \left(\tfrac{\alpha}{3} \right)<br />
\end{aligned}
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  3. #3
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    Quote Originally Posted by eumyang View Post
    Use the sine double-angle identity.
    \begin{aligned}<br />
\tfrac{1}{2}\sin \left(\tfrac{2\alpha}{3} \right) &= \tfrac{1}{2}\sin \left(2 \cdot \tfrac{\alpha}{3} \right) \\<br />
&= \tfrac{1}{2} \cdot 2 \sin \left(\tfrac{\alpha}{3} \right) \cos \left(\tfrac{\alpha}{3} \right) \\<br />
&= \sin \left(\tfrac{\alpha}{3} \right) \cos \left(\tfrac{\alpha}{3} \right)<br />
\end{aligned}
    In the second step how did you get Cos to appear?
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  4. #4
    Senior Member eumyang's Avatar
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    Like I said, I used the sine double angle identity:
    \sin\,2\theta = 2\sin\,\theta\,\cos\,\theta.
    You should have learned this. If not, you could still verify the problem using the sine of a sum identity instead:
    \sin\,(u + v) = \sin\,u\,\cos\,v + \cos\,u\,\sin\,v.
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