Evaluate the square of csc 22.5∘in surd form
回答 (3)
2007-06-09 3:05 am
✔ 最佳答案
cos2x = 1 - 2 sin^2 xcos45∘ = 1 - 2 sin^2 22.5∘
1/sqrt(2) = 1 - 2 sin^2 22.5∘
sin^2 22.5∘= 1/2 ( 1 - 1/sqrt(2) )
sin 22.5∘= sqrt[ 1/2 ( 1 - 1/sqrt(2) ) ]
2007-06-08 20:48:06 補充：
做少左, 補返比你csc^2 22.5∘= 2sqrt(2) / ( sqrt(2) - 1)= 2sqrt(2)( sqrt(2) + 1) / 1= 4 + 2sqrt(2)
2007-06-09 3:16 am
cos45 = cos2(22.5)
cos45 = 1 – 2sin222.5
√2/2 = 1 - 2 sin222.5
2 sin222.5 = 1 - √2/2
sin222.5 = (1 - √2/2)/2
sin222.5 = (2 - √2)/4
sin22.5 =√(2 - √2)/2
csc22.5 = 2 / √(2 - √2)
= 2√(2 - √2) / (2 - √2)
= 2(2 + √2)√(2 - √2) / (4 - 2)
= (2 + √2)√(2 - √2)
= √(2 + √2) √[(2 - √2)(2 + √2)]
csc22.5 = √(2 + √2)√2
cos45 = 1 – 2sin222.5
√2/2 = 1 - 2 sin222.5
2 sin222.5 = 1 - √2/2
sin222.5 = (1 - √2/2)/2
sin222.5 = (2 - √2)/4
sin22.5 =√(2 - √2)/2
csc22.5 = 2 / √(2 - √2)
= 2√(2 - √2) / (2 - √2)
= 2(2 + √2)√(2 - √2) / (4 - 2)
= (2 + √2)√(2 - √2)
= √(2 + √2) √[(2 - √2)(2 + √2)]
csc22.5 = √(2 + √2)√2
2007-06-09 3:07 am
cosec²22.5°
= 1 / sin²22.5°
= 1 / {1/2 [1 - cos2(22.5°)]}
= 2 / (1 - cos45°)
= 2 / (1 - √2 / 2)
= 2 / [(2 - √2) / 2]
= 4 / (2 - √2) X (2 + √2)/(2 + √2) (Rationalize the answer.)
= 4(2 + √2) / [(2)² - (√2)²]
= 4(2 + √2) / 2
= 2(2 + √2)
= 1 / sin²22.5°
= 1 / {1/2 [1 - cos2(22.5°)]}
= 2 / (1 - cos45°)
= 2 / (1 - √2 / 2)
= 2 / [(2 - √2) / 2]
= 4 / (2 - √2) X (2 + √2)/(2 + √2) (Rationalize the answer.)
= 4(2 + √2) / [(2)² - (√2)²]
= 4(2 + √2) / 2
= 2(2 + √2)
參考: Myself~~~
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