Cho alpha+ beta+ gamma = pi. Chứng minh A) sin alpha + sin beta + sin gamma = 4cos alpha/2*cos beta/2*cos gamma /2

Question

Cho alpha+ beta+ gamma = pi. Chứng minh A) sin alpha + sin beta + sin gamma = 4cos alpha/2*cos beta/2*cos gamma /2

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Verity 6 months 2021-08-31T10:16:07+00:00 1 Answers 0 views 0

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    2021-08-31T10:17:09+00:00

    Answer:

    It is proved .

    Step-by-step explanation:

    \alpha +\beta +\gamma =\pi.....(1)\\\\sin\alpha + sin\beta +sin\gamma=4 cos\frac{\alpha}{2}cos\frac{\beta}{2}cos\frac{\gamma}{2}

    Take LHS

    =sin\alpha + sin\beta +sin\gamma\\\\=2 sin \frac{\alpha +\beta }{2} cos\frac{\alpha -\beta }{2} + sin\gamma\\\\=2 sin\frac{180-\gamma}{2}cos \frac{\alpha - \beta}{2}+2 sin\frac{\gamma}{2}cos\frac{\gamma}{2}\\\\=2 cos\frac{\gamma}{2}cos \frac{\alpha - \beta}{2}+2 sin\frac{\gamma}{2}cos\frac{\gamma}{2}\\\\=2 cos\frac{\gamma}{2}\left [ cos \frac{\alpha - \beta}{2}+sin\frac{180-\alpha-\beta}{2}\\ \right ]\\\\=2 cos\frac{\gamma}{2}\times 2 cos\frac{\alpha}{2}cos\frac{\beta}{2}\\

    =4 cos\frac{\alpha}{2}cos\frac{\beta}{2}cos\frac{\gamma}{2}

    Hence proved.

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