Problem 58 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r=3(1-\sin \theta)$$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + y^2 + 3y = 3\sqrt{x^2 + y^2} \).

Step 1: Recall Basic Equations

To convert from polar coordinates to rectangular coordinates, recall the following relationships: - The rectangular coordinate for the radius is given by: \( r = \sqrt{x^2 + y^2} \).- The rectangular coordinate for the angle is given by: \( \tan\theta = \frac{y}{x} \).- Also remember that \( x = r\cos\theta \) and \( y = r\sin\theta \).

Step 2: Express Known Variables

Given the polar equation \( r = 3(1 - \sin\theta) \), express \( \sin\theta \) in rectangular coordinates: - \( y = r\sin\theta \) implies \( \sin\theta = \frac{y}{r} \).

Step 3: Substitute \( \sin\theta \)

Substitute \( \sin\theta = \frac{y}{r} \) into the polar equation:\[ r = 3\left(1 - \frac{y}{r}\right) \]Multiply through by \( r \) to eliminate the fraction:\[ r^2 = 3r - 3y \]

Step 4: Substitute \( r^2 = x^2 + y^2 \)

Replace \( r^2 \) with \( x^2 + y^2 \) in the equation:\[ x^2 + y^2 = 3r - 3y \]

Step 5: Isolate \( r \)

We know \( r = \sqrt{x^2 + y^2} \), so substitute it back:\[ x^2 + y^2 = 3\sqrt{x^2 + y^2} - 3y \]

Step 6: Rearrange the Equation

Rearrange the equation by adding \( 3y \) to both sides of the equation:\[ x^2 + y^2 + 3y = 3\sqrt{x^2 + y^2} \]This is the rectangular form of the equation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of specifying the position of a point in a 2-dimensional plane using two values, usually represented as \( (x, y) \). These values correspond to the horizontal and vertical distances from the origin (0,0) to the point. This system is very intuitive because it breaks down complex placements into simple straight lines.

Here are a few key points about rectangular coordinates:

• The x-coordinate represents the horizontal distance. A positive x value indicates a position to the right of the origin, while a negative x value indicates a position to the left.
• The y-coordinate signifies the vertical distance, where a positive y value is above the origin and a negative one is below.
• It's the most frequently used coordinate system for graphing because it aligns with our visual understanding of space.

Understanding how to switch between different coordinate systems, like from polar to rectangular, allows you to harness the benefits of each, depending on your needs.

Polar Coordinates

Polar coordinates offer a different way of describing a point in a plane compared to rectangular coordinates. Instead of using horizontal and vertical distances, polar coordinates use a radius and angle. The position is given as \( (r, \theta) \), where \( r \) is the distance from the origin to the point, and \( \theta \) is the angle from the positive x-axis.

A few relevant details about polar coordinates include:

• The radius \( r \) must be non-negative, and it measures how far the point is from the origin.
• The angle \( \theta \) is typically measured in radians, but can also be in degrees.
• If \( \theta \) is zero, the point lies on the positive x-axis, moving counterclockwise as \( \theta \) increases.

Polar coordinates are particularly useful for problems involving circular and rotational symmetry. They simplify equations and calculations linked to circles, arcs, and spirals.

Trigonometric Identities

Trigonometric identities are formulas involving trigonometric functions like sine, cosine, and tangent. These identities are very useful in mathematics, especially when converting coordinates or simplifying expressions. They allow relationships between angles and lengths to be easily understood and manipulated.

In the context of converting polar to rectangular coordinates, some fundamental trigonometric identities include:

• The relationship \( x = r\cos\theta \) helps find the rectangular x-coordinate from the polar coordinates.
• The identity \( y = r\sin\theta \) allows us to derive the y-coordinate from the polar representation.
• The Pythagorean identity \( r = \sqrt{x^2 + y^2} \) links the rectangular distance to the polar radius.
• Additionally, \( \tan\theta = \frac{y}{x} \) connects the angle with the ratio of the y-coordinate to the x-coordinate.

These identities are essential for performing the conversions with accuracy and for simplifying expressions in trigonometry.