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

+3
811
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Write the product as a sum: 6sin(47y)cos(21y) =

May 16, 2015

#1
+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
+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

fiora May 16, 2015
#2
+101431
0

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

May 16, 2015
#3
+22358
+10

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

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

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

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

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May 16, 2015