Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$

Step-by-Step Solution

Answer

The particular exercise has been converted to the rectangular form as required using the relationships between polar and rectangular coordinates.

Step 1: Substitute for cos theta

Our starting equation is \(r = 1/(1 - \cos\theta)\). We now use the formula for cos theta in polar coordinates. Thus, replace \(\cos\theta\) by \(x/r\) Getting: \(r = 1/(1 - x/r)\)

Step 2: Simplify the equation

Clear the denominator by multiplying the entire equation by \((1 - x/r)\), then by r\n\(r^2 = r - x\)

Step 3: Substitute for r squared

Now, we utilize the formula \(r^2 = x^2 + y^2\) from polar to rectangular conversion. Thus, replace \(r^2\) by \(x^2 + y^2\) Resulting in: \(x^2 + y^2 = r - x\)

Step 4: Final Simplification

We need to get rid of r in the equation. So we replace r with its equivalent in the rectangular form r = sqrt(x^2 + y^2). Substituting this into our equation we obtain the following: \(x^2 + y^2 = \sqrt{x^2 + y^2} - x\), which can now be solved for x and y