See my notes. You've done some excellent work here.
Sir,
This problem wants one to transform a polar equation into its rectangular form.
Question:
rcos(theta)= 8 r csc (theta)
Answer:
If I divide by r then I get
cos theta=8* sin^-1(theta)
When you divide by r remember that you have to exclude r = 0, since you can't divide by 0. So if r = 0 gives us one or more solutions (as it does), the 'new' equation won't indicate it.
Not a good idea to use sin^-1 to indicate the reciprocal of the sine function, since sin^-1 can be used to represent the inverse sine, or arcsin, function. The reciprocal could be written as (sin(theta))^-1, or as just 1 / sin(theta).
Therefore:
8*1/sin(theta)=cos(theta)
8=cos(theta)*sin(theta)
2(sin theta*cos theta)=16
sin 2 (theta)=16
Good to this point.
sin (theta)=8
Division of both sides by 2 does not extend to expressions inside the sine function.
theta = arcsin(8) imaginary !!!!
Actually it's 2 theta = arcsin(16) -- still not a real number.
There is only one point at which a solution to the original equation exists, namely the origin (r = 0 makes the equation true for any value of theta except those at which csc(theta) is not defined).
Now:
rcos(theta)= 8 r csc (theta)
x=8(r sin^-1(theta))
x/8 =r *1/sin (theta)
r*1/sin(theta)=x/8
r = x/8 *sin theta
r(r = x/8 *sin theta)
r^2=(x/8)r*sin(theta)
x^2+y^2 = (x/8)*y
x^2+y^2=xy/8
x^2 – xy/8+y^2=0
(x-y/16)^2 +y^2= y^2/256
(x-y/16)^2+ 255y^2/256=0
Very nice work. At this point you have again proven that except for the origin x = y = 0, there are no real solutions. This is the case since the equation states that the sum of two perfect squares is zero.
((x-y/16)^2+ 255y^2/256)/y^2=0
((x-y/16)^2 )/y^2 + 255/256=0
((x-y/16)^2 )/y^2)=-255/256
Sqrt ((x-y/16)^2 )/y^2) = Sqrt(-255/256)
The square root is imaginary, and I believe you recognize.
The left-hand side of this equation will be real for real x and y, provided y is not zero.
So again you have shown that there are no real solutions except for x = y = 0.
x/y = 1/16 +- Sqrt(-255/256)
y= (x) / 16^-1 +- Sqrt(-255/256)