How to find the area of a triangle, when only one side and angle is known?

Question

Recently I thought, could you calculate the area of a triangle (scalene) when you have:

1. 2 sides of a triangle with the angle between them.
2. 2 sides of a triangle with the angle opposite to any 1 side of the triangle.
3. 1 side and it's adjacent angles.

I found out, yes you could using trigonometry.

Please see this first.

https://drive.google.com/file/d/15oQ5a_OfCxIjmYx_zqjJhjbVqGNk9fxY/view?usp=drivesdk

The formula for case 1,

$ \frac{xyq}{2}$

The formula for case 2,

$ \frac{(xp)(xqr+s\sqrt{(yq)^2-(xp)^2})}{sq^2} $

The formula for case 3,

$ \frac{x^2sq}{2ps+2qr} $

But can you find the area of the triangle if 1 side and 1 angle was given to you?

Answer 1

No. Consider the triangle with vertices $$ (0,2) \\ (0,0) \\ (n, 0) $$ It's got one sides equal to 2, and one angle that's 90 degrees, But its area is $n$, which can be any number.

Answer 2

No we can't, indeed in general to determine the area of a triangle we need at least 2 sides and the angle between/opposite or 1 side and 2 adjacent angles or three sides or simila coditions with at least three information (with at least a side). Thus we can't with only 2 information.

Answer 3

No, you can't. You can check this yourself: draw a line of fixed lenth, draw a fixed angle in one of its ends, and try to draw triangles of different area using only those two parameters. It is easy, so those two don't determine the area.