Question 830809
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To change from either polar form to rectangular form, or from
rectangular to polar form, always begin by drawing the graph
below, and out beside it write out the four equations, the 
three basic trig ratios and the Pythagorean theorem equation:

{{{drawing(200,200,-.5,1.5,-.5,1.5,
line(.8,.6,.8,0), locate(.84,.4,y),locate(.4,0,x),locate(.35,.5,r),locate(.28,.2,theta),
line(-10,0,10,0),line(0,-10,0,10), line(0,0,.8,.6) )}}}{{{matrix(4,2,"(1)", 
sin(theta)=y/r,"(2)",cos(theta)=x/r,"(3)",tan(theta)=y/x,"(4)",x^2+y^2=r^2)}}}

To change 

{{{x - y}}}{{{""=""}}}{{{4}}}

Solve (2) for x by clearing {{{cos(theta)=x/r}}} of fractions
by multiplying both sides by r: {{{r*cos(theta)=x}}}
So replace {{{x}}} by {{{r*cos(theta)}}}

{{{r*cos(theta) - y}}}{{{""=""}}}{{{4}}}

Solve (1) for y by clearing {{{sin(theta)=y/r}}} of fractions
by multiplying both sides by r: {{{r*sin(theta)=y}}}
So replace {{{y}}} by {{{r*sin(theta)}}}

{{{r*cos(theta) - r*sin(theta)}}}{{{""=""}}}{{{4}}}

Solve for r by first factoring out r on the left:

{{{r(cos(theta) - sin(theta))}}}{{{""=""}}}{{{4}}}

{{{r}}}{{{""=""}}}{{{4/(cos(theta)-sin(theta))}}}

----------------------------------- 

{{{r}}}{{{""=""}}}{{{2sin(theta)}}} 

{{{drawing(200,200,-.5,1.5,-.5,1.5,
line(.8,.6,.8,0), locate(.84,.4,y),locate(.4,0,x),locate(.35,.5,r),locate(.28,.2,theta),
line(-10,0,10,0),line(0,-10,0,10), line(0,0,.8,.6) )}}}{{{matrix(4,2,"(1)", 
sin(theta)=y/r,"(2)",cos(theta)=x/r,"(3)",tan(theta)=y/x,"(4)",x^2+y^2=r^2)}}}

{{{r = 2sin(theta)}}}

Always substitute for the trig functions first, and for r last:

Substitute {{{(y/r)}}} for {{{sin(theta)}}}

{{{r = 2(y/r)}}}

{{{r=2y/r}}}

Clear of fractions by multiplying both sides by r:

{{{r^2=2y}}}

Now use (4) to replace {{{r^2}}} by {{{x^2+y^2}}}

{{{x^2+y^2=2y}}}

---------------------

{{{r = 2/(1-cos(theta))}}}

Clear of fractions by multiplying both sides by {{{1-cos(theta)}}}

{{{r(1-cos(theta))}}}{{{""=""}}}{{{2}}}

{{{drawing(200,200,-.5,1.5,-.5,1.5,
line(.8,.6,.8,0), locate(.84,.4,y),locate(.4,0,x),locate(.35,.5,r),locate(.28,.2,theta),
line(-10,0,10,0),line(0,-10,0,10), line(0,0,.8,.6) )}}}{{{matrix(4,2,"(1)", 
sin(theta)=y/r,"(2)",cos(theta)=x/r,"(3)",tan(theta)=y/x,"(4)",x^2+y^2=r^2)}}}

Again, always substitute for the trig functions first, and for r last: 

Use (2) to substitute {{{x/r}}}for {{{cos(theta)}}}

{{{r(1-x/r)}}}{{{""=""}}}{{{2}}}

Distribute to remove the parentheses on the left side:

{{{r-x}}}{{{""=""}}}{{{2}}}

Solve (4) for {{{r}}}:   {{{r=sqrt(x^2+y^2)}}} and replace
r by that:

{{{sqrt(x^2+y^2)-x}}}{{{""=""}}}{{{2}}}

Isolate the square root term on the left by adding x to both sides:

{{{sqrt(x^2+y^2)}}}{{{""=""}}}{{{2+x}}}

Square both sides

{{{(sqrt(x^2+y^2))^2}}}{{{""=""}}}{{{(2+x)^2}}}

{{{x^2+y^2}}}{{{""=""}}}{{{(2+x)(2+x)}}}

{{{x^2+y^2}}}{{{""=""}}}{{{4+4x+x^2}}}

Simplify by subtracting {{{x^2}}} from both sides:

{{{y^2}}}{{{""=""}}}{{{4+4x}}}

Edwin
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[BTW, this symbol {{{theta}}} is not "data", it's the Greek letter
"theta".  It stands for the angle in the graph above.]

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