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Prove that the average of the numbers 2 sin 2◦ , 4 sin 4◦ , 6 sin 6◦ , . . . , 180 sin 180◦ is cot 1◦.

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asked Sep 8, 2015 in K-12 by ans (3,000 points)

1 Answer

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we have cos (n-1) - cos (n+1) = 2 sin n sin 1 ,,,1
now 180 sin 180 = 180 sin 0
178 sin 178 = 180 sin 2
so total sum = 180 ( sin 0 + sin 2 + sin 4 + ... + sin 88) + 90 sin 90
= 90 ( 2 sin 0 + 2 sin 2 + 2 sin 4 + 2 sin 88) + 90 sin 90
as there are 90 numbers average
= 2 sin 2 + 2 sin 4 + 2 sin 88 + sin 90 as
= ( cos 1 - cos 3)/ sin 1 + (cos 3 - cos 5)/ sin 1 + ( cos 87- cos 89 / sin 1 + 1
= ( cos 1 - cos 89)/ sin 1 + 1
= cot 1 - sin 1/sin 1 + 1 as cos 89 = sin 1
= cot 1 - 1 + 1
= cot 1
answered Oct 28, 2015 by anonymous
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