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Write the product as a sum: 6sin(47y)cos(21y) =

 May 16, 2015

Best Answer 

 #1
avatar+575 
+13

sin(x+y)=sinx*cosy+cosx*siny

sin(x-y)=sinx*cosy-cosx*siny

so, sin(x+y)+sin(x-y)=sinx*cosy+cosx*siny+sinx*cosy-cosx*siny=2sinx*cosy

    sinx*cosy=1/2[sin(x+y)+sin(x-y)]

orginal qustion=6*sin(47y)*cos(21y)=(1/2)*6*[sin(47y+21y)+sin(47y-21y)=3*[sin68y+sin26y]

                     =3sin68y+3sin26y

 May 16, 2015
 #1
avatar+575 
+13
Best Answer

sin(x+y)=sinx*cosy+cosx*siny

sin(x-y)=sinx*cosy-cosx*siny

so, sin(x+y)+sin(x-y)=sinx*cosy+cosx*siny+sinx*cosy-cosx*siny=2sinx*cosy

    sinx*cosy=1/2[sin(x+y)+sin(x-y)]

orginal qustion=6*sin(47y)*cos(21y)=(1/2)*6*[sin(47y+21y)+sin(47y-21y)=3*[sin68y+sin26y]

                     =3sin68y+3sin26y

fiora May 16, 2015
 #2
avatar+101431 
0

Very nicely done,  fiora.......!!!  Impressive....

 

 

 May 16, 2015
 #3
avatar+22358 
+10

Write the product as a sum: 6sin(47y)cos(21y) = ?

 

$$\boxed{
\small{\text{
Formula:
$
\begin{array}{rcl}
\sin u+\sin v=2\sin \frac{u+v}{2}\cos \frac{u-v}{2}
\end{array}
$}}
}$$

$$\small{\text{$
\begin{array}{rcl}
47y = \dfrac{u+v}{2} \qquad 21y=\dfrac{u-v}{2}\\\\
u=\dfrac{u+v}{2}+\dfrac{u-v}{2}=47y+21y=68y\\\\
v=\dfrac{u+v}{2}-\dfrac{u-v}{2}=47y-21y=26y\\\\
\end{array}
$}}$$

 

$$\small{\text{$
\begin{array}{rcl}
\sin u+\sin v&=&2\sin \frac{u+v}{2}\cos \frac{u-v}{2}\\\\
\sin {(68y)}+\sin {(26y)} &=&2\sin{( 47y )}\cos{( 21y )} \quad | \quad \cdot 3 \\\\
3\sin {(68y)}+3\sin {(26y)} &=&6\sin{( 47y )}\cos{( 21y )}
\end{array}
$}}$$

.
 May 16, 2015

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