Question 1004654
the equation in polar coordinates is r = 2/sin(theta)


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:


<a href = "http://www.desmos.com/calculator" target = "_blank">http://www.desmos.com/calculator</a>


the graph i made is shown below:


<img src = "http://theo.x10hosting.com/2015/111805.jpg" alt="$$$" </>


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.


<img src = "http://theo.x10hosting.com/2015/111806.jpg" alt="$$$" </>


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.