in rectangular coordinates, you would get:
x = r * cos(theta)
y = r * sin(theta)
using pythagorus, the hypotenuse of the triangle formed by x and y is equal to:
h^2 = x^2 + y^2 = r^2 * cos^2(theta) + r^2 * sin^2(theta) which becomes:
h^2 = r^2 * (sin^2(theta) + cos^2(theta)) which becomes:
h^2 = r^2 which becomes:
hypotenuse = j = r
you are given that r = 2 / sin(theta)
this means that the hypotenuse is equal to 2 / sin(theta).
the sine of theta is equal to y / r
r is equal to 2 / sin(theta)
this means that r is equal to 2 / (y / r)
since 2 / (y / r) is the same as 2 * (r / y), then:
r is equal to 2 * (r / y)
multiply both sides of this equation by y and divide both sides of this equation by r and you get:
y = 2
what this says is that y is going to be equal to 2 regardless of the value of x.
that means that the graph of the equation of r = 2 / sin(theta) is a straight line that is parallel to the x-axis and has a value of 2.
you can verify this graphically.
just plot the graph of y = 2 / sin(theta) and you will see that it is a straight line at y = 2.
you need graphing software that can graph equations in polar form.
the following calculator can do that:
http://www.desmos.com/calculator
the graph i made is shown below:
you can see that the graph is a horizontal line at y = 2.
the following picture may help clarify the relationship between the polar coordinate form and the rectangular form of the equation.
what you see in the pictures is that r = 2 / sin(theta) is the hypotenuse of a right triangle that has theta as the angle and the opposite side as y and the adjacent side as x.
since sin(theta) = y/r, then y = r * sin(theta)
since cos(theta) = x/r, then x = r * cos(theta)
since r is given as being equal to 2/sin(theta), you can replace r in the equations for x and y with 2/sin(theta) and solve for x and y.
you get y = 2
you get x = 2 cot(theta).
by the pythagorean formula, r = sqrt(x^2 + y^2)
replace x^2 with (2cot(theta)^2 and replace y with 2^2 and you wind up with:
r = sqrt(4cot^2(theta) + 4)
this is also equivalent to r = sqrt(4/tan^2(theta) + 4)
that last form is easier to calculate from when you use your calculator.
what i think you were trying to do is manually plot the equation.
that's a chore.
you can do it, but it's much easier to graph it using appropriate graphing software.
also knowing the translations from polar to rectangular helps.
i used graphing software to see that the graph was a line at y = 2.
once i saw that, i was able to figure out how it became that way using algebra.
calculating for y said that y was always going to be equal to 2.
that corresponded with the graph and so i was reasonably confident that the equation of r = 2 / sin(theta) was the same as the equation y = 2.
what you were doing was trying to calculate r which would not be the same each time but would always wind up creating a line where the y value of the coordinate point would be equal to 2.
you would only have been able to see this after connecting all those points.