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!
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!
No, the question is asking to start with 0 and go to q.
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.
To show that you can get any positive rational number using the given buttons, we will use the concept of continued fractions. The basic idea is to express a positive rational number as an infinite sequence of integers.
Let's suppose we want to express the positive rational number q = p/q, where p and q are positive integers, in terms of the buttons on the calculator.
To do this, we will use the following steps:
Step 1: Start with the given number q.
Step 2: Take the floor function of q, denoted by [q], which gives the largest integer less than or equal to q. If [q] = 0, then we have reached our desired number. Otherwise, continue to the next step.
Step 3: Press the button corresponding to the inverse trigonometric function of [q]. For example, if [q] = 3, press the sin^{-1} button three times. This will give us an angle in radians that we will use in the next step.
Step 4: Apply the sequence of buttons that transforms x into 1/x (which you have already found in part a)). Apply this sequence of buttons to the angle obtained in step 3. Let's call the resulting value x.
Step 5: Repeat steps 2 to 4 using x instead of q. Continue this process until you reach a value of 0. At each step, you will obtain a finite sequence of button presses.
Step 6: The final result will be the composition of all the button sequences obtained in step 5. Press these buttons in the order from the initial value q to 0.
By following these steps, you can obtain any positive rational number.
It is important to note that this method does not guarantee the shortest sequence of button presses to obtain a specific rational number. However, it guarantees that such a sequence exists and can be obtained using the given buttons.