Question

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.

I have finished a), and b), but now I need c). Thanks!

Answer 1

Assuming (c) is the equivalent of the original question:
"given any positive rational number q, show that pressing some finite sequence of buttons will yield q"

then we need a transformation from q -> q using the given functions.
We look for mapping R to R,
or equivalently R-> [a,b] -> R
A suitable pair could be the composition of tan(x)o atan(x), or
f(x)=tan(atan(x))
or tan(tan-1(x)) assuming the ( ) keys work
If the parentheses keys do not work, then the ans key has to work, such as
tan-1 x tan ans.
If the ans key does not work, or if there is none, then it needs to be a reverse-polish calculator,
x enter
tan-1 enter
tan enter.

If even the enter key does not work, get a new calculator!

Answer 2

No, the question is asking to start with 0 and go to q.

Answer 3

Since it is a rational number, the numerator and denominator can be replaced by integers.
Using (b), you can create any integer, and using (a), you can divide the two numbers, which means you can create any rational number.
Hope that helps.