Prostaferese

por Rbragaf Sex 08 Mar 2024, 16:01

Se a+b+c=0, determine m tal que a identidade é válida:

sena + senb + senc = msen(a/2)sen(b/2)sen(c/2)

Resp.: -4

por **Giovana Martins** Sex 08 Mar 2024, 17:35

[latex]\mathrm{Dado\ que\ a+b+c=0,logo,a=-b-c.\ Assim:}[/latex]

[latex]\mathrm{sin(a)+sin(b)+sin(c)=sin(-b-c)+sin(b)+sin(c)}[/latex]

[latex]\mathrm{A\ func\tilde{a}o\ seno\ \acute{e}\ impar,logo,sin(-x)=-sin(x).\ Portanto:}[/latex]

[latex]\mathrm{sin(a)+sin(b)+sin(c)=sin(b)+sin(c)-sin(b+c)}[/latex]

[latex]\mathrm{Por\ Prostaf\acute{e}rese:sin(x)+sin(y)=2sin\left ( \frac{x+y}{2} \right )cos\left ( \frac{x-y}{2} \right )}[/latex]

[latex]\mathrm{sin(a)+sin(b)+sin(c)=2sin\left ( \frac{b+c}{2} \right )cos\left ( \frac{b-c}{2} \right )-sin(b+c)}[/latex]

[latex]\mathrm{Do\ arco\ duplo\ tem-se:sin(x)=2sin\left ( \frac{x}{2} \right )cos\left ( \frac{x}{2} \right )\ tal\ que:}[/latex]

[latex]\mathrm{sin(a)+sin(b)+sin(c)=2sin\left ( \frac{b+c}{2} \right )cos\left ( \frac{b-c}{2} \right )-2sin\left ( \frac{b+c}{2} \right )cos\left ( \frac{b+c}{2} \right )}[/latex]

[latex]\mathrm{Por\ Prostaf\acute{e}rese:cos(x)-cos(y)=2sin\left ( \frac{x+y}{2} \right )sin\left ( \frac{y-x}{2} \right ).\ Portanto:}[/latex]

[latex]\mathrm{sin(a)+sin(b)+sin(c)=2sin\left ( \frac{b+c}{2} \right )\left [ 2sin\left ( \frac{b}{2} \right )sin\left ( \frac{c}{2} \right ) \right ]}[/latex]

[latex]\mathrm{Dado\ que\ a=-b-c,tem-se:sin\left ( \frac{b+c}{2} \right )=sin\left ( -\frac{a}{2} \right )=-sin\left ( \frac{a}{2} \right )\ tal\ que:}[/latex]

[latex]\mathrm{sin ( a )+sin( b )+sin ( c )= - 4 sin\left ( \frac{a}{2} \right ) sin\left ( \frac{b}{2} \right ) sin\left ( \frac{c}{2} \right ),\ o\ que\ acarreta\ m = - 4 }[/latex]

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