Solve for x r=8cos(x)

Problem

r=8*cos(x)

Solution

1. Identify the goal, which is to convert the polar equation r=8*cos(x) into a rectangular (Cartesian) equation using the standard conversion formulas.

2. Recall the relationships between polar and rectangular coordinates: x=r*cos(x) y=r*sin(x) and r^2=x^2+y^2

3. Multiply both sides of the equation by r to create terms that are easily substituted.

r^2=8*r*cos(x)

4. Substitute x^2+y^2 for r^2 and x for r*cos(x)

x^2+y^2=8*x

5. Rearrange the equation by moving all terms to one side to prepare for completing the square.

x^2−8*x+y^2=0

6. Complete the square for the x terms by adding ((−8)/2)^2=16 to both sides.

(x^2−8*x+16)+y^2=16

7. Simplify the expression into the standard form of a circle equation (x−h)^2+(y−k)^2=R^2

(x−4)^2+y^2=16

Final Answer

r=8*cos(x)⇒(x−4)^2+y^2=16

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