Trigonometry

數學小子

聽日要考啦,有幾條exercise唔多識做!
1)
Show that sinx+sin2x+sin3x+...+sinnx={sin(1/2 nx)sin[1/2 (n+1)x]}/sin(1/2 x) for all  positive integers n
2)
Express cos^n x in terms of multiple angles, for positive integer n.
3)
Express sin(nx) in terms of sinx and cosx, for positive integer n.

(第一題最好唔好用MI啦,我想睇下有冇其他方法,希望有人答到我!)

-終場ソ使者-

岩岩考完試@@
1)
Let and ,


               (geometric series)




Hence,


(PS: an extra series for cosine can be obviously found as )

2)
Since



Hence,
呢條唔多肯定,
3)(諗緊,一陣先做)

p445hkk20001

我唔多明第3條想問咩 有無中文翻譯?

-終場ソ使者-

求sin nx,答案以sinx,cosx表示
eg: sin2x=2sinxcosx

p445hkk20001

sin3X = 3sinX-4sin^3 (X)
好難寫關係@@''

順帶一提 你個簽名 即係334個網條LINK係死

[H]bunbunbunbun

third question use de moivre's theorem?

-終場ソ使者-

咁就用De Moivre's thm@@

p445hkk20001

For the sake of convenience,we set C =cosθ S=sinθ
by the  moivre's theorem,
(C+iS)^n=cosnθ+isinnθ
The binomial expansion of the left hand side is
(C+iS)^n
=C^n+  nC1 (C^n-1)(iS)+(nC2)(C^n-2)(iS)^2+(nC3)(C^n-3)(iS)^3....+....
+(nCn-1)C x (iS)^n-1   +(iS)^n
={C^n - (nC2)C^(n-2)S^2+(nC4)C^(n-4)S^4 - .......}
+i{(nC1)C^(n-1)S-(nC3)C^(n-3)S^3+(nC5)C^(n-5)S^5 -........}

equating real and imaginary parts of both sides ,we get
sinnθ=(nC1)C^(n-1)S-(nC3)C^(n-3)S^3+(nC5)C^(n-5)S^5 -........  (可能錯)

話時話m2 會考咁多i既野咩?