Math Is Fun Forum

Hopecantid

Member

Registered: 2008-04-08

Posts: 13

Finite Sum

Determine the exact numerical value of:

I'm completely lost on this.  Any help please?  It'd be much appreciated. 

Last edited by Hopecantid (2008-04-08 15:47:06)

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Finite Sum

wow, thats a fun one!

first note, we can change the initial index to zero without effecting the sum:

   

because that just adds a term of sin(0), which is zero.
Now if we multiply the sum by 2

 

note I obtained twice the sum by adding each term twice. But I arranged the second set of terms from top to bottom instead of bottom to top.

However, if you look carefully you will observe that the above sum could be written as follows:

 

now, since sin(90-k) = cos(k),  we make this substitution to get

thus you divide by 2 to get:

You should be mindful that the above summation will have 90+1 itterations, and therefore the sum is 91. So you're answer is 91/2 = 45.5.

Last edited by mikau (2008-04-08 16:07:58)

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A logarithm is just a misspelled algorithm.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Finite Sum

Ans:  45 1/2
pair them up knowing special right triangles.
89 & 1 right triangle.    smallLeg^2 + bigLeg^2 = 1 for any right triangle.
88 & 2
87 & 3
...
46 & 44
45 goes to 1/2 because sine of 45 is sqRoot2/2, so square that one, and 2/4 is 1/2.
90 goes to 1 because the unit circle is one high, that's how I learned trig, the "unit circle", radius = 1.
Add 44 ones + 1/2 + 1 = answer.

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Hopecantid

Member

Registered: 2008-04-08

Posts: 13

Re: Finite Sum

Wow, thank you guys!

You both explained it differently, but I think it actually helped me understand it a lot better.  Thank you. 

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Finite Sum

very often, its helpful to multiply the sum by 2 by adding it to itself backwards, as I did.

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A logarithm is just a misspelled algorithm.