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. Start by recalling the relationships between polar and rectangular coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2\).
2. Given the polar equation \(r = 12 \cos \theta\), multiply both sides by \(r\) to get \(r^2 = 12r \cos \theta\).
3. Substitute \(r^2\) with \(x^2 + y^2\) and \(r \cos \theta\) with \(x\) to convert the equation to rectangular form: \(x^2 + y^2 = 12x\).
4. Rearrange the equation to form a standard circle equation: \(x^2 - 12x + y^2 = 0\).
5. Complete the square for the \(x\) terms: \((x - 6)^2 - 36 + y^2 = 0\). Simplify to get the equation of a circle: \((x - 6)^2 + y^2 = 36\).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to interpret and manipulate these coordinates is essential for converting polar equations to rectangular form.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes (x and y) to define the position of points in a plane. The conversion from polar to rectangular coordinates involves using the relationships x = r cos(θ) and y = r sin(θ). This understanding is crucial for graphing equations in the rectangular coordinate system.

Conversion Formulas

The conversion from polar to rectangular coordinates relies on specific formulas that relate the two systems. For a polar equation of the form r = f(θ), the corresponding rectangular equation can be derived by substituting r and θ with their rectangular equivalents. Mastery of these formulas is necessary to accurately transform and graph polar equations.