Why parametric graphs are rectangular shape - Texas Instruments TI-84 Plus Calculator

Your function is not correctly defined. If graph is function y=f(x), variable must be x; if graph is polar, variable must be theta, and if graph is parametric, variable must be t. This does not cover all possible cases. Check the graph type and the definition of the function.

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In parametric functions, the variables x and y are both functions of an independent variable, called t, and that usually represents the time. The default variable in parametric functions must be called t. You can enter the limits you want. However if you type in 2 Pi it will be converted on the entry line to 6.28. The default tstep is 0.13 which is about pi/24. I fail to see what the commotion is about.
In this representation of the witch of Agnesi, the variable t is between 0 and 4Pi, and the step tstep is 0.13, a little less that pi/24
Reset your calculator and restart

That would depend on what you want to do with it. If you want to graph a polar plot, for example, you'd set it for polar plotting. If you want to graph a parametric plot instead, you'd set it for parametric plotting.

There is nothing to convert, because if T is a real number so is 40sin(4000T).
40sin(4000T)=40abs(sin(4000T))*sign(sin(4000T))
where abs() is the absolute value of it argument, and sign(sin(4000T)) is positive or negative.
So in (radius, angle) polar coordinates
r= 40 abs(sin(4000T)),
theta =(0, or PI), depending on sign(sin(4000T))

If you are trying to graph parametric functions, this calculator does not do graphs, and you need another function of T

The key allows you to enter different variables, X for function graphs, T for parametric graphs, Theta for polar graphs and n fro sequence graphs.

Your calculator does not have graphing capabilities. On a graphing calculator you would first select the graph type 1. function y= f(x), 2. parametric x=x(t), y=y(t), 3.polar r=function(theta).

The variable t is reserved for parametric graphs. If you want to use functions of the form y=, then the default variable is x. You can enter it by pressing the X,T,Theta key.
Besides, what you name a variable is purely a matter of style. Call it what you may, it will not change the shape of the graph. The full name of a variable is dummy variable, meaning that the symbol you use is irrelevant.
You do have to conform to the idiosyncrasies of your calculator, otherwise it will complain and send you error messages.

At first, you have to use t symbol for parametric functions when you operating with TI 89 Titanium. In your case you must input your relations as x1=3cos(t) and y1=2sin(2t). Then graphing it. See captured image

See carefully captured images below