Not all decimal representations have equivalent fractions. Take pi. Its approximate value as given by the Casio FX9750II is 3.141592654. This is just an approximation because the string of decimal digits in pi has no end. It continues . Such a number that cannot be represented by a ratio of two integers is not rational, not a fraction. It is called irrational.Other such numbers are ?2, ?3,?5, e the base of natural logarithms.
As a matter of principle, if a number is called irrational, it is because it is not rational (not a fraction), and it makes no sense to try to cast it into fraction form.
However, if you truncate an irrational number so as to keep only a finite number of decimal digits, then that truncated number can be represented by a fraction, but qualitatively speaking, it is not the original irrational number.
Thus if you decide to use 3.141500000000000000....0 as an approximate value of pi then you write
3.1415=31415/10000 and the calculator will convert it to the mixed fraction 3u283u2000. To effect this conversion, you press the [FD] key to the right of [a b/c]. If you want an improper fraction 6283u2000 you press [SHIFT][[FD].
As to the value of sin 45 you quote, you should be aware that it is valid for an angle unit in radians.
It is an approximation because the radian unit is based on the value of pi, which, as I said above, is an irrational number.
If you use the degree angle unit, sin45 deg is actually equal to 1/?2 or ?2/2 and this is clearly irrational, and it cannot be represented as a fraction. In conclusion it makes no sense to try to cast it into such form.
On the other hand, sin 30 deg or cos 60 deg are both exactly equal to 1/2. The calculator will convert them to a fraction.
On a more general note, the calculator will always be able to convert a fraction into an approximate decimal number, but the converse may not be possible. It depends on the number of display space available on the screen.
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