Trigonometric functions as fractions on TI-84 Plus.

As the other answer says, you simply can't express all values of trigonometric functions as fractions.

I am guessing that on your TI-84, if you set the calculator to give you the fractions, then $\sin(30^\circ) = \sin(\pi /6)$ will give you $\frac{1}{2}$ (that is how it works on my TI-84). But you also know that, for example, $\sin(45^\circ) = \sin(\pi / 4)$ is equal to $\frac{\sqrt{2}}{2}$. This is not a rational number and can't be written as a ratio of two integers.

You calculator can only find these exact fractions when the answer is a rational number.