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Trigonometry - Cosec summation

  • Thread starter Saitama
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  • #1
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Homework Statement


If
$$\csc\frac{\pi}{32}+\csc\frac{\pi}{16}+\csc\frac{\pi}{8}+\csc\frac{\pi}{4}+\csc\frac{\pi}{2}$$
has the value equal to ##\cot\frac{\pi}{A}## then find A.
A)61
B)62
C)63
D)64


Homework Equations





The Attempt at a Solution


Writing cosec in terms of sin and taking the LCM to make a common denominator doesn't seem to be of any help.

I can find the value of each term but that would be tedious and of no use.

I honestly cannot figure out how should I proceed here.

Any help is appreciated. Thanks!
 
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Answers and Replies

  • #2
Dick
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Homework Helper
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618

Homework Statement


If
$$\csc\frac{\pi}{32}+\csc\frac{\pi}{16}+\csc\frac{\pi}{2}+\csc\frac{\pi}{4}+\csc\frac{\pi}{2}$$
has the value equal to ##\cot\frac{\pi}{A}## then find A.
A)61
B)62
C)63
D)64


Homework Equations





The Attempt at a Solution


Writing cosec in terms of sin and taking the LCM to make a common denominator doesn't seem to be of any help.

I can find the value of each term but that would be tedious and of no use.

I honestly cannot figure out how should I proceed here.

Any help is appreciated. Thanks!
Try showing cot(x)-cot(2x)=csc(2x).
 
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  • #3
Simon Bridge
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##\csc(\frac{\pi}{2})## appears twice.
All the angles are successive doublings of ##\pi/32##
 
  • #4
3,812
92
Try showing cot(x)-cot(2x)=csc(2x).
Wow! Thanks a lot Dick! :smile:

How did you come up with that?

##\csc(\frac{\pi}{2})## appears twice.
All the angles are successive doublings of ##\pi/32##
Very sorry for the typo, its ##\pi/8## instead of the second ##\pi/2##.
 
  • #5
Dick
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Wow! Thanks a lot Dick! :smile:

How did you come up with that?
I guessed the series must telescope somehow. So somehow cot(2x) must be related to cot(x) with the difference related to a csc. Seems obvious in retrospect, yes?
 
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  • #6
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92
Seems obvious in retrospect, yes?
Yes. I liked the way you came up with cot(x)-cot(2x) and solved the problem in few seconds where I was stuck for a week.

Thank you again! :)
 
  • #7
Dick
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Yes. I liked the way you came up with cot(x)-cot(2x) and solved the problem in few seconds where I was stuck for a week.

Thank you again! :)
You're welcome, but it took me more than a "few seconds". Still keeping that strategy in mind might help in the future. If you've got the sum of a bunch of csc's equaling a cot, then if you can express each csc as a difference of two cot's you might be able to sum the series easily. Substitute any other functions you want for 'csc' and 'cot'.
 
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