How do i solve the following sin(theta)=-0.6 0<(theta)<360degrees on my graphics calculator?

Since this calculator cannot perform symbolic manipulations (algebra) you never need to type in sin(theta) or cos(theta). To calculate the sine of an angle (whatever the name of the angle may be) just press the sin key followed by an angle value and the function will be calculated. same thing with any other trigonometric function.
Make sure that the angle units is set to the unit required by your calculation: degree, radian, or grad.

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

{r, theta} is equivalent to r*exp(i*theta)=r*cos(theta) +i*sin(theta)
That is all you need.

You have several types of graphs
Function graphs
Y_1=f(x), Y1=3X^2-4, [X, T, Theta, n] key types X
Polar graphs r=F(theta), r=r_o*ln(theta). [X, T, Theta, n] types Theta
Parametric graphs X_1=f(T) and Y_1=g(T). [X.T, Theta, n] types T
Examples: X_1= cos(T). Y_1= 2(1--sin(T))
Sequence graphs u_n+1= f(u_n), [X,T,Theta,n] types n

You have several types of graphs
Function graphs
Y_1=f(x), Y1=3X^2-4, [X, T, Theta, n] key types X
Polar graphs r=F(theta), r=r_o*ln(theta). [X, T, Theta, n] types Theta
Parametric graphs X_1=f(T) and Y_1=g(T). [X.T, Theta, n] types T
Examples: X_1= cos(T). Y_1= 2(1--sin(T))
Sequence graphs u_n+1= f(u_n), [X,T,Theta,n] types n

Press 2nd SIN 3 2 / 1 4 5 ) =

Make sure you have the desired angular mode set before you do this.

You can enter it several ways.
1/3[^][X, theta, T] or
(1/3)[^][X, theta, T] or
( 1 [a b/c] 3)[^][X, theta, T]
In the following screen capture the small negative sign is the change sign (-)

The variable theta is reserved for polar plots. So if you want to graph functions of the form y= expression, you must use the default variable x. Thus draw y=sin(X). Make sure your calculator is configured with the correct angle unit (degree or radian).
If you use degrees the window dimensions for x must cover an interval of 360 degrees, to have a full period. If angle unit is radian, then choose an x-interval between [-pi and pi] or between [0 and 2pi]

Graph 2X PLUS 1 for X in open interval ]-1,0[

It should be entered as follows
(2X plus 1) (X larger than negative 1) (X less than zero)

In [Y=] editor and on line Y1= type your right hand side between parentheses (2X Plus 1). I use the Plus instead of the usual sign because the parser of the web site removes the sign.
[( ]2 [X,T,theta,n] [Plus] 1 [)] [(] [X,T,theta,n] [2nd][MATH] [3: larger than] [(-)] 1 [)] [(] [X,T,Theta,n] [2nd][MATH][5: less than] 0 [)]
Here are some screen captures to help you

Hello,
The e is the same, it is the exponential. According to Euler's relation
e^(i theta) = cos(theta) + i sin(theta), where i is the imaginary unit.
When represented on the complex plane (x,iy) the point (cos(theta), sin(theta)) is at the extremity of a vector of length 1 and making an angle theta with the real axis.

In (plane) polar coordinates, a point is defined by the radius r, and the angle, theta, it makes with the x axis, measured in the trigonometric (counterclockwise) direction. It is structurally equaivalent to representing it in the complex plane as r*e^(i*theta). Since r is the measure ot is radius, and the theta is it argument (angle). The complex notation is used for its convenience when adding vectors (as is AC circuits)
That is the theory.
I am inserting a clipping from the book to show you how to convert between polar and rectangular coordinates.