Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation to rectangular form. $$r=2 \cos \theta$$

Step-by-Step Solution

Answer

The rectangular form of the given polar equation is \(x^2 - 2x + y^2 = 0\).

Step 1: Apply the Polar to Rectangular Conversion Formulas

Start by applying the formula for conversion from polar to rectangular coordinates. Specifically, replace r in the given equation with \(x^2 + y^2\). The polar equation is \(r = 2 \cos \theta\). Replacing r gives: \(x^2 + y^2 = 2 \cos \theta\).

Step 2: Simplify \(\cos \theta\) in rectangular terms

Next, the goal is to express \(\cos \theta\) in terms of x and y. To this end, recall that \(\cos \theta = \frac{x}{r}\), and replace \(\cos \theta\) in the equation obtained in Step 1 by \(\frac{x}{r}\): \(x^2 + y^2 = 2 \frac{x}{\sqrt{x^2 + y^2}}\).

Step 3: Conversion to rectangular form

Now, simplify the equation from Step 2 to arrive at the equation in rectangular form. First, cross-multiply to eliminate the denominator giving: \(x^2 + y^2 = 2x\). Then, subtract 2x from both sides to isolate the terms: \(x^2 - 2x + y^2 = 0\).