Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=6 \cos \theta $$

Short answer

The rectangular form is \((x - 3)^2 + y^2 = 9\), a circle centered at \((3, 0)\) with radius \(3\).

Step-by-step solution

Understand Polar and Rectangular Coordinates

Polar coordinates use the radius \(r\) and angle \(\theta\), whereas rectangular coordinates use \(x\) and \(y\). We will use transformation formulas to convert the polar equation to rectangular form.

Apply Polar to Rectangular Transformation

The transformation is given by: \(x = r \cos \theta\) and \(y = r \sin \theta\). We need to express \(r\) and \(\cos \theta\) in terms of \(x\) and \(y\).

Substitute and Simplify

Start with the polar equation \(r = 6 \cos \theta\). Substitute \(\cos \theta = \frac{x}{r}\) into the equation: \(r = 6 \frac{x}{r}\). Multiply both sides by \(r\) to clear the fraction, yielding \(r^2 = 6x\).

Express in Terms of Rectangular Coordinates

Remember that \(r^2 = x^2 + y^2\). Substitute this into the equation: \(x^2 + y^2 = 6x\). This is the rectangular form of the given polar equation.

Rearrange the Rectangular Equation

Rearrange \(x^2 + y^2 = 6x\) to complete the square for the \(x\) terms: \((x - 3)^2 + y^2 = 9\). This shows the equation of a circle with center at \((3, 0)\) and radius \(3\).

Sketch the Graph

Plot the circle on a coordinate plane. The center is at \((3, 0)\), the radius is \(3\). Draw the circle with its boundary points at \((0, 0)\) and \((6, 0)\) along the x-axis.

Key concepts

Polar Coordinates

Polar coordinates represent the position of a point in a plane by using a radius and an angle. This system is different from the standard Cartesian system, where points are described with an x and y value. In polar coordinates, \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
Polar coordinates are especially useful in situations where directions and distances are more relevant than horizontal and vertical positions, such as when dealing with circular paths or angles.

• \( r \: \) Radius - the distance from the point to the origin
• \( \theta \: \) Angle - measured in radians or degrees from the positive x-axis

Understanding and converting between these systems is crucial in graphing and solving problems effectively, especially those involving circular and rotational movement.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the standard system for defining points in a plane using x and y values. Each point is located by its perpendicular distances, or projections, on the x-axis and y-axis, respectively. This system is intuitive for graphing equations that describe curves such as lines, parabolas, and ellipses.
In rectangular coordinates:

• \( x \: \) The horizontal distance from the y-axis
• \( y \: \) The vertical distance from the x-axis

Converting polar coordinates to rectangular coordinates involves using trigonometric relationships derived from the unit circle. For any point, \( x = r \cos \: \theta \) and \( y = r \sin \: \theta \). These transformations help in seamlessly translating problems from the world of circles and angles to lines and coordinates.

Circle Equation

The equation of a circle in rectangular coordinates is typically expressed as \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is its radius. This equation comes from the definition of a circle as the set of all points equidistant from a given point (the center).
In the solution of our exercise, starting with \( x^2 + y^2 = 6x \), we completed the square, arriving at \( (x - 3)^2 + y^2 = 9 \). This shows us:

• The center is at \( (3, 0) \)
• The radius is \( 3 \)

Graphically, this represents a circle centered rightward on the x-axis, which is a direct outcome of translating the polar form to its rectangular counterpart.

Coordinate Transformation

Coordinate transformation is the process of converting one form of coordinates to another, such as polar to rectangular or vice versa. This is essential in various fields, from engineering to physics, as it allows us to describe the same point or object with different systems, each offering unique insights.
For transforming polar coordinates \( (r, \: \theta) \) to rectangular coordinates \( (x, \: \y) \):

• Calculate \( x = r \cos \: \theta \)
• Calculate \( y = r \sin \: \theta \)
• Using the relationship \( r^2 = x^2 + y^2 \) is useful in equations involving circles.

In our example, starting from \( r = 6 \cos \: \theta \), we used these transformations to express the equation as \( x^2 + y^2 = 6x \), illustrating how these systems are interconnected. This transformation allows equations of curves like circles to be analyzed in terms of horizontal and vertical displacements.