Formula used in joint and polar calculations

You cannot add rectangular coordinates to polar coordinates. You can only add rectangular coordinates to other rectangular coordinates. For polar coordinates you must use some specific trigonometric formulas.

Since you have the coordinates of the three vertices, the most straightforward method is to calculate the length of the sides using the distance formula
d(P_1,P_2)=SQRT((X_1-X_2)^2+(Y_1-Y_2)^2)
where SQRT is the square root function, X_1, Y_1) are the coordinates of point P_1, etc.
With the three lengths available, use Heron's (sometimes called Hero's) to find the area.
Here is Heron's formula.
Let's call the lengths a, b, and c
Let p be the semi-perimeter p= (a+b+c)/2
Then
Area= SQRT [ p(p-a)(p-b)(p-c) ]
Make sure that there is a matching ) parenthesis to the one in the SQRT.

Alternatively,
You can choose the base as the side opposite the vertex (0,0)
Calculate the equation of the line that supports the base.
Calculate the equation of the line issuing from (0,0) and perpendicular t the base.
Calculate the coordinates of the intersection point , call it H, of the base and its perpendicular line (coming from (0,0)).
Calculate the distance OH, that is the height relative to the chosen base.
Use the formula Area= base*height/2

Now it is up to you to choose one of the two methods and calculate the area of that triangle. The second method involves more calculations than the first, and more possibilities of errors. Good Luck

if by theta function you mean the change to polar coordinates function, type in your x and y coordinate for example (3; 4) to do this press 3 (x,y) key 4 then 2nd F (the orange button) and then 8 (the theta function) and then r will be 5 and theta will be 53.13010235. If you have the polar coordinates and want to change to rectangular coordinates type in the hypotenuse (x,y) button and then the angle and then press 2nd F and 9 and it will give you the x and y coordinates.

If you have any more questions there is a dedicated website http://www.mathsatsharp.co.za/ where you can leave questions about the SHARP calculators on the student page and they will be answered within a week maximum.

Convert the Cartesian coordinates to polar coordinates (r, Theta). The calculator can do that. Press the SHIFT [+] key combination. Calculator will display Pol(. Type in the Cartesian coordinates (x,y), close the right parenthesis and press [=]. The calculator will display the radius r and the angle theta.
If you want to convert from polar to Cartesian coordinates, use the SHIFT [-] (Rec) key combination. Enter the radius and the angle separated by a comma, close the right parenthesis and press [=]
Be careful, the angle unit you use (radian or degree) will affect the result.

The Pol function converts coordinates from rectangular to polar. Pol(x,y) converts the rectangular coordinates (x,y) to polar coordinates (r, theta).

For examples and instructions, see page E-37 of the manual and example #036 in the Appendix.

No, your calculator does not know a variable called theta. But what is theta but a symbol to represent a variable. Usually it is used to represent an angle in polar coordinates. It is just the usage and it is not really essential. You can do conversion from polar to rectangular coordinates and from rectangular to polar coordinates but it does not matter what the angle variable is called.
Go ahead, do all the calculations you want that involve an angle. If the calculator can perform them , they will be evaluated correctly and the name you give the angle value (alpha, beta, theta, iota...) is immaterial.

Are you serious?
No calculator purchase should be solely based on such a trivial matter.
But if you are looking for a very good scientific calculator, I would suggest the new Write View EL-W506/W516/W546 calculators by Sharp, or the FX-115ES/991ES Natural Display by Casio.

With these calculators you can see what you enter displayed as in textbooks. Naturally, they all do coordinates conversions from Polar to Rectangular and from Rectangular to Polar.

To avoid errors make sure that the angle unit is THE ONE required by your problem. To verify or to change the angle unit press [SIFT][MODE] and select [3: Deg] for degrees, [4:Rad] for radians, or [5:Grad] for grads.

Convert from rectangular to polar

[SHIFT][+] (Pol) . Screen displays Pol(
You enter the X-coordinate, and the Y coordinate separated by a comma, then you close the parenthesis and press [=].

Exemple: Polar coordinate of the point (1,1)
Pol( 1, 1) gives
r=1.414213562 (square root of 2)
θ=45

Here is how you enter it
Press [SHIFT][MODE][2:LineIO]
[SHIFT][POL] 1 [SHIF] [,] 1 [)] [=]

The screen displays
Pol(1,1)
r= 1.414213562
θ= 45

Convert from polar to rectangular coordinates

Rec ( radius, angle) =

Exemple: Recatngular coordinates of (r=3, θ=60°)
[SHIFT][Rec] 3 [SHIFT][,] 60 [ ) ] [=]
The screen displays
Rec(3,60)
X= 1.5
Y= 2.598076211

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.