Relation between root and coeff @ 飛航百科

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Relation between root and coeff

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(a) cot6α= ( cot 6 α-15 cot4α+15cot2α-1)/(6 cot 5α- 20 cot3α+6 cot α)(I can do this )(b)(i)Put α= ? to (a) then replace cot α by x , we get x 6 - 6x5 cot 6? – 15x4 +20 x3 cot 6? +15x2-6x cot 6? -1=0 by considering the above eqt , prove Σ (k=0 to 5)cot (?+k(pi)/6)= 6 cot 6?(I know using sum of roots... 顯示更多 (a) cot6α= ( cot 6 α-15 cot4α+15cot2α-1)/(6 cot 5α- 20 cot3α+6 cot α) (I can do this ) (b)(i)Put α= ? to (a) then replace cot α by x , we get x 6 - 6x5 cot 6? – 15x4 +20 x3 cot 6? +15x2-6x cot 6? -1=0 by considering the above eqt , prove Σ (k=0 to 5)cot (?+k(pi)/6)= 6 cot 6? (I know using sum of roots can get RHS, how about LHS?) (ii) Hence, deduce Σ (k=1 to 5) csc2 (? +k(pi)/6)= 36 csc2 6? -csc2? (iii) Find Σ(k=1 to 5)cot 2(4k+3)(pi)/24

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難度是有﹐不過還是比較好做﹐因為有些更難做的 (a) cot6α= ( cot 6 α-15 cot4α+15cot2α-1)/(6 cot 5α- 20 cot3α+6 cot α) (b)(i)Put α= ? to (a) then replace cot α by x , we get x 6 - 6x5 cot 6? – 15x4 +20 x3 cot 6? +15x2-6x cot 6? -1=0 Σ (k=0 to 5)cot (?+k(pi)/6)= 6 cot 6? (ii) Hence, deduce Σ (k=1 to 5) csc2 (? +k(pi)/6)= 36 csc2 6? -csc2? (iii) Find Σ(k=1 to 5)cot 2(4k+3)(pi)/24 SOLUTION (a) Have been done (b) (i) Put α= ? cot6?= ( cot 6 ?-15 cot4?+15cot2?-1)/(6 cot 5?- 20 cot3?+6 cot ?) Put cot α by x cot6?= ( x^ 6-15 x^4+15 x^2-1)/(6 x^5- 20 x^3+6 x) So (cot6?)(6 x^5- 20 x^3+6 x)= ( x^ 6-15 x^4+15 x^2-1) Or x 6 - 6x5 cot 6? – 15x4 +20 x3 cot 6? +15x2-6x cot 6? -1=0 Since x=cot? is a root of this equation. But using fundamental theorem of algebra, there are 6 roots So we want to find 5 more roots. We notice that cot 6(?+k(pi)/6)=cot (6?+k(pi))=cot 6? So cot(?+k(pi)/6) k=1 to 5 is the other 5 roots You can just think that you want to find a value such that cot6x=0 if ? is one then the other five is ?+k(pi)/6 (ii) Σ (k=0 to 5)cot (?+k(pi)/6)= 6 cot 6? [Σ (k=0 to 5)cot (?+k(pi)/6)]^2= 36 cot^2 6? Σ (k=0 to 5) cot2 (? +k(pi)/6)-30= 36 cot2 6? Since csc^2x=cot^2x+1 Σ (k=0 to 5) [csc2 (? +k(pi)/6)-1]-30= 36 (csc2 6?-1) Σ (k=1 to 5) csc2 (? +k(pi)/6)= 36 csc2 6?-csc^2? (iii) sub ?=3pi/24 Σ (k=0 to 5) cot2 ((4k+3)(pi)/24) -30= 36 cot2 (3/4)pi=36 Σ (k=0 to 5) cot2 ((4k+3)(pi)/24) =6 Σ(k=1 to 5)cot 2(4k+3)(pi)/24=6-cot^2(pi/8)=6-[1/(√2-1)]^2=66-(3+2√2)=63-2√2

其他解答:

(b) (i) 圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Jan08/tinhoi5.jpg (ii) 圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Jan08/tinhoi6.jpg (iii) 圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Jan08/tinhoi7.jpg