Problem 60 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The rectangular equation is \\(x^2 + y^2 = 2\sqrt{x^2 + y^2} - x\\).

Step 1: Recall Polar to Rectangular Coordinate Conversion Formulas

To convert from polar coordinates \(r, \theta\) to rectangular coordinates \(x, y\), use the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, recall that \(r^2 = x^2 + y^2\).

Step 2: Express \\(r\\) in terms of \\(x\\), \\(y\\), and \\(\cos \theta\\)

We know \(r = x/\cos \theta\), so we substitute \(\cos \theta = x/r\) into the given equation \(r = 2 - \cos \theta\).

Step 3: Substitute and Rearrange the Equation

Substitute \(\cos \theta = x/r\) into the polar equation: \(r = 2 - x/r\). Multiply through by \(r\) to clear the fraction: \(r^2 = 2r - x\).

Step 4: Replace \\(r^2\\) with \\(x^2 + y^2\\)

Replace \(r^2\) with \(x^2 + y^2\) in the equation: \(x^2 + y^2 = 2r - x\).

Step 5: Solve for \\(r\\)

We have \(r = \sqrt{x^2 + y^2}\). Substitute this into the equation: \(x^2 + y^2 = 2\sqrt{x^2 + y^2} - x\).

Step 6: Simplify the Equation

Square both sides to remove the square root: \((x^2 + y^2)^2 = (2\sqrt{x^2 + y^2} - x)^2\). Expand and simplify to get the final rectangular equation.

Key Concepts

Coordinate Conversion Formulas

Converting between polar and rectangular coordinates involves specific formulas. Rectangular coordinates use the familiar

• \(x\)
• \(y\)

axes that we often encounter in a Cartesian plane. In contrast, polar coordinates use

• \(r\)
• \(\theta\)

— the radius from the origin and the angle from the positive x-axis, respectively. The critical formulas for conversion include

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r^2 = x^2 + y^2\)

These relationships allow one to switch from one set of coordinates to the other. Recognizing and applying these formulas is crucial in solving problems across both coordinate systems. Whenever you're dealing with such conversions, think about these expressions that bridge the gap between polar and rectangular worlds.

Rectangular Coordinates

Rectangular coordinates are probably what you're most familiar with from high school geometry. They express locations in the plane using

• \(x\)
• \(y\)

values. These values tell you how far a point is from the accepted origin along horizontal and vertical axes. This coordinate system is like a grid where each point has an exact placement, making it easy to understand and visualize. Rectangular coordinates are perfect for detailed space and graph plotting. When converting an equation from polar to rectangular form, your goal is to reframe the expression in terms of these

• \(x\)
• \(y\)

coordinates, thus translating a spatial relationship into a more linear format.

Polar Coordinates

Imagine standing in a circle's center and describing where something is relative to you, that's the idea behind polar coordinates. They use

• \(r\), the distance from a central point (radius)
• \(\theta\), an angle measured in radians or degrees

to determine a point's position. Polar coordinates are particularly effective in situations involving circular or rotational systems. For instance, they're the natural choice when dealing with angles and arcs. In polar equations,

• \(r\)
• \(\theta\)

work together to plot curves and circles, among other shapes, illustrating relationships in a way that rectangular coordinates cannot easily match. Understanding how these components intertwine aids in visualizing complex geometrical shapes.

Equation Conversion

Equation conversion moves equations from one coordinate system to another. This process allows us to interpret equations using different lenses, potentially simplifying the math. In our problem, we're converting \(r=2-\cos \theta\) from polar to rectangular form. This involves using conversion formulas to express all components in terms of

• \(x\)
• \(y\)
• \(r\)
• \(\theta\)

Rectangular form is often easier for graphing and finding intersections, so converting equations can give us clarity on how a shape interacts with the Cartesian plane. In conversion, ensure you rearrange terms properly and use algebraic manipulation to simplify expressions. As seen in the step-by-step solution, these skills culminate in translating the spatial interpretation of polar equations into the straightforward, numerical understanding of rectangular coordinates.