567 = 500+60+7
4*567 = 4*(500+60*7)
4*567 = 4*500 + 4*60 + 4*7 ... see note below
4*567 = 2000 + 240 + 28
4*567 = 2268
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note: multiply the outer term 4 by each term inside the parenthesis to use the distributive property. The general distributive property is a*(b+c) = a*b+a*c. This can be extended to a*(b+c+d) = a*b+a*c+a*d. You can have as many terms as you like inside the parenthesis.
We'll use the following properties of sine and cosine to prove this:
![sin(a)\cdot sin(b) = \frac12(cos(a-b) - cos(a+b))](https://tex.z-dn.net/?f=sin%28a%29%5Ccdot%20sin%28b%29%20%3D%20%5Cfrac12%28cos%28a-b%29%20-%20cos%28a%2Bb%29%29)
![cos(a)\cdot cos(b) = \frac12(cos(a+b) + cos(a-b))](https://tex.z-dn.net/?f=cos%28a%29%5Ccdot%20cos%28b%29%20%3D%20%5Cfrac12%28cos%28a%2Bb%29%20%2B%20cos%28a-b%29%29)
![cos(180+a) = cos(180-a)](https://tex.z-dn.net/?f=cos%28180%2Ba%29%20%3D%20cos%28180-a%29)
Then it's just a matter of filling it in...
sin20sin40 * sin60sin80 = 1/2(cos20 - cos60) * 1/2 (cos20 - cos140) =
1/8( cos40 + 1 - cos160 - cos120 - cos40 - cos80 + cos80 + cos200) =
1/8(1 - cos160 - cos120 + cos200) =
1/8(1 - cos160 - cos120 + cos160) =
1/8(1 - cos120 ) = 1/8( 1 + 1/2 ) = 3/16
Multiply by 0.4 on both sides to cancel the original 0.4 and change the 10
Answer:
pattern
Step-by-step explanation:
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