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 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.
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.
Thanks a lot man ^_^ Much appreciated :)
(a) To transform x into 1/x, you can follow these steps:
1. Press the sin button twice, which will give you sin(sin(x)).
2. Press the cos button twice, which will give you cos(cos(sin(sin(x)))).
3. Press the sin^-1 button, which will give you sin^-1(cos(cos(sin(sin(x))))).
4. Press the cos^-1 button, which will give you cos^-1(sin^-1(cos(cos(sin(sin(x)))))).
The final result of this sequence of buttons will be 1/x.
(b) To transform sqrt(x) into sqrt(x+1), you can follow these steps:
1. Press the sin button twice, which will give you sin(sin(sqrt(x)))).
2. Press the cos button twice, which will give you cos(cos(sin(sin(sqrt(x)))))).
3. Press the sin^-1 button, which will give you sin^-1(cos(cos(sin(sin(sqrt(x))))))).
4. Press the tan button, which will give you tan(sin^-1(cos(cos(sin(sin(sqrt(x)))))))).
The final result of this sequence of buttons will be sqrt(x+1).
(c) To show that you can get any positive rational number, we can use the fact that any positive rational number q can be written as a fraction p/q, where p and q are positive integers.
1. Start with the initial display showing 0.
2. Press the sin button p times, which will give sin(sin(sin(sin(...sin(0))))) p number of times. This will give you sin^p(0).
3. Press the cos button q times, which will give sin^p(cos(cos(...cos(0))))) q number of times. This will give you sin^p(cos^q(0)).
4. Press the sin^-1 button, which will give sin^p(cos^q(sin^-1(0))). Since sin^-1(0) = 0, this becomes sin^p(cos^q(0)).
5. Press the cos^-1 button, which will give cos^-1(sin^p(cos^q(0))).
The final result of this sequence of buttons will be p/q, which is the desired positive rational number.
By following these steps, you can transform any positive rational number q using a finite sequence of buttons on the broken calculator.
Sure, I'd be happy to help! Let's tackle each part one by one:
(a) To transform x into 1/x, you can use the following sequence of buttons:
1. Press the sin button to calculate sin(x).
2. Press the sin^-1 button to calculate sin^-1(sin(x)), which simplifies to x.
3. Press the cos^-1 button to calculate cos^-1(x), which will give you π/2 - x.
4. Press the sin^-1 button to calculate sin^-1(cos^-1(x)), which simplifies to π/2 - x.
5. Divide π/2 - x by π/2 to get 1 - (2x/π).
6. Press the sin button to calculate sin(1 - (2x/π)), which simplifies to sin(2x/π).
7. Divide sin(2x/π) by x to get the final result of 1/x.
(b) To transform sqrt(x) into sqrt(x+1), you can use the following sequence of buttons:
1. Press the sin button to calculate sin(x).
2. Square sin(x) to get sin^2(x).
3. Press sin^-1 button to calculate sin^-1(sin^2(x)), which simplifies to sin^2(x).
4. Subtract 1 from sin^2(x) to get sin^2(x) - 1.
5. Press the sin button to calculate sin(sin^2(x) - 1), which simplifies to sin(cos^-1(x) - 1).
6. Divide sin(cos^-1(x) - 1) by sqrt(x) to get the final result of sqrt(x+1).
(c) Now, let's show that any positive rational number q can be obtained. Here's a general approach:
1. Start with the number 1.
2. Multiply by q to get q.
3. Take the square root to get sqrt(q).
4. Use the sequence of buttons from part (b) to transform sqrt(q) into sqrt(q+1).
5. Repeat steps 2-4 until you reach the desired rational number.
By following this approach, you can transform the initial value of 1 into any positive rational number q.
I hope that helps! Let me know if you have any further questions.