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Topic: Broken Calculator (Read 967 times) |
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FiBsTeR
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Broken Calculator
« on: May 16th, 2008, 3:23pm » |
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A calculator is broken so that the only keys that still work are the sin, cos, tan, arcsin, arccos, and arctan buttons. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.
Source: USAMO
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Eigenray
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Re: Broken Calculator
« Reply #1 on: May 18th, 2008, 10:55am » |
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In fact, you can compute sqrt(q) for any q.
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Barukh
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Re: Broken Calculator
« Reply #2 on: May 18th, 2008, 11:01am » |
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on May 18th, 2008, 10:55am, Eigenray wrote:| In fact, you can compute sqrt(q) for any q. |
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That's what I arrived at also.
Does that mean the problem at USAMO was invalid?
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FiBsTeR
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Re: Broken Calculator
« Reply #3 on: May 18th, 2008, 11:14am » |
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on May 18th, 2008, 11:01am, Barukh wrote:| Does that mean the problem at USAMO was invalid? |
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It's not invalid, it's just not asking for the most specific set of numbers that you can attain with this calculator. They may have thought that asking about the square roots of the rationals would have hinted at the proof.
EDIT: The fact that Eigenray hid that part of the statement suggests that this may have been the reason behind it.
Good job in your solutions, by the way. 
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| « Last Edit: May 18th, 2008, 11:16am by FiBsTeR » |
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Barukh
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Re: Broken Calculator
« Reply #4 on: May 18th, 2008, 9:45pm » |
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Yes, of course! The queastion in my previous post is a complete nonsense.
Here’s my solution (in short):
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I will use short notations S for sine, S-1 for arcsine, and similarly for cosine and tangent functions.
Using the following formulas, we obtain the transformations:
TS-1TC-1(z) = 1/z
CT-1( z) = 1/(z+1)
We also have SC-1(0) = 1.
It is then sufficient to use the fact that every rational number can be represented as a finite simple continued fraction.
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| « Last Edit: May 18th, 2008, 11:28pm by Barukh » |
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ThudnBlunder
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Re: Broken Calculator
« Reply #5 on: May 19th, 2008, 5:00pm » |
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Using only the addition, multiplication, division, square root, and '2' key (plus Store, Recall, and Sum), how can you approximate  to any required precision?
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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FiBsTeR
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Re: Broken Calculator
« Reply #6 on: May 19th, 2008, 5:31pm » |
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on May 18th, 2008, 9:45pm, Barukh wrote:
I'm assuming by your notation that I apply arccos to z first? But then how do you know that z is in the domain of arccos?
on May 18th, 2008, 9:45pm, Barukh wrote:| We also have SC-1(0) = 1. |
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C(0)=1, as well. 
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Barukh
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Re: Broken Calculator
« Reply #7 on: May 20th, 2008, 6:31am » |
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on May 19th, 2008, 5:31pm, FiBsTeR wrote:| I'm assuming by your notation that I apply arccos to z first? But then how do you know that z is in the domain of arccos? |
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Not only that, the transformation is not correct!
The correct one IMHO is TS-1CT-1.
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Barukh
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Re: Broken Calculator
« Reply #8 on: May 20th, 2008, 6:40am » |
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on May 19th, 2008, 5:00pm, ThudanBlunder wrote:| Using only the addition, multiplication, division, square root, and '2' key (plus Store, Recall, and Sum), how can you approximate to any required precision? |
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(69)?
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ThudnBlunder
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Re: Broken Calculator
« Reply #9 on: May 20th, 2008, 8:27am » |
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on May 20th, 2008, 6:40am, Barukh wrote:
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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Barukh
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Re: Broken Calculator
« Reply #10 on: May 20th, 2008, 10:46am » |
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on May 20th, 2008, 8:27am, ThudanBlunder wrote:
Vieta!
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ThudnBlunder
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Re: Broken Calculator
« Reply #11 on: May 20th, 2008, 11:00am » |
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on May 20th, 2008, 6:40am, Barukh wrote:
on May 20th, 2008, 10:46am, Barukh wrote:
I knew you were well-informed, Barukh, but surely you are not also familiar with Viete's sexual proclivities? 
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Barukh
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Re: Broken Calculator
« Reply #12 on: May 22nd, 2008, 2:14am » |
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on May 19th, 2008, 5:00pm, ThudanBlunder wrote:| Using only the addition, multiplication, division, square root, and '2' key (plus Store, Recall, and Sum), how can you approximate to any required precision? |
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BTW, according to the following source, formula (69), division is not needed.
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FiBsTeR
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Re: Broken Calculator
« Reply #13 on: May 23rd, 2008, 8:34am » |
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My original thought was using (6 in your link, but I couldn't do it without using a second store variable.
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ThudnBlunder
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Re: Broken Calculator
« Reply #14 on: May 23rd, 2008, 8:54am » |
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on May 23rd, 2008, 8:34am, FiBsTeR wrote:| My original thought was using (6 in your link, but I couldn't do it without using a second store variable. |
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That was the formula I had in mind. It, together with the required iteration, can be found here on page 140.
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