Let k=sin 1°×sin 3°×sin 5°×…×sin 89° Find log


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Question Number 116887 by ZiYangLee last updated on 07/Oct/20

$Let k = \sin 1 ° \times \sin 3 ° \times \sin 5 ° \times \dots \times \sin 89 °$ $Find \log_{2} k^{2} .$
Commented by MJS_new last updated on 07/Oct/20

$answer is - 89$
	Answered by TANMAY PANACEA last updated on 07/Oct/20

	$89 = 1 + (n - 1) 2 \to n = 45$ $m i d d l e t e r m 23 r d t e r m = s i n 45$ $k = s i n 1 s i n 3 s i n 5 . . . . s i n 45 s i n 47 s i n 49 . . . s i n 89$ $k = (s i n 1 s i n 89) (s i n 3 s i n 87) (s i n 5 s i n 85) . . . (s i n 43 s i n 47) s i n 45$ $k \times 2^{22}$ $= (2 s i n 1 c o s 1) (2 s i n 3 c o s 3) (2 s i n 5 c o s 5) . . . (2 s i n 43 c o s 43) s i n 45$ $= (s i n 2) (s i n 6) (s i n 10) . . . (s i n 86) s i n 45$ $w a i t$
	Answered by Bird last updated on 08/Oct/20

	$k = s i n (\frac{π}{180}) . s i n (\frac{3 π}{180}) . . . s i n (\frac{89 π}{180})$ $= \prod_{k = 0}^{44} s i n (\frac{(2 k + 1) π}{180})$ ${l o g}_{2} k^{2} = \frac{{l n k}^{2}}{l n 2} = \frac{2}{l n 2} l n k$ $= \frac{2}{l n 2} \sum_{n = 0}^{44} s i n (\frac{(2 n + 1) π}{180})$ $b u t \sum_{n = 0}^{44} s i n (\frac{(2 n + 1) π}{180})$ $= I m (\sum_{n = 0}^{44} e^{\frac{i (2 n + 1) π}{180}})$ $a n d$ $\sum_{n = 0}^{44} e^{\frac{i (2 n + 1) π}{180}} = e^{\frac{i π}{180}} \sum_{n = 0}^{44} {(e^{\frac{i π}{90}})}^{n}$ $= e^{\frac{i π}{180}} \times \frac{1 - {(e^{\frac{i π}{90}})}^{45}}{1 - e^{\frac{i π}{90}}}$ $= e^{\frac{i π}{180}} \times \frac{1 - i}{1 - e^{\frac{i π}{90}}}$ $= (1 - i) e^{\frac{i π}{180}} \times \frac{1}{1 - c o s (\frac{π}{90}) - i s i n (\frac{π}{90})}$ $= (1 - i) e^{\frac{i π}{180}} \times \frac{1 - c o s (\frac{π}{90}) + i s i n (\frac{π}{90})}{{(1 + c o s (\frac{π}{90}))}^{2} + {s i n}^{2} (\frac{π}{90})}$ $n o w i t s e a z y t o e x t r a c t I m (o f$ $t h i s q u s n t i t y . . .)$
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