數學知識交流---PolyU math 8~9JAN2011

2011-11-08 1:32 am

8JAN
find sec(pi/7)sec(2pi/7)sec(3pi/7)

9JAN
x liters of mixture A and y liters of mixture B are poured a task to from a final mixture C.
The ratios of volume of juice to that of water is mixture A,B&C are 1:3,2:3&1:2 respectively.
If x and y are integers,find the least [two] positive value x+y.

回答 (1)

用戶2053

2011-11-08 3:26 am

✔ 最佳答案

8JANsec(π/7) sec(2π/7) sec(3π/7)= 1 / ( cos(π/7) cos(2π/7) cos(3π/7) )= - 1 / ( cos(π/7) cos(2π/7) cos(4π/7) )= - 2 sin(π/7) / ( 2 sin(π/7) cos(π/7) cos(2π/7) cos(4π/7) )= - 2 sin(π/7) / ( sin(2π/7) cos(2π/7) cos(4π/7) )= - 4 sin(π/7) / ( 2 sin(2π/7) cos(2π/7) cos(4π/7) )= - 4 sin(π/7) / ( sin(4π/7) cos(4π/7) )= - 8 sin(π/7) / ( 2 sin(4π/7) cos(4π/7) )= - 8 sin(π/7) / sin(8π/7) = 8
9JAN(1/3)x + (2/3)y = (1/2)(x + y)x = yThe least [two] positive value x + y = 1 + 1 = 2 and 2 + 2 = 4 .

2011-11-07 20:41:11 補充：
Corrections :

9JAN

( 1 / (3+1) )x + ( 2 / (2+3) )y= ( 1 / (2+1) ) (x + y)

x/4 + 2y/5 = (x + y)/3

15x + 24y = 20(x + y)

5x = 4y

So

x = 4k (where k is an integer)

y = 5k (where k is an integer)

The least [two] positive value x + y = 4 + 5 = 9 and 8 + 10 = 18.

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