A calculator is broken so that the only keys that still work are the sin, cos, tan, cot, sin^-1, cos^-1, and tan^-1 buttons. The display initially shows 0. In this problem, we will prove that 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.)

Question

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

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

A calculator is broken so that the only keys that still work are the \sin, \cos, \tan, \cot, \sin^{-1}, \cos^{-1}, and \tan^{-1} buttons. The display initially shows 0. In this problem, we will prove that 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.)

(a) Find a sequence of buttons that will transform x into \frac{1}{x}.

(b) Find a sequence of buttons that will transform \sqrt x into \sqrt{x+1}.

(c) Now show that you can get any positive rational number.

Thanks for all the help before it was really appreciated :) But I have trouble trying to do these questions: if you can only answer one that's fine but I kinda need help. Thanks.

Answer 1

I assume that the calculator initially displays q.

What's wrong with tan(arctan(q)) ?

cot(arctan(q)) = 1/q

A triangle with legs 1 and √x has hypotenuse √(x+1) so,

cos(arctan(√x)) = 1/√(x+1)
Now just convert that into its reciprocal, as above.

Answer 2

Thanks a lot man ^_^ Much appreciated :)