Trigonometry - compound angles

用戶225241

2008-03-09 7:24 pm

ai) Show that csc C = cot (C/2) - cot C
ii) Using i, deduce the values of cot 3.14/8 and cot 5*3.14/12 in surd form
b) using a, express cscC + csc2C + csc4C + csc8C as the difference of two
contangents.
hence,prove that, without using tables or calculator, that
scs 2*3.14/5 + csc 4*3.14/5 +csc 8*3.14/5 + csc16*3.14/5 = 0

回答 (1)

魏王將張遼

2008-03-09 8:17 pm

✔ 最佳答案

(a) (i) Consider csc C + cot C = (1 + cos C)/sin C
= [1 + cos 2(C/2)]/[sin 2(C/2)]
= 2 cos2 (C/2)/[2 sin (C/2) cos (C/2)]
= cos (C/2)/sin (C/2)
= cot (C/2)
SO csc C = cot (C/2) - cot C
(ii) csc 3π/4 = cot 3π/8 - cot 3π/4
√2 = cot 3π/8 + 1
cot 3π/8 = √2 - 1
csc 5π/6 = cot 5π/12 - cot 5π/6
2 = cot 5π/12 + √3
cot 5π/12 = 2 - √3
(b) csc C = cot (C/2) - cot C
csc 2C = cot C - cot 2C
csc 4C = cot 2C - cot 4C
csc 8C = cot 4C - cot 8C
So, csc C + csc 2C + csc 4C + csc 8C = cot (C/2) - cot 8C
Put C = 2π/5:
csc 2π/5 + csc 4π/5 + csc 8π/5 + csc 16π/5 = cot π/5 - cot 16π/5
= cot π/5 - cot (3π + π/5)
= cot π/5 - cot π/5
= 0

參考: My Maths knowledge

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