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This is a repost, as I have not gotten an answer and the first post is on like the 4th page

 

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).

 

(a) has already been solved by asinus here, but feel free to post your own answers!

Edit: (b) also solved

 Jul 17, 2019
edited by Davis  Jul 17, 2019
 #1
avatar+8759 
+5

(b)

 

We start with  sqrt(x)  on the display.

 

Then take the  arctan  of what is on the display.

 

Then take the  cos  of what is on the display.  This gives us  \(\frac{1}{\sqrt{x+1}}\)

 

Then...just like from part (a).....we can take the  arctan  of what is on the display.

 

Then take the  cot  of what is on the display.

 

Altogether:

 

\(\begin{array}{ccc} &\cot( \arctan( &\cos( \arctan(\sqrt{x}\,)\,) & ) \quad ) \\~\\ =&\cot( \arctan( &\frac{1}{\sqrt{x+1}}& ) \quad ) \\~\\ =&\sqrt{x+1} \end{array}\)

 

Check: https://www.wolframalpha.com/input/?i=cot(arctan(cos(arctan(sqrt(x)))))

 Jul 17, 2019
 #2
avatar+507 
+3

Thanks!

Davis  Jul 17, 2019
 #3
avatar+8759 
+4

(c)

 

Let's define functions  a,  b,  and  c  like this:

 

a(x)  =  cot( arctan(x) )   =   \(\frac1x\)       where  x  is any real number except  0

 

b(x)  =  cos( arctan(x) )  =  \(\sqrt{\frac{1}{x^2+1}}\)       where  x  is any real number

 

c(x)  =  cos( arcsin(x) )   =   \(\sqrt{1-x^2}\)       where  x  is a real number the interval  [-1, 1]

 

 

We start with  0  on the display.

 

Then we can take the  cos  to get  1

 

Then we can do  b  to get  \(\sqrt{\frac12}\)

 

Then we can do  b  to get  \(\sqrt{\frac23}\)

 

Then we can do  c  to get  \(\sqrt{\frac13}\)

 

Then we can do  b  to get  \(\sqrt{\frac34}\)

 

Then we can do  c  to get  \(\sqrt{\frac14}\)

 

Then we can do  b  to get  \(\sqrt{\frac45}\)

 

Then we can do  b  to get  \(\sqrt{\frac59}\)

 

Then we can do  a  to get  \(\frac{3}{\sqrt5}\)

 

Check: https://www.wolframalpha.com/input/?i=cot(arctan(cos(arctan(cos(arctan(. . .

 Jul 17, 2019
edited by hectictar  Jul 17, 2019
edited by hectictar  Jul 17, 2019
 #4
avatar+507 
+2

Thanks!

Just wondering, did you use a computer to find this?

 Jul 18, 2019
edited by Davis  Jul 18, 2019

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