Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation \(r=\cos \theta+3 \sin \theta\) to rectangular form and identify the graph.

Step-by-Step Solution

Answer

In rectangular coordinates the polar equation \(r=\cos \theta+3 \sin \theta\) becomes the equation of a circle \((x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4}\). The graph represents a circle centered at \((\frac{1}{2},\frac{3}{2})\) with a radius of \(\frac{\sqrt{5}}{2}\).

Step 1: Convert Polar to Rectangular

In polar coordinates, \(r=\cos \theta+3 \sin \theta\). To convert this into rectangular coordinates, use the relationships \(x=r\cos \theta\) and \(y=r\sin \theta\). Substitute into the given polar equation: \(r = \sqrt{x^2 + y^2} = x/r + 3y/r\). Multiply through by r, recognizing that in rectangular coordinates, \(r^2 = x^2 + y^2\). This results in \(x^2 + y^2 = x + 3y\).

Step 2: Rearrange to a Standard Form

To identify the curve, it helps to have the equation in a standard form. Bring every term to one side of the equation: \(x^2 -x + y^2 -3y = 0 \). Complete the square for the x and y terms. This results into: \( (x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4} \). This is the equation of a circle in standard form.

Step 3: Identify the Graph

With the equation written in standard form, the graph can be identified. The equation \( (x - h)^2 + (y - k)^2 = r^2 \) represents a circle whose center is at the point \((h, k)\) and whose radius is \(r\). Here, \(h = \frac{1}{2}\), \(k = \frac{3}{2}\), and \(r = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}\). Therefore, the equation represents a circle centered at \((\frac{1}{2}, \frac{3}{2})\) with radius \(\frac{\sqrt{5}}{2}\).