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A calculator is broken so that the only keys that still work are the sin, cos, tan, cot, asin, acos, and atan buttons. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

(a) Find, with proof, a sequence of buttons that will transform x into 1/x.

(b) Find, with proof, a sequence of buttons that will transform sqrt(x) into sqrt(x+1).

(c) The display initially shows 0. Prove that there is a sequence of buttons that will produce 3/sqrt(5).

Davis Jul 2, 2019

#1**+5 **

A calculator is broken so that the only keys that still work are the sin, cos, tan, cot, asin, acos, and atan buttons. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

(a) Find, with proof, a sequence of buttons that will transform x into 1/x.

(b) Find, with proof, a sequence of buttons that will transform sqrt(x) into sqrt(x+1).

(c) The display initially shows 0. Prove that there is a sequence of buttons that will produce 3/sqrt(5).

Provided x can be entered, and

if the calculator uses the reverse polish notation then:

a) \( \frac{1}{x}\) \(x \Rightarrow atan \Rightarrow cot\) \(\frac{1}{x}=cot(atan (x)) \) \(x \in \{x|x\in\mathbb {R}\}\)

Works with DEG and RAD.

I pass.

!

asinus Jul 3, 2019

#3**+2 **

There's probably something quicker than this !

With thanks to asinis and hectictar, we have the two sequences

sequence [ 1 ] \(\displaystyle \cot(\arctan(x))=1/x,\)

and

sequence [ 2 ] \(\displaystyle \cot(\arctan(\cos(\arctan(\sqrt{x}))))=\sqrt{x+1}.\)

Beginning with a zero on the display, selecting cos, will produce the number 1, which can be thought of the square root of 1.

So, then

sequence [ 2 ] will get us sqrt(1 + 1 ) = sqrt( 2 ),

sequence [ 2 ] will get us sqrt(2 + 1) = sqrt( 3 ),

sequence [ 2 ] will get us sqrt(3 + 1) = sqrt( 4 ),

sequence [ 1 ] will get us 1/sqrt(4) = sqrt(1/4),

sequence [ 2 ] will get us sqrt(1/4 + 1) = sqrt(5/4),

sequence [ 1 ] will get us 1/sqrt(5/4) = sqrt(4/5),

and finally,

sequence [ 2 ] will get us sqrt(4/5 + 1) = sqrt( 9/5) = 3/sqrt(5).

Guest Jul 17, 2019

#4**+5 **

Out of curiosity, I wanted to see a length comparison between your answer and my answer:

\(\cot(\arctan(\cos(\arctan(\cot(\arctan(\cot(\arctan(\cos(\arctan(\cot(\arctan(\cot(\arctan(\cos(\arctan(\cot(\arctan(\cos(\arctan(\cot(\arctan(\cos(\arctan(\cos(0)))))))))))))))))))))))))\)

25 total basic functions

Here is WolframAlpha's result: https://www.wolframalpha.com/input/?i=cot(arctan(. . .

versus

\(\cot(\arctan(\cos(\arctan(\cos(\arctan(\cos(\arcsin(\cos(\arctan(\cos(\arcsin(\cos(\arctan(\cos(\arctan(\cos(0)))))))))))))))))\)

17 total basic functions

Here is WolframAlpha's result: https://www.wolframalpha.com/input/?i=cot(arctan(. . .

.

.

hectictar
Jul 17, 2019