In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 12 cos θ

Verified step by step guidance

1. Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
2. Start with the given polar equation: \(r = 12 \cos \theta\).
3. Multiply both sides of the equation by \(r\) to eliminate the \(\cos \theta\) term: \(r \cdot r = 12 r \cos \theta\), which gives \(r^2 = 12 r \cos \theta\).
4. Substitute the rectangular coordinate equivalents: replace \(r^2\) with \(x^2 + y^2\) and \(r \cos \theta\) with \(x\), resulting in the equation \(x^2 + y^2 = 12x\).
5. Rearrange the equation to standard form by bringing all terms to one side: \(x^2 - 12x + y^2 = 0\). Then, complete the square for the \(x\) terms to express the equation in the form of a circle.

Key Concepts

Polar to Rectangular Coordinate Conversion

Polar coordinates (r, θ) relate to rectangular coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Converting a polar equation to rectangular form involves substituting these expressions to eliminate r and θ, enabling analysis and graphing in the Cartesian plane.

Trigonometric Identities and Relationships

Understanding trigonometric functions like cosine and sine is essential for converting and simplifying equations. For example, recognizing that r = 12 cos θ can be rewritten using x = r cos θ helps isolate variables and transform the equation into a rectangular form.

Graphing in the Rectangular Coordinate System

Once the equation is converted to rectangular form, graphing involves plotting points (x, y) that satisfy the equation. Familiarity with the Cartesian plane and the shapes represented by different equations, such as circles or lines, aids in accurately sketching the graph.